@article{sprenger_bridges_shearer_2023, title={Traveling Wave Solutions of the Kawahara Equation Joining Distinct Periodic Waves}, volume={33}, ISSN={["1432-1467"]}, DOI={10.1007/s00332-023-09922-0}, number={5}, journal={JOURNAL OF NONLINEAR SCIENCE}, author={Sprenger, Patrick and Bridges, Thomas J. and Shearer, Michael}, year={2023}, month={Oct} } @article{barker_gray_schaeffer_shearer_2023, title={Well-posedness and ill-posedness of single-phase models for suspensions}, volume={954}, ISSN={["1469-7645"]}, url={https://doi.org/10.1017/jfm.2022.1004}, DOI={10.1017/jfm.2022.1004}, abstractNote={Classical theories for suspensions have been formulated by starting from the Navier–Stokes equations describing pure liquid flow and then introducing additional dependencies to account for the presence of suspended particles. These models are often accurate for low particle concentrations but have lacked a convincing description of the frictional interactions of particles, which are important at larger solid volume fractions. The $\mu (J), \varPhi (J)$ rheology, which draws a direct analogy between suspension flow and dry granular flow, is a recent theory that addresses this issue, but is shown here to be dynamically ill-posed for large solid volume fractions. An alternative well-posed theory is introduced that includes additional dependence on the particle-phase dilation and compression. The new theory, denoted vCIDR, is tested numerically to show grid convergence for problems in which the $\mu (J), \varPhi (J)$ rheology instead suffers from catastrophic blow-up. A further well-posed extension provides a framework for handling the transition between viscous and inertial flows.}, journal={JOURNAL OF FLUID MECHANICS}, author={Barker, T. and Gray, J. M. N. T. and Schaeffer, D. G. and Shearer, M.}, year={2023}, month={Jan} } @article{congy_el_hoefer_shearer_2021, title={Dispersive Riemann problems for the Benjamin-Bona-Mahony equation}, volume={8}, ISSN={["1467-9590"]}, url={https://doi.org/10.1111/sapm.12426}, DOI={10.1111/sapm.12426}, abstractNote={Abstract}, journal={STUDIES IN APPLIED MATHEMATICS}, author={Congy, T. and El, G. A. and Hoefer, M. A. and Shearer, M.}, year={2021}, month={Aug} } @article{bardall_chen_daniels_shearer_2020, title={Gradient-induced droplet motion over soft solids}, volume={85}, ISBN={1464-3634}, DOI={10.1093/imamat/hxaa015}, abstractNote={Abstract}, number={3}, journal={IMA Journal of Applied Mathematics}, publisher={Oxford University Press (OUP)}, author={Bardall, Aaron and Chen, Shih-Yuan and Daniels, Karen E. and Shearer, Michael}, year={2020}, month={Jun}, pages={495–512} } @article{schaeffer_barker_tsuji_gremaud_shearer_gray_2019, title={Constitutive relations for compressible granular flow in the inertial regime}, volume={874}, ISSN={["1469-7645"]}, DOI={10.1017/jfm.2019.476}, abstractNote={Granular flows occur in a wide range of situations of practical interest to industry, in our natural environment and in our everyday lives. This paper focuses on granular flow in the so-called inertial regime, when the rheology is independent of the very large particle stiffness. Such flows have been modelled with the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$ -rheology, which postulates that the bulk friction coefficient $\unicode[STIX]{x1D707}$ (i.e. the ratio of the shear stress to the pressure) and the solids volume fraction $\unicode[STIX]{x1D719}$ are functions of the inertial number $I$ only. Although the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$ -rheology has been validated in steady state against both experiments and discrete particle simulations in several different geometries, it has recently been shown that this theory is mathematically ill-posed in time-dependent problems. As a direct result, computations using this rheology may blow up exponentially, with a growth rate that tends to infinity as the discretization length tends to zero, as explicitly demonstrated in this paper for the first time. Such catastrophic instability due to ill-posedness is a common issue when developing new mathematical models and implies that either some important physics is missing or the model has not been properly formulated. In this paper an alternative to the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$ -rheology that does not suffer from such defects is proposed. In the framework of compressible $I$ -dependent rheology (CIDR), new constitutive laws for the inertial regime are introduced; these match the well-established $\unicode[STIX]{x1D707}(I)$ and $\unicode[STIX]{x1D6F7}(I)$ relations in the steady-state limit and at the same time are well-posed for all deformations and all packing densities. Time-dependent numerical solutions of the resultant equations are performed to demonstrate that the new inertial CIDR model leads to numerical convergence towards physically realistic solutions that are supported by discrete element method simulations.}, journal={JOURNAL OF FLUID MECHANICS}, author={Schaeffer, D. G. and Barker, T. and Tsuji, D. and Gremaud, P. and Shearer, M. and Gray, J. M. N. T.}, year={2019}, month={Sep}, pages={926–951} } @article{chen_bardall_shearer_daniels_2019, title={Distinguishing deformation mechanisms in elastocapillary experiments}, volume={15}, ISSN={1744-683X 1744-6848}, url={http://dx.doi.org/10.1039/C9SM01756A}, DOI={10.1039/C9SM01756A}, abstractNote={Soft materials are known to deform due to a variety of mechanisms, including capillarity, buoyancy, and swelling. The choice of liquid plays a significant role in the outcome of experiments.}, number={46}, journal={Soft Matter}, publisher={Royal Society of Chemistry (RSC)}, author={Chen, Shih-Yuan and Bardall, Aaron and Shearer, Michael and Daniels, Karen E.}, year={2019}, pages={9426–9436} } @article{brown_shearer_2018, title={A SCALAR CONSERVATION LAW FOR PLUME MIGRATION IN CARBON SEQUESTRATION}, volume={78}, ISSN={["1095-712X"]}, DOI={10.1137/17m1127089}, abstractNote={A quasi-linear hyperbolic partial differential equation with a discontinuous flux models geologic carbon dioxide (CO$_2$) migration and storage [M. Hesse, F. Orr, and H. Tchelepi, Fluid Mech., 611 ...}, number={3}, journal={SIAM JOURNAL ON APPLIED MATHEMATICS}, author={Brown, Elisabeth and Shearer, Michael}, year={2018}, pages={1823–1841} } @article{congy_el_hoefer_shearer_2018, title={Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure}, volume={142}, ISSN={0022-2526}, url={http://dx.doi.org/10.1111/sapm.12247}, DOI={10.1111/sapm.12247}, abstractNote={Abstract}, number={3}, journal={Studies in Applied Mathematics}, publisher={Wiley}, author={Congy, T. and El, G.A. and Hoefer, M.A. and Shearer, M.}, year={2018}, month={Nov}, pages={241–268} } @article{tang_brzinski_shearer_daniels_2018, title={Nonlocal rheology of dense granular flow in annular shear experiments}, volume={14}, ISSN={1744-683X 1744-6848}, url={http://dx.doi.org/10.1039/c8sm00047f}, DOI={10.1039/c8sm00047f}, abstractNote={Experimental measurements of boundary stresses and flow fields of a quasi-2D granular material under steady shear validate two nonlocal rheological models.}, number={16}, journal={Soft Matter}, publisher={Royal Society of Chemistry (RSC)}, author={Tang, Zhu and Brzinski, Theodore A. and Shearer, Michael and Daniels, Karen E.}, year={2018}, pages={3040–3048} } @article{bardall_daniels_shearer_2017, title={Deformation of an elastic substrate due to a resting sessile droplet}, volume={29}, ISSN={0956-7925 1469-4425}, url={http://dx.doi.org/10.1017/S0956792517000134}, DOI={10.1017/s0956792517000134}, abstractNote={On a sufficiently soft substrate, a resting fluid droplet will cause significant deformation of the substrate. This deformation is driven by a combination of capillary forces at the contact line and the fluid pressure at the solid surface. These forces are balanced at the surface by the solid traction stress induced by the substrate deformation. Young's Law, which predicts the equilibrium contact angle of the droplet, also indicates an a priori radial force balance for rigid substrates, but not necessarily for soft substrates that deform under loading. It remains an open question whether the contact line transmits a non-zero force tangent to the substrate surface in addition to the conventional normal (vertical) force. We present an analytic Fourier transform solution technique that includes general interfacial energy conditions, which govern the contact angle of a 2D droplet. This includes evaluating the effect of gravity on the droplet shape in order to determine the correct fluid pressure at the substrate surface for larger droplets. Importantly, we find that in order to avoid a strain singularity at the contact line under a non-zero tangential contact line force, it is necessary to include a previously neglected horizontal traction boundary condition. To quantify the effects of the contact line and identify key quantities that will be experimentally accessible for testing the model, we evaluate solutions for the substrate surface displacement field as a function of Poisson's ratio and zero/non-zero tangential contact line forces.}, number={2}, journal={European Journal of Applied Mathematics}, publisher={Cambridge University Press (CUP)}, author={Bardall, Aaron and Daniels, Karen E. and Shearer, Michael}, year={2017}, month={Jun}, pages={281–300} } @article{el_hoefer_shearer_2017, title={Dispersive and Diffusive-Dispersive ShockWaves for Nonconvex Conservation Laws}, volume={59}, ISSN={["1095-7200"]}, DOI={10.1137/15m1015650}, abstractNote={We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion is known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is nonconvex. This review compares the structure of solutions of Riemann problems for a conservation law with nonconvex, cubic flux regularized by two different mechanisms: (1) dispersion in the modified Korteweg--de Vries (mKdV) equation; and (2) a combination of diffusion and dispersion in the mKdV--Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs), and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both class...}, number={1}, journal={SIAM REVIEW}, author={El, G. A. and Hoefer, M. A. and Shearer, M.}, year={2017}, pages={3–61} } @article{shearer_el_hoefer_2017, title={Stationary Expansion Shocks for a Regularized Boussinesq System}, volume={139}, ISSN={0022-2526}, url={http://dx.doi.org/10.1111/sapm.12191}, DOI={10.1111/sapm.12191}, abstractNote={Abstract}, number={4}, journal={Studies in Applied Mathematics}, publisher={Wiley}, author={Shearer, Michael and El, Gennady A. and Hoefer, Mark A.}, year={2017}, month={Sep}, pages={1–22} } @article{barker_schaeffer_shearer_gray_2017, title={Well-posed continuum equations for granular flow with compressibility and μ ( I )-rheology}, volume={473}, ISSN={1364-5021 1471-2946}, url={http://dx.doi.org/10.1098/rspa.2016.0846}, DOI={10.1098/rspa.2016.0846}, abstractNote={ Continuum modelling of granular flow has been plagued with the issue of ill-posed dynamic equations for a long time. Equations for incompressible, two-dimensional flow based on the Coulomb friction law are ill-posed regardless of the deformation, whereas the rate-dependent μ ( I )-rheology is ill-posed when the non-dimensional inertial number I is too high or too low. Here, incorporating ideas from critical-state soil mechanics, we derive conditions for well-posedness of partial differential equations that combine compressibility with I -dependent rheology. When the I -dependence comes from a specific friction coefficient μ ( I ), our results show that, with compressibility, the equations are well-posed for all deformation rates provided that μ ( I ) satisfies certain minimal, physically natural, inequalities. }, number={2201}, journal={Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences}, publisher={The Royal Society}, author={Barker, T. and Schaeffer, D. G. and Shearer, M. and Gray, J. M. N. T.}, year={2017}, month={May}, pages={20160846} } @article{bostwick_dijksman_shearer_2017, title={Wetting dynamics of a collapsing fluid hole}, volume={2}, ISSN={["2469-990X"]}, DOI={10.1103/physrevfluids.2.014006}, abstractNote={The collapse dynamics of an axisymmetric fluid cavity that wets the bottom of a rotating bucket bound by vertical sidewalls are studied. Lubrication theory is applied to the governing field equations for the thin film to yield an evolution equation that captures the effect of capillary, gravitational, and centrifugal forces on this converging flow. The focus is on the quasistatic spreading regime, whereby contact-line motion is governed by a constitutive law relating the contact-angle to the contact-line speed. Surface tension forces dominate the collapse dynamics for small holes with the collapse time appearing as a power law whose exponent compares favorably to experiments in the literature. Gravity accelerates the collapse process. Volume dependence is predicted and compared with experiment. Centrifugal forces slow the collapse process and lead to complex dynamics characterized by stalled spreading behavior that separates the large and small hole asymptotic regimes.}, number={1}, journal={PHYSICAL REVIEW FLUIDS}, author={Bostwick, J. B. and Dijksman, J. A. and Shearer, M.}, year={2017}, month={Jan} } @article{el_hoefer_shearer_2016, title={Expansion shock waves in regularized shallow-water theory}, volume={472}, ISSN={1364-5021 1471-2946}, url={http://dx.doi.org/10.1098/rspa.2016.0141}, DOI={10.1098/rspa.2016.0141}, abstractNote={We identify a new type of shock wave by constructing a stationary expansion shock solution of a class of regularized shallow-water equations that include the Benjamin–Bona–Mahony and Boussinesq equations. An expansion shock exhibits divergent characteristics, thereby contravening the classical Lax entropy condition. The persistence of the expansion shock in initial value problems is analysed and justified using matched asymptotic expansions and numerical simulations. The expansion shock's existence is traced to the presence of a non-local dispersive term in the governing equation. We establish the algebraic decay of the shock as it is gradually eroded by a simple wave on either side. More generally, we observe a robustness of the expansion shock in the presence of weak dissipation and in simulations of asymmetric initial conditions where a train of solitary waves is shed from one side of the shock.}, number={2189}, journal={Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science}, publisher={The Royal Society}, author={El, Gennady A. and Hoefer, Mark A. and Shearer, Michael}, year={2016}, month={May}, pages={20160141} } @book{shearer_levy_2015, title={Partial differential equations: An introduction to theory and applications}, ISBN={9780691161297}, publisher={Princeton: Princeton University Press}, author={Shearer, M. and Levy, R.}, year={2015} } @article{strickland_shearer_daniels_2015, title={Spatiotemporal measurement of surfactant distribution on gravity-capillary waves}, volume={777}, ISSN={["1469-7645"]}, DOI={10.1017/jfm.2015.352}, abstractNote={Materials adsorbed onto the surface of a fluid – for instance, crude oil, biogenic slicks or industrial/medical surfactants – will move in response to surface waves. Owing to the difficulty of non-invasive measurement of the spatial distribution of a molecular monolayer, little is known about the dynamics that couple the surface waves and the evolving density field. Here, we report measurements of the spatiotemporal dynamics of the density field of an insoluble surfactant driven by gravity–capillary waves in a shallow cylindrical container. Standing Faraday waves and travelling waves generated by the meniscus are superimposed to create a non-trivial surfactant density field. We measure both the height field of the surface using moiré imaging, and the density field of the surfactant via the fluorescence of NBD-tagged phosphatidylcholine, a lipid. Through phase averaging stroboscopically acquired images of the density field, we determine that the surfactant accumulates on the leading edge of the travelling meniscus waves and in the troughs of the standing Faraday waves. We fit the spatiotemporal variations in the two fields using an ansatz consisting of a superposition of Bessel functions, and report measurements of the wavenumbers and energy damping factors associated with the meniscus and Faraday waves, as well as the spatial and temporal phase shifts between them. While these measurements are largely consistent for both types of waves and both fields, it is notable that the damping factors for height and surfactant in the meniscus waves do not agree. This raises the possibility that there is a contribution from longitudinal waves in addition to the gravity–capillary waves.}, journal={JOURNAL OF FLUID MECHANICS}, publisher={Cambridge University Press (CUP)}, author={Strickland, Stephen L. and Shearer, Michael and Daniels, Karen E.}, year={2015}, month={Aug} } @article{swanson_strickland_shearer_daniels_2015, title={Surfactant spreading on a thin liquid film: reconciling models and experiments}, volume={94}, ISSN={["1573-2703"]}, DOI={10.1007/s10665-014-9735-0}, abstractNote={The spreading dynamics of surfactant molecules on a thin fluid layer is of both fundamental and practical interest. A mathematical model formulated by Gaver and Grotberg [J Fluid Mech 235:399–414, 1992] describing the spreading of a single layer of insoluble surfactant has become widely accepted, and several experiments on axisymmetric spreading have confirmed its predictions for both the height profile of the free surface and the spreading exponent (the radius of the circular area covered by surfactant grows as $$t^{1/4}$$ ). However, these prior experiments utilized primarily surfactant quantities exceeding (sometimes far exceeding) a monolayer. In this paper, we report that this regime is characterized by a mismatch between the timescales of the experiment and model and, additionally, find that the spatial distribution of surfactant molecules differs substantially from the model prediction. For experiments performed in the monolayer regime for which the model was developed, the surfactant layer is observed to have a spreading exponent of less than $$1/10$$ , far below the predicted value, and the surfactant distribution is also in disagreement. These findings suggest that the model is inadequate for describing the spreading of insoluble surfactants on thin fluid layers.}, number={1}, journal={JOURNAL OF ENGINEERING MATHEMATICS}, publisher={Springer Nature}, author={Swanson, Ellen R. and Strickland, Stephen L. and Shearer, Michael and Daniels, Karen E.}, year={2015}, month={Oct}, pages={63–79} } @article{shearer_spayd_swanson_2015, title={Traveling waves for conservation laws with cubic nonlinearity and BBM type dispersion}, volume={259}, ISSN={["1090-2732"]}, DOI={10.1016/j.jde.2015.04.019}, abstractNote={Scalar conservation laws with non-convex fluxes have shock wave solutions that violate the Lax entropy condition. In this paper, such solutions are selected by showing that some of them have corresponding traveling waves for the equation supplemented with dissipative and dispersive higher-order terms. For a cubic flux, traveling waves can be calculated explicitly for linear dissipative and dispersive terms. Information about their existence can be used to solve the Riemann problem, in which we find solutions for some data that are different from the classical Lax–Oleinik construction. We consider dispersive terms of a BBM type and show that the calculation of traveling waves is somewhat more intricate than for a KdV-type dispersion. The explicit calculation is based upon the calculation of parabolic invariant manifolds for the associated ODE describing traveling waves. The results extend to the p-system of one-dimensional elasticity with a cubic stress–strain law.}, number={7}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Shearer, Michael and Spayd, Kimberly R. and Swanson, Ellen R.}, year={2015}, month={Oct}, pages={3216–3232} } @article{bostwick_shearer_daniels_2014, title={Elastocapillary deformations on partially-wetting substrates: rival contact-line models}, volume={10}, ISSN={1744-683X 1744-6848}, url={http://dx.doi.org/10.1039/C4SM00891J}, DOI={10.1039/C4SM00891J}, abstractNote={A partially-wetting liquid can deform the underlying elastic substrate upon which it rests. This situation requires the development of theoretical models to describe the wetting forces imparted by the drop onto the solid substrate, particularly those at the contact-line. We construct a general solution using a displacement potential function for the elastic deformations within a finite elastic substrate associated with these wetting forces, and compare the results for several different contact-line models. Our work incorporates internal contributions to the surface stress from both liquid/solid Σls and Σsg solid/gas solid surface tensions (surface stress), which results in a non-standard boundary-value problem that we solve using a dual integral equation. We compare our results to relevant experiments and conclude that the generalization of solid surface tension Σls ≠ Σsg is an essential feature in any model of partial-wetting. The comparisons also allow us to systematically eliminate some proposed contact-line models.}, number={37}, journal={Soft Matter}, publisher={Royal Society of Chemistry (RSC)}, author={Bostwick, Joshua B. and Shearer, Michael and Daniels, Karen E.}, year={2014}, month={Jul}, pages={7361} } @misc{strait_shearer_levy_cueto-felgueroso_juanes_2014, title={Two Fluid Flow in a Capillary Tube}, volume={109}, ISBN={9783319111247 9783319111254}, ISSN={2194-1009 2194-1017}, url={http://dx.doi.org/10.1007/978-3-319-11125-4_14}, DOI={10.1007/978-3-319-11125-4_14}, journal={Collaborative Mathematics and Statistics Research}, publisher={Springer International Publishing}, author={Strait, Melissa and Shearer, Michael and Levy, Rachel and Cueto-Felgueroso, Luis and Juanes, Ruben}, year={2014}, month={Oct}, pages={149–161} } @article{shearer_2013, title={Smooth Periodic Solutions of a 2x2 System of Nonlinear Hyperbolic Conservation Laws}, volume={7}, ISSN={1687-1200 1687-1197}, url={http://dx.doi.org/10.1093/amrx/abt006}, DOI={10.1093/amrx/abt006}, abstractNote={The Keyfitz–Kranzer system of two conservation laws has the property that it can be written as a triangular system of equations that can be solved successively. In this paper, we show that the system reduces to a linearly degenerate system if the tension T is given by Hooke’s law. This linearly degenerate system has a formula for time-periodic smooth solutions that are easily generated numerically.}, journal={Applied Mathematics Research eXpress}, publisher={Oxford University Press (OUP)}, author={Shearer, M.}, year={2013}, month={Jul} } @inproceedings{shearer_2013, title={Two Fluid Flow in Porous Media}, volume={8}, booktitle={Proceedings of HYP2012}, publisher={American Institute of Mathematical Sciences}, author={Shearer, Michael}, year={2013}, pages={212–232} } @article{peterson_shearer_2012, title={Simulation of spreading surfactant on a thin liquid film}, volume={218}, ISSN={0096-3003}, url={http://dx.doi.org/10.1016/j.amc.2011.11.002}, DOI={10.1016/j.amc.2011.11.002}, abstractNote={The spreading of insoluble surfactant on a thin liquid film is modeled by a pair of nonlinear partial differential equations for the height of the free surface and the surfactant concentration. A numerical method is developed in which the leading edge of the surfactant is tracked. In the absence of higher order regularization the system becomes hyperbolic/degenerate-parabolic, introducing jumps in the height of the free surface and the surfactant concentration gradient. We compare numerical simulations to those of a hybrid Godunov method in which the height equation is treated as a scalar conservation law and a parabolic solver is used for the surfactant equation. We show how the tracking method applies to the full equations with realistic gravity and capillarity terms included, even though the disturbance in the height of the free surface extends beyond the support of the surfactant concentration.}, number={9}, journal={Applied Mathematics and Computation}, publisher={Elsevier BV}, author={Peterson, Ellen R. and Shearer, Michael}, year={2012}, month={Jan}, pages={5157–5167} } @article{spayd_shearer_hu_2012, title={Stability of plane waves in two-phase porous media flow}, volume={91}, ISSN={["0003-6811"]}, DOI={10.1080/00036811.2011.618128}, abstractNote={We examine the Saffman–Taylor instability for oil displaced by water in a porous medium. The model equations are based on Darcy's law for two-phase flow, with dependent variables pressure and saturation. Stability of plane wave solutions is governed by the hyperbolic/elliptic system obtained by ignoring capillary pressure, which adds diffusion to the hyperbolic equation. Interestingly, the growth rate of perturbations of unstable waves is linear in the wave number to leading order, whereas a naive analysis would indicate quadratic dependence. This gives a sharp boundary in the state space of upstream and downstream saturations separating stable from unstable waves. The role of this boundary, derived from the linearized hyperbolic/elliptic system, is verified by numerical simulations of the full nonlinear parabolic/elliptic equations.}, number={2}, journal={APPLICABLE ANALYSIS}, author={Spayd, Kim and Shearer, Michael and Hu, Zhengzheng}, year={2012}, pages={295–308} } @article{spayd_shearer_2011, title={THE BUCKLEY-LEVERETT EQUATION WITH DYNAMIC CAPILLARY PRESSURE}, volume={71}, ISSN={["1095-712X"]}, DOI={10.1137/100807016}, abstractNote={The Buckley–Leverett equation for two-phase flow in a porous medium is modified by including dependence of the capillary pressure on the rate of change of saturation. This model, due to Hassanizadeh and Gray, results in a nonlinear pseudoparabolic partial differential equation. Phase plane analysis, including a separation function to measure the distance between invariant manifolds, is used to determine when the equation supports traveling waves corresponding to undercompressive shocks. The Riemann problem for the underlying conservation law is solved, and the structures of the various solutions are confirmed with numerical simulations of the partial differential equation.}, number={4}, journal={SIAM JOURNAL ON APPLIED MATHEMATICS}, author={Spayd, K. and Shearer, M.}, year={2011}, pages={1088–1108} } @article{mcintyre_rowe_shearer_gray_thornton_2010, title={Evolution of a Mixing Zone in Granular Avalanches}, volume={7}, ISSN={1687-1200 1687-1197}, url={http://dx.doi.org/10.1093/amrx/abm008}, DOI={10.1093/amrx/abm008}, abstractNote={A nonlinear first-order partial differential equation in two space variables and time describes the process of kinetic sieving in an avalanche, in which larger particles tend to rise to the surface while smaller particles descend, quickly leading to completely segregated layers. The interface between layers is a shock wave satisfying its own nonlinear equation. When the interface becomes vertical, it loses stability, and develops a mixing zone. The mixing zone is described explicitly under idealized initial conditions, and verified with numerical simulation. The problem and its solution are similar to twodimensional Riemann problems for scalar first-order conservation laws; the difference here is that the equation is not scale-invariant, due to shear in the avalanche, an essential ingredient of kinetic sieving.}, journal={Applied Mathematics Research eXpress}, publisher={Oxford University Press (OUP)}, author={McIntyre, M. and Rowe, E. L. and Shearer, M. and Gray, J. M. N. T. and Thornton, A. R.}, year={2010}, month={Jul} } @article{shearer_dafermos_2010, title={FINITE TIME EMERGENCE OF A SHOCK WAVE FOR SCALAR CONSERVATION LAWS}, volume={7}, ISSN={["0219-8916"]}, DOI={10.1142/s0219891610002037}, abstractNote={ For a convex conservation law [Formula: see text] bounded initial data u0(x), are considered that take on constant values u- to the left of a bounded interval, and u+ to the right, with u- > u+. The solution of the initial value problem is shown to collapse in finite time to a single shock wave joining u- to u+. The proof involves comparison with a solution having piecewise constant initial data, for which the evolution to a single shock involves straightforward rarefaction-shock interactions. This result has a significant application to steady granular flow in a chute, and the result is reformulated to apply to the Lighthill–Whitham–Richards equation of traffic flow. }, number={1}, journal={JOURNAL OF HYPERBOLIC DIFFERENTIAL EQUATIONS}, author={Shearer, Michael and Dafermos, Constantine}, year={2010}, month={Mar}, pages={107–116} } @article{peterson_shearer_2010, title={Radial Spreading of a Surfactant on a Thin Liquid Film}, volume={9}, ISSN={1687-1200 1687-1197}, url={http://dx.doi.org/10.1093/amrx/abq015}, DOI={10.1093/amrx/abq015}, abstractNote={When a surfactant is placed on a layer of fluid, it reduces surface tension locally, creating a surface stress imbalance that sets the fluid in motion. The lubrication approximation is applied to axisymmetric spreading, yielding a coupled system of nonlinear partial differential equation for the height of the fluid free surface and the distribution of the surfactant. For a simplified system ignoring the effects of gravity and capillarity, as well as diffusion of surfactant molecules, the location of the surfactant can be tracked numerically. The free surface height converges quickly to a similarity form [Jensen, “Self-similar, surfactant-driven flows.” Physics of Fluids 6 (1994): 1084–94] away from the origin. Near the origin, a self-similar solution is identified, but it differs qualitatively from long-time numerical solutions. Including nonself-similar terms in an expansion around the origin corrects this inconsistency.}, journal={Applied Mathematics Research eXpress}, publisher={Oxford University Press (OUP)}, author={Peterson, E. R. and Shearer, M.}, year={2010}, month={Sep} } @article{shearer_giffen_2010, title={SHOCK FORMATION AND BREAKING IN GRANULAR AVALANCHES}, volume={27}, ISSN={["1553-5231"]}, DOI={10.3934/dcds.2010.27.693}, abstractNote={In this paper, we explore properties of shock wave solutions of the Gray-Thornton model for particle size segregation in granular avalanches. The model equation is a nonlinear scalar conservation law expressing conservation of mass under shear for the concentration of small particles in a bidisperse mixture. Shock waves are weak solutions of the partial differential equation across which the concentration jumps. We give precise criteria on smooth initial conditions under which a shock wave forms in the interior of the avalanche in finite time. Shocks typically lose stability as they are sheared by the flow, giving way to a complex structure in which a two-dimensional rarefaction wave interacts dynamically with a pair of shocks. The rarefaction represents a mixing zone, in which small and large particles are mixed as they are transported up and down (respectively) through the zone. The mixing zone expands and twice changes its detailed structure before reaching the boundary.}, number={2}, journal={DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS}, author={Shearer, Michael and Giffen, Nicholas}, year={2010}, month={Jun}, pages={693–714} } @article{may_shearer_daniels_2010, title={Scalar Conservation Laws with Nonconstant Coefficients with Application to Particle Size Segregation in Granular Flow}, volume={20}, ISSN={0938-8974 1432-1467}, url={http://dx.doi.org/10.1007/s00332-010-9069-7}, DOI={10.1007/s00332-010-9069-7}, abstractNote={Granular materials will segregate by particle size when subjected to shear, as occurs, for example, in avalanches. The evolution of a bidisperse mixture of particles can be modeled by a nonlinear first order partial differential equation, provided the shear (or velocity) is a known function of position. While avalanche-driven shear is approximately uniform in depth, boundary-driven shear typically creates a shear band with a nonlinear velocity profile. In this paper, we measure a velocity profile from experimental data and solve initial value problems that mimic the segregation observed in the experiment, thereby verifying the value of the continuum model. To simplify the analysis, we consider only one-dimensional configurations, in which a layer of small particles is placed above a layer of large particles within an annular shear cell and is sheared for arbitrarily long times. We fit the measured velocity profile to both an exponential function of depth and a piecewise linear function which separates the shear band from the rest of the material. Each solution of the initial value problem is nonstandard, involving curved characteristics in the exponential case, and a material interface with a jump in characteristic speed in the piecewise linear case.}, number={6}, journal={Journal of Nonlinear Science}, publisher={Springer Science and Business Media LLC}, author={May, Lindsay B. H. and Shearer, Michael and Daniels, Karen E.}, year={2010}, month={May}, pages={689–707} } @article{may_golick_phillips_shearer_daniels_2010, title={Shear-driven size segregation of granular materials: Modeling and experiment}, volume={81}, ISSN={["1550-2376"]}, DOI={10.1103/physreve.81.051301}, abstractNote={Granular materials segregate by size under shear, and the ability to quantitatively predict the time required to achieve complete segregation is a key test of our understanding of the segregation process. In this paper, we apply the Gray-Thornton model of segregation (developed for linear shear profiles) to a granular flow with an exponential shear profile, and evaluate its ability to describe the observed segregation dynamics. Our experiment is conducted in an annular Couette cell with a moving lower boundary. The granular material is initially prepared in an unstable configuration with a layer of small particles above a layer of large particles. Under shear, the sample mixes and then resegregates so that the large particles are located in the top half of the system in the final state. During this segregation process, we measure the velocity profile and use the resulting exponential fit as input parameters to the model. To make a direct comparison between the continuum model and the observed segregation dynamics, we map the local concentration (from the model) to changes in packing fraction; this provides a way to make a semiquantitative comparison with the measured global dilation. We observe that the resulting model successfully captures the presence of a fast mixing process and relatively slower resegregation process, but the model predicts a finite resegregation time, while in the experiment resegregation occurs only exponentially in time.}, number={5}, journal={PHYSICAL REVIEW E}, publisher={American Physical Society (APS)}, author={May, Lindsay B. H. and Golick, Laura A. and Phillips, Katherine C. and Shearer, Michael and Daniels, Karen E.}, year={2010}, month={May} } @inproceedings{shearer_may_giffen_daniels_goddard_giovine_jenkins_2010, title={The Gray-Thornton Model of Granular Segregation}, ISSN={0094-243X}, url={http://dx.doi.org/10.1063/1.3435407}, DOI={10.1063/1.3435407}, abstractNote={In this paper, we explore properties of the Gray‐Thornton model for particle size segregation in granular avalanches. The model equation is a single conservation law expressing conservation of mass under shear for the concentration of the smaller of two types of particle in a bidisperse mixture. Sharp interfaces across which the concentration jumps are shock wave solutions of the partial differential equation. We show that they can form internally from smooth data, as well as propagate in from boundaries of the domain. We prove a general stability result that expresses the physically reasonable notion that an interface should be stable only if the concentration of small particles is larger below the interface than above. Once shocks form, they are sheared by the flow, leading to loss of stability when an interface becomes vertical. The subsequent evolution of a mixing zone, a two‐dimensional rarefaction solution of the equation that replaces the unstable part of the shock can be tracked explicitly for a s...}, publisher={AIP}, author={Shearer, Michael and May, Lindsay B. H. and Giffen, Nicholas and Daniels, Karen E. and Goddard, Joe and Giovine, Pasquale and Jenkins, James T.}, year={2010} } @article{shearer_gremaud_kleiner_2009, title={Periodic motion of a mass-spring system}, volume={74}, ISSN={["0272-4960"]}, DOI={10.1093/imamat/hxp032}, abstractNote={The equations of planar motion of a mass attached to two anchored massless springs form a symmetric Hamiltonian system. The system has a single dimensionless parameter L, corresponding to the spacing between the anchors. For L > 1, there is a stable equilibrium at which the springs are in tension and lie on a line, but for L < 1, this equilibrium has both springs in compression and is unstable. However, there are then two stable equilibria at which both springs carry no force. Oscillations are studied in both regimes, but more systematically in the tension case, where techniques of bifurcation theory, numerical approximation and numerical simulation are used to explore the rich variety of periodic solutions.}, number={6}, journal={IMA JOURNAL OF APPLIED MATHEMATICS}, author={Shearer, Michael and Gremaud, Pierre and Kleiner, Kristoph}, year={2009}, month={Dec}, pages={807–826} } @misc{peterson_shearer_witelski_levy_2009, title={Stability of traveling waves in thin liquid films driven by gravity and surfactant}, volume={67}, ISSN={0160-7634 2324-7088}, url={http://dx.doi.org/10.1090/psapm/067.2/2605281}, DOI={10.1090/psapm/067.2/2605281}, abstractNote={A thin lay er of fluid flowing down a solid planar surface has a free sur face height described by a nonlinear POE derived via the lubrication ap­ proximat ion from the Navi er St okes equations. For th in films , sur face tension plays an important rol e both in providing a significant driving force and in smoothi ng the free surface. Sur fac tant molecules on the free surface tend to reduce surfac e tensio n, set t ing up grad ients that modify th e shape of the free surface. In ear lier work [12, 13J a traveling wave was found in which the free sur fac e undergoes three sharp transitions, or in ternal layers , and the surfactant is d istributed ove r a bounded region . T his triple-step traveling wave sa t is fies a system of POE, a hyperbolic conservation law for the free sur face height , and a degenerate parabolic equation descr ibing t he surfac t ant distribution. As such, th e traveling wave is overco rnpressive. An ex am ination of the lin­ earized equat ions indicates the direction and growt h rates of one-dimensiona l waves generated by small perturbat ion s in va r ious parts of the wave. Numeri­ cal si mulat ions o f the nonlinea r eq uat ions o ffer further evide nce of stability t o one-d ime nsiona l perturbations.}, journal={Hyperbolic Problems: Theory, Numerics and Applications}, publisher={American Mathematical Society}, author={Peterson, Ellen and Shearer, Michael and Witelski, Thomas P. and Levy, Rachel}, year={2009}, pages={855–868} } @article{shearer_gray_thornton_2008, title={Stable solutions of a scalar conservation law for particle-size segregation in dense granular avalanches}, volume={19}, ISSN={["1469-4425"]}, DOI={10.1017/S0956792507007280}, abstractNote={Dense, dry granular avalanches are very efficient at sorting the larger particles towards the free surface of the flow, and finer grains towards the base, through the combined processes of kinetic sieving and squeeze expulsion. This generates an inversely graded particle-size distribution, which is fundamental to a variety of pattern formation mechanisms, as well as subtle size-mobility feedback effects, leading to the formation of coarse-grained lateral levees that create channels in geophysical flows, enhancing their run-out. In this paper we investigate some of the properties of a recent model [Gray, J. M. N. T. & Thornton, A. R. (2005) A theory for particle size segregation in shallow granular free-surface flows. Proc. R. Soc. 461, 1447–1473]; [Thornton, A. R., Gray, J. M. N. T. & Hogg, A. J. (2006) A three-phase mixture theory for particle size segregation in shallow granular free-surface flows. J. Fluid. Mech. 550, 1–25] for the segregation of particles of two sizes but the same density in a shear flow typical of shallow avalanches. The model is a scalar conservation law in space and time, for the volume fraction of smaller particles, with non-constant coefficients depending on depth within the avalanche. It is proved that for steady flow from an inlet, complete segregation occurs beyond a certain finite distance down the slope, no matter what the mixture at the inlet. In time-dependent flow, dynamic shock waves can develop; they are interfaces separating different mixes of particles. Shock waves are shown to be stable if and only if there is a greater concentration of large particles above the interface than below. Constructions with shocks and rarefaction waves are demonstrated on a pair of physically relevant initial boundary value problems, in which a region of all small particles is penetrated from the inlet by either a uniform mixture of particles or by a layer of small particles over a layer of large particles. In both cases, and under a linear shear flow, solutions are constructed for all time and shown to have similar structure for all choices of parameters.}, journal={EUROPEAN JOURNAL OF APPLIED MATHEMATICS}, author={Shearer, M. and Gray, J. M. N. T. and Thornton, A. R.}, year={2008}, month={Feb}, pages={61–86} } @article{levy_shearer_taylor_2007, title={Automated Review of Prerequisite Material for Intermediate-Level Undergraduate Mathematics}, volume={17}, ISSN={1051-1970 1935-4053}, url={http://dx.doi.org/10.1080/10511970601131555}, DOI={10.1080/10511970601131555}, abstractNote={Abstract We describe a program that provides structured practice of prerequisite material to students in an ordinary differential equations course using an existing automated homework system originally designed for precalculus and calculus classes. The goal of the program is to improve students' comprehension of material presented in class by timing the review to occur just before the prerequisite material is relevant in class. The review is designed to gain class time that would otherwise be dedicated to review, with minimal involvement of the instructor. We report positive responses from both instructors and students in two sections of the course.}, number={2}, journal={PRIMUS}, publisher={Informa UK Limited}, author={Levy, Rachel and Shearer, Michael and Taylor, Padraic}, year={2007}, month={May}, pages={167–180} } @article{schaeffer_shearer_witelski_2007, title={Boundary-value problems for hyperbolic equations related to steady granular flow}, volume={12}, ISSN={["1741-3028"]}, DOI={10.1177/1081286506067325}, abstractNote={ Boundary value problems for steady-state flow in elastoplasticity are a topic of mathematical and physical interest. In particular, the underlying PDE may be hyperbolic, and uncertainties surround the choice of physically appropriate stress and velocity boundary conditions. The analysis and numerical simulations of this paper address this issue for a model problem, a system of equations describing antiplane shearing of an elastoplastic material. This system retains the relevant mathematical structure of elastoplastic planar flow. Even if the flow rule is associative, two significant phenomena appear: (i) For boundary conditions suggestive of granular flow in a hopper, in which it seems physically natural to specify the velocity everywhere along a portion of the boundary, no such solutions of the equations exist; rather, we construct a solution with a shear band (velocity jump) along part of the boundary, and an appropriate relaxed boundary condition is satisfied there. (ii) Rigid zones appear inside deforming regions of the flow, and the stress field in such a zone is not uniquely determined. For a nonassociative flow rule, an extreme form of nonuniqueness—both velocity and stress—is encountered. }, number={6}, journal={MATHEMATICS AND MECHANICS OF SOLIDS}, author={Schaeffer, David G. and Shearer, Michael and Witelski, Thomas P.}, year={2007}, month={Dec}, pages={665–699} } @article{levy_shearer_witelski_2007, title={Gravity-driven thin liquid films with insoluble surfactant: smooth traveling waves}, volume={18}, ISSN={0956-7925 1469-4425}, url={http://dx.doi.org/10.1017/s0956792507007218}, DOI={10.1017/S0956792507007218}, abstractNote={The flow of a thin layer of fluid down an inclined plane is modified by the presence of insoluble surfactant. For any finite surfactant mass, traveling waves are constructed for a system of lubrication equations describing the evolution of the free-surface fluid height and the surfactant concentration. The one-parameter family of solutions is investigated using perturbation theory with three small parameters: the coefficient of surface tension, the surfactant diffusivity, and the coefficient of the gravity-driven diffusive spreading of the fluid. When all three parameters are zero, the nonlinear PDE system is hyperbolic/degenerate-parabolic, and admits traveling wave solutions in which the free-surface height is piecewise constant, and the surfactant concentration is piecewise linear and continuous. The jumps and corners in the traveling waves are regularized when the small parameters are nonzero; their structure is revealed through a combination of analysis and numerical simulation.}, number={6}, journal={European Journal of Applied Mathematics}, publisher={Cambridge University Press (CUP)}, author={Levy, Rachel and Shearer, Michael and Witelski, Thomas P.}, year={2007}, month={Dec}, pages={679–708} } @article{witelski_shearer_levy_2006, title={Growing surfactant waves in thin liquid films driven by gravity}, volume={1}, ISSN={1687-1200 1687-1197}, url={http://dx.doi.org/10.1155/amrx/2006/15487}, DOI={10.1155/amrx/2006/15487}, abstractNote={The dynamics of a gravity-driven thin film flow with insoluble surfactant are described in the lubrication approximation by a coupled system of nonlinear PDEs. When the total quantity of surfactant is fixed, a traveling wave solution exists. For the case of constant flux of surfactant from an upstream reservoir, global traveling waves no longer exist as the surfactant accumulates at the leading edge of the thin film profile. The dynamics can be described using matched asymptotic expansions for t →∞ . The solution is constructed from quasistatically evolving traveling waves. The rate of growth of the surfactant profile is shown to be O( √ t) and is supported by numerical simulations.}, journal={Applied Mathematics Research eXpress}, publisher={Oxford University Press (OUP)}, author={Witelski, T. P. and Shearer, M. and Levy, R.}, year={2006}, month={Jan} } @article{levy_shearer_2006, title={The Motion of a Thin Liquid Film Driven by Surfactant and Gravity}, volume={66}, ISSN={0036-1399 1095-712X}, url={http://dx.doi.org/10.1137/050637030}, DOI={10.1137/050637030}, abstractNote={We investigate wave solutions of a lubrication model for surfactant-driven flow of a thin liquid film down an inclined plane. We model the flow in one space dimension with a system of nonlinear PDEs of mixed hyperbolic-parabolic type in which the effects of capillarity and surface diffusion are neglected. Numerical solutions reveal distinct patterns of waves that are described analytically by combinations of traveling waves, some with jumps in height and surfactant concentration gradient. The various waves and combinations are strikingly different from what is observed in the case of flow on a horizontal plane. Jump conditions admit new shock waves sustained by a linear surfactant wave traveling upstream. The stability of these waves is investigated analytically and numerically. For initial value problems, a critical ratio of upstream to downstream height separates two distinct long-time wave patterns. Below the critical ratio, there is also an exact solution in which the height is piecewise constant and ...}, number={5}, journal={SIAM Journal on Applied Mathematics}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Levy, R. and Shearer, M.}, year={2006}, month={Jan}, pages={1588–1609} } @article{gray_shearer_thornton_2006, title={Time-dependent solutions for particle-size segregation in shallow granular avalanches}, volume={462}, ISSN={["1471-2946"]}, DOI={10.1098/rspa.2005.1580}, abstractNote={Rapid shallow granular free-surface flows develop in a wide range of industrial and geophysical flows, ranging from rotating kilns and blenders to rock-falls, snow slab-avalanches and debris-flows. Within these flows, grains of different sizes often separate out into inversely graded layers, with the large particles on top of the fines, by a process called kinetic sieving. In this paper, a recent theory is used to construct exact time-dependent two-dimensional solutions for the development of the particle-size distribution in inclined chute flows. The first problem assumes the flow is initially homogeneously mixed and is fed at the inflow with homogeneous material of the same concentration. Concentration shocks develop during the flow and the particles eventually separate out into inversely graded layers sufficiently far downstream. Sections with a monotonically decreasing shock height, between these layers, steepen and break in finite time. The second problem assumes that the material is normally graded, with the small particles on top of the coarse ones. In this case, shock waves, concentration expansions, non-centred expanding shock regions and breaking shocks develop. As the parameters are varied, nonlinearity leads to fundamental topological changes in the solution, and, in simple-shear, a logarithmic singularity prevents a steady-state solution from being attained.}, number={2067}, journal={PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES}, author={Gray, JMNT and Shearer, M and Thornton, AR}, year={2006}, month={Mar}, pages={947–972} } @article{levy_shearer_2005, title={Kinetics and nucleation for driven thin film flow}, volume={209}, ISSN={0167-2789}, url={http://dx.doi.org/10.1016/j.physd.2005.07.003}, DOI={10.1016/j.physd.2005.07.003}, abstractNote={The lubrication theory of thin liquid films, driven by a constant surface stress opposing gravity, is described by a scalar fourth order PDE for the film height h:ht+(h2−h3)x=−γ(h3hxxx)x, in which γ is a positive constant related to surface tension. In this paper, the wave structure of solutions observed in numerical simulations with γ>0 is related to the recent hyperbolic theory of the underlying scalar conservation law, in which γ=0. This theory involves a kinetic relation, describing possible undercompressive shocks, and a nucleation condition, governing the transition from classical to non-classical solution of the Riemann problem. The kinetic relation and nucleation condition are derived from consideration of traveling wave solutions (with γ>0). The kinetic relation is identified with a codimension-one bifurcation of the corresponding vector field, for which there is a traveling wave approximating an undercompressive shock. The nucleation condition is identified as a transition in the vector field at which there is no traveling wave connecting upstream and downstream heights. The thresholds defined by these conditions are incorporated into a Riemann solver map, which is tested for initial value problems for the full PDE. It is found that the parameter γ determines a limit to the applicability of the hyperbolic theory, in which the fourth order diffusion can dominate short-time transients, resulting in long-time convergence to the classical solution when the hyperbolic theory would predict a non-classical solution.}, number={1-4}, journal={Physica D: Nonlinear Phenomena}, publisher={Elsevier BV}, author={Levy, Rachel and Shearer, Michael}, year={2005}, month={Sep}, pages={145–163} } @article{behringer_shearer_2005, title={Non-linear dynamics of thin films and fluid interfaces - Preface}, volume={209}, ISSN={["1872-8022"]}, DOI={10.1016/j.physd.2005.08.003}, number={1-4}, journal={PHYSICA D-NONLINEAR PHENOMENA}, author={Behringer, RP and Shearer, M}, year={2005}, month={Sep}, pages={VII-VIII} } @article{levy_shearer_2004, title={Comparison of two dynamic contact line models for driven thin liquid films}, volume={15}, ISSN={0956-7925 1469-4425}, url={http://dx.doi.org/10.1017/s0956792504005741}, DOI={10.1017/S0956792504005741}, abstractNote={The modeling of the motion of a contact line, the triple point at which solid, liquid and air meet, is a major outstanding problem in the fluid mechanics of thin films [2, 9]. In this paper, we compare two well-known models in the specific context of Marangoni driven films. The precursor model replaces the contact line by a sharp transition between the bulk fluid and a thin layer of fluid, effectively pre-wetting the solid; the Navier slip model replaces the usual no-slip boundary condition by a singular slip condition that is effective only very near the contact line. We restrict attention to traveling wave solutions of the thin film PDE for a film driven up an inclined planar solid surface by a thermally induced surface tension gradient. This involves analyzing third order ODE that depend on several parameters. The two models considered here have subtle differences in their description, requiring a careful treatment when comparing traveling waves and effective contact angles. Numerical results exhibit broad agreement between the two models, but the closest comparison can be done only for a rather restricted range of parameters. The driven film context gives contact angle results quite different from the case of a film moving under the action of gravity alone. The numerical technique for exploring phase portraits for the third order ODE is also used to tabulate the kinetic relation and nucleation condition, information that can be used with the underlying hyperbolic conservation law to explain the rich combination of wave structures observed in simulations of the PDE and in experiments [3, 15].}, number={6}, journal={European Journal of Applied Mathematics}, publisher={Cambridge University Press (CUP)}, author={Levy, Rachel and Shearer, Michael}, year={2004}, month={Dec}, pages={625–642} } @article{lefloch_shearer_2004, title={Non-classical Riemann solvers with nucleation}, volume={134}, ISSN={["1473-7124"]}, DOI={10.1017/s0308210500003577}, abstractNote={We introduce a new non-classical Riemann solver for scalar conservation laws with concave–convex flux-function. This solver is based on both a kinetic relation, which determines the propagation speed of (under-compressive) non-classical shock waves, and a nucleation criterion, which makes a choice between a classical Riemann solution and a non-classical one. We establish the existence of (non-classical entropy) solutions of the Cauchy problem and discuss several examples of wave interactions. We also show the existence of a class of solutions, called splitting–merging solutions, which are made of two large shocks and small bounded-variation perturbations. The nucleation solvers, as we call them, are applied to (and actually motivated by) the theory of thin-film flows; they help explain numerical results observed for such flows.}, number={2004}, journal={PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS}, author={LeFloch, PG and Shearer, M}, year={2004}, pages={961–984} } @article{shearer_schaeffer_witelski_2003, title={Stability of shear bands in an elastoplastic model for granular flow: The role of discreteness}, volume={13}, ISSN={["0218-2025"]}, DOI={10.1142/S0218202503003069}, abstractNote={ Continuum models for granular flow generally give rise to systems of nonlinear partial differential equations that are linearly ill-posed. In this paper we introduce discreteness into an elastoplasticity model for granular flow by approximating spatial derivatives with finite differences. The resulting ordinary differential equations have bounded solutions for all time, a consequence of both discreteness and nonlinearity. }, number={11}, journal={MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES}, author={Shearer, M and Schaeffer, DG and Witelski, TP}, year={2003}, month={Nov}, pages={1629–1671} } @article{buckingham_shearer_bertozzi_2003, title={Thin film traveling waves and the Navier slip condition}, volume={63}, DOI={10.1137/s0036139902401409}, abstractNote={We consider the lubrication model for a thin film driven by competing gravitational forces and thermal gradients on an inclined plane. We are interested in the general traveling wave problem when the Navier slip boundary condition is used. We contrast (1) gravity dominated flow, (2) Marangoni dominated flow, and (3) flow in which the two driving effects balance. For a "singular slip" model we show that when Marangoni forces are present the resulting traveling wave ODE reduces locally near the contact line to a case not considered previously in the literature. We compute an asymptotic expansion of the solution near the contact line and compare with numerical simulations of the full problem. Using numerical simulations and phase space analysis involving Poincare sections, we show that for all three problems there is a finite range of admissible contact angles for which traveling wave solutions exist. Even in the well-studied case (1), this is a new observation that has ramifications for the use of constitut...}, number={2}, journal={SIAM Journal on Applied Mathematics}, author={Buckingham, R. and Shearer, Michael and Bertozzi, A.}, year={2003}, pages={722–744} } @article{segal_martonen_kim_shearer_2002, title={Computer simulations of particle deposition in the lungs of chronic obstructive pulmonary disease patients}, volume={14}, ISSN={["1091-7691"]}, DOI={10.1080/08958370290084593}, abstractNote={Epidemiology data show that mortality rates for chronic obstructive pulmonary disease (COPD) patients increase with an increase in concentration of ambient particulate matter (PM). This is not seen for normal subjects. Therefore, the U.S. Environmental Protection Agency (EPA) has identified COPD patients as a susceptible subpopulation to be considered in regulatory standards. In the present study, a computer model was used to calculate deposition fractions of PM within the lungs of COPD patients. The morphology of COPD lungs was characterized by two distinct components: obstruction of airways (chronic bronchitis component), and degeneration of alveolar structure (emphysema component). The chronic bronchitis component was modeled by reducing airway diameters using airway resistance measurements in vivo, and the emphysema component was modeled by increasing alveolar volumes. Calculated results were compared with experimental data obtained from COPD patients for controlled breathing trials (tidal volume of 500 ml, respiratory time of 1 s) with a particle size of 1 µm. The model successfully depicts PM deposition patterns and their dependence on the severity of disease. The findings indicate that airway obstructions are the main cause for increased deposition in the COPD lung.}, number={7}, journal={INHALATION TOXICOLOGY}, author={Segal, RA and Martonen, TB and Kim, CS and Shearer, M}, year={2002}, month={Jul}, pages={705–720} } @article{witelski_schaeffer_shearer_2001, title={A discrete model for an ill-posed nonlinear parabolic PDE}, volume={160}, ISSN={["0167-2789"]}, DOI={10.1016/S0167-2789(01)00350-5}, abstractNote={We study a finite-difference discretization of an ill-posed nonlinear parabolic partial differential equation. The PDE is the one-dimensional version of a simplified two-dimensional model for the formation of shear bands via anti-plane shear of a granular medium. For the discretized initial value problem, we derive analytically, and observed numerically, a two-stage evolution leading to a steady-state: (i) an initial growth of grid-scale instabilities, and (ii) coarsening dynamics. Elaborating the second phase, at any fixed time the solution has a piecewise linear profile with a finite number of shear bands. In this coarsening phase, one shear band after another collapses until a steady-state with just one jump discontinuity is achieved. The amplitude of this steady-state shear band is derived analytically, but due to the ill-posedness of the underlying problem, its position exhibits sensitive dependence. Analyzing data from the simulations, we observe that the number of shear bands at time t decays like t−1/3. From this scaling law, we show that the time-scale of the coarsening phase in the evolution of this model for granular media critically depends on the discreteness of the model. Our analysis also has implications to related ill-posed nonlinear PDEs for the one-dimensional Perona–Malik equation in image processing and to models for clustering instabilities in granular materials.}, number={3-4}, journal={PHYSICA D-NONLINEAR PHENOMENA}, author={Witelski, TP and Schaeffer, DG and Shearer, M}, year={2001}, month={Dec}, pages={189–221} } @article{hayes_shearer_2001, title={A nonconvex scalar conservation law with trilinear flux}, volume={59}, ISSN={["0033-569X"]}, DOI={10.1090/qam/1866551}, abstractNote={The focus of this paper is on traveling wave solutions of the equation \[ u t + f ( u ) x = ϵ u x x + ϵ 2 γ u x x x {u_t} + f{\left ( u \right )_x} = \epsilon {u_{xx}} + {\epsilon ^2}\gamma {u_{xxx}} \] , in which the flux function f f is trilinear and nonconvex. In particular, it is shown that for combinations of parameters in certain ranges, there are traveling waves that converge as ϵ → 0 \epsilon \to 0 to undercompressive shocks, in which the characteristics pass through the shock. The analysis is based on explicit solutions of the piecewise linear ordinary differential equation satisfied by traveling waves. The analytical results are illustrated by numerical solutions of the Riemann initial value problem, and are compared with corresponding explicit results for the case of a cubic flux function.}, number={4}, journal={QUARTERLY OF APPLIED MATHEMATICS}, author={Hayes, BT and Shearer, M}, year={2001}, month={Dec}, pages={615–635} } @article{bertozzi_munch_shearer_zumbrun_2001, title={Stability of compressive and undercompressive thin film travelling waves}, volume={12}, ISSN={["0956-7925"]}, DOI={10.1017/s0956792501004466}, abstractNote={Recent studies of liquid films driven by competing forces due to surface tension gradients and gravity reveal that undercompressive travelling waves play an important role in the dynamics when the competing forces are comparable. In this paper, we provide a theoretical framework for assessing the spectral stability of compressive and undercompressive travelling waves in thin film models. Associated with the linear stability problem is an Evans function which vanishes precisely at eigenvalues of the linearized operator. The structure of an index related to the Evans function explains computational results for stability of compressive waves. A new formula for the index in the undercompressive case yields results consistent with stability. In considering stability of undercompressive waves to transverse perturbations, there is an apparent inconsistency between long-wave asymptotics of the largest eigenvalue and its actual behaviour. We show that this paradox is due to the unusual structure of the eigenfunctions and we construct a revised long-wave asymptotics. We conclude with numerical computations of the largest eigenvalue, comparisons with the asymptotic results, and several open problems associated with our findings.}, number={2001 June}, journal={EUROPEAN JOURNAL OF APPLIED MATHEMATICS}, author={Bertozzi, AL and Munch, A and Shearer, M and Zumbrun, K}, year={2001}, month={Jun}, pages={253–291} } @article{bertozzi_shearer_2000, title={Existence of undercompressive traveling waves in thin film equations}, volume={32}, ISSN={["1095-7154"]}, DOI={10.1137/S0036141099350894}, abstractNote={We consider undercompressive traveling wave solutions ofthe partial differential equation ∂th + ∂xf (h )= −∂x(h 3 ∂ 3h )+ D∂x(h 3 ∂xh), when the flux function f has the nonconvex form f (h )= h2 − h3. In numerical simulations, these waves appear to play a central role in the dynamics ofthe PDE; they also explain unusual phenomena in experiments ofdriven contact lines modeled by the PDE. We prove existence ofan undercom- pressive traveling wave solution for sufficiently small nonnegative D and nonexistence when D is sufficiently large.}, number={1}, journal={SIAM JOURNAL ON MATHEMATICAL ANALYSIS}, author={Bertozzi, AL and Shearer, M}, year={2000}, month={Jun}, pages={194–213} } @article{segal_guan_shearer_martonen_2000, title={Mathematical Model of Airflow in the Lungs of Children I: Effects of Tumor Sizes and Locations}, volume={2}, ISSN={1027-3662 1607-8578}, url={http://dx.doi.org/10.1080/10273660008833046}, DOI={10.1080/10273660008833046}, abstractNote={To contribute to the development of more effective aerosol therapy protocols in pediatric medicine, we examined airflow patterns in the lung of a four-year-old child. In particular, we addressed how the presence of tumors in airways affected the character of airflow patterns. To study the effects of tumors we employed a computational fluid dynamics package, FIDAP, to define flow conditions within a model lung. The results indicated that tumors have a pronounced affect on both (i) localized velocity profiles in airways and (ii) bulk flow distribution within the lung. By identifying the effects of physical factors on flow conditions the findings will lead to improved drug delivery regimens.}, number={3}, journal={Journal of Theoretical Medicine}, publisher={Hindawi Limited}, author={Segal, R. A. and Guan, X. and Shearer, M. and Martonen, T. B.}, year={2000}, pages={199–213} } @article{guan_segal_shearer_martonen_2000, title={Mathematical Model of Airflow in the Lungs of Children II: Effects of Ventilatory Parameters}, volume={3}, ISSN={1027-3662 1607-8578}, url={http://dx.doi.org/10.1080/10273660008833064}, DOI={10.1080/10273660008833064}, abstractNote={In an effort to develop more effective aerosol therapy procedures, we examined airflow patterns in the lung of a child (age four years). In particular, we were concerned with how ventilatory parameters (i.e., breathing rate and tidal volume) affected the patterns of airflow around tumors. To conduct the study, a computational fluid dynamics package, FIDAP was used to define a model lung. The results of simulations show the extent to which changing ventilatory parameters can affect flow patterns in the neighborhood of the tumors as well as drug distribution throughout the lung.}, number={1}, journal={Journal of Theoretical Medicine}, publisher={Hindawi Limited}, author={Guan, X. and Segal, R. A. and Shearer, M. and Martonen, T. B.}, year={2000}, pages={51–62} } @article{gremaud_schaeffer_shearer_2000, title={Numerical determination of flow corrective inserts for granular materials in conical hoppers}, volume={35}, ISSN={["0020-7462"]}, DOI={10.1016/S0020-7462(99)00064-5}, abstractNote={The flow of granular materials in hoppers is studied. In industrial applications, inserts of various sizes and shapes are often used to improve the flow properties and get rid of undesirable effects such as material sticking to the walls, funnel flow, arching, etc. We study the case of inverted conical inserts in conical hoppers. In spite of the complexity of the phenomenon as observed in practice, existing methods assume in general a radial structure of the stress and velocity fields. A new numerical approach to the problem of designing “optimal” inserts is proposed and tested. It allows for non-purely radial solutions. General comments about the overall approach and its relationship with experiments are offered.}, number={5}, journal={INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS}, author={Gremaud, PA and Schaeffer, DG and Shearer, M}, year={2000}, month={Sep}, pages={869–882} } @misc{gremaud_matthews_shearer_2000, title={Similarity solutions for granular flows in hoppers}, ISBN={9780821810521 9780821878453}, ISSN={1098-3627 0271-4132}, url={http://dx.doi.org/10.1090/conm/255/03975}, DOI={10.1090/conm/255/03975}, abstractNote={Similarity solutions for the steady state equations governing flow of granular materials under gravity in a conical or wedge-shaped hopper are constructed. Such solutions can be considered for various plasticity models. Results obtained under the classical Mohr-Coulomb yield condition are com- pared with those obtained when a von Mises yield condition is imposed. A thorough stability study in the case of Mohr-Coulomb materials, for which several simplications take place, is proposed.}, journal={Nonlinear PDE’s, Dynamics and Continuum Physics}, publisher={American Mathematical Society}, author={Gremaud, Pierre-Alain and Matthews, John V. and Shearer, Michael}, year={2000}, pages={79–95} } @article{hayes_shearer_1999, title={Undercompressive shocks and Riemann problems for scalar conservation laws with non-convex fluxes}, volume={129}, ISSN={["0308-2105"]}, DOI={10.1017/s0308210500013111}, abstractNote={The Riemann initial value problem is studied for scalar conservation laws whose fluxes have a single inflection point. For a regularization consisting of balanced diffusive and dispersive terms, the travelling wave criterion is used to select admissible shocks. In some cases, the Riemann problem solution contains an undercompressive shock. The analysis is illustrated by exploring parameter space for the Buckley–Leverett flux. The boundary of the set of parameters for which there is a physical solution of the Riemann problem for all data is computed. Within the region of acceptable parameters, the solution hasseveral different forms, depending on the initial data; the different forms are illustrated by numerical computations. Qualitatively similar behaviour is observed in Lax–Wendroff approximations of solutions of the Buckley–Leverett equation with no dissipation or dispersion.}, number={1999}, journal={PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS}, author={Hayes, B and Shearer, M}, year={1999}, pages={733–754} } @article{schulze_shearer_1999, title={Undercompressive shocks for a system of hyperbolic conservation laws with cubic nonlinearity}, volume={229}, ISSN={["0022-247X"]}, DOI={10.1006/jmaa.1998.6186}, number={1}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Schulze, MR and Shearer, M}, year={1999}, month={Jan}, pages={344–362} } @article{bertozzi_munch_shearer_1999, title={Undercompressive shocks in thin film flows}, volume={134}, ISSN={["1872-8022"]}, DOI={10.1016/s0167-2789(99)00134-7}, abstractNote={Equations of the type ht+(h2−h3)x=−ϵ3(h3hxxx)x arise in the context of thin liquid films driven by the competing effects of a thermally induced surface tension gradient and gravity. In this paper, we focus on the interaction between the fourth order regularization and the nonconvex flux. Jump initial data, from a moderately thick film to a thin precurser layer, is shown to give rise to a double wave structure that includes an undercompressive wave. This wave, which approaches an undercompressive shock as ϵ→0, is an accumulation point for a countable family of compressive waves having the same speed. The family appears through a series of bifurcations parameterized by the shock speed. At each bifurcation, a pair of traveling waves is produced, one being stable for the PDE, the other unstable. The conclusions are based primarily on numerical results for the PDE, and on numerical investigations of the ODE describing traveling waves. Fourth order linear regularization is observed to produce a similar bifurcation structure of traveling waves.}, number={4}, journal={PHYSICA D-NONLINEAR PHENOMENA}, author={Bertozzi, AL and Munch, A and Shearer, M}, year={1999}, month={Dec}, pages={431–464} } @inbook{shearer_bertozzi_munch_1999, title={Undercompressive waves in driven thin film flow: Theory, computation, and experiment}, volume={13}, ISBN={9780821820063 9781470438043}, ISSN={1089-3288 2472-5153}, url={http://dx.doi.org/10.1090/amsip/013/04}, DOI={10.1090/amsip/013/04}, booktitle={Trends in Mathematical Physics}, publisher={American Mathematical Society}, author={Shearer, Michael and Bertozzi, A.L. and Munch, A.}, editor={Alexiades, V. and Siopsis, G.Editors}, year={1999}, month={Jul}, pages={43–68} } @article{schaeffer_shearer_1998, title={A simple model for stress fluctuations in plasticity with application to granular materials}, volume={58}, ISSN={["0036-1399"]}, DOI={10.1137/S0036139996309746}, abstractNote={When granular material is modeled as a continuum, plastic constitutive behavior is often assumed. The use of plasticity amounts to replacing a complicated micromechanical system by its average behavior. Recent experiments have shown that, at least for small-scale systems, stress ∞uctuations may be of the same order, or even much larger, than average stresses. In this paper a rst generation of discrete models for stress ∞uctuations is discussed. These models consist of many spring-slider elements in parallel. The sliders all obey the same law for frictional resistance, and this resistance varies with the position, but not the velocity, of the slider. The initial positions (and hence the initial frictional resistances) of the sliders are taken to be random. The usual elastoplastic response emerges as the ensemble average over all possible initial positions of the sliders. The stress response resulting from any particular choice of initial conditions exhibits ∞uctuations similar to those in the experiments. It is shown that the magnitude of ∞uctuations is governed by two parameters, namely, the system size and the roughness, the latter dened as the ratio of particle contact length to particle size. In numerical simulations, it is observed that the roughness parameter controls the shape of the stress response as a function of applied strain.}, number={6}, journal={SIAM JOURNAL ON APPLIED MATHEMATICS}, author={Schaeffer, DG and Shearer, M}, year={1998}, month={Dec}, pages={1791–1807} } @article{garaizar_gordon_shearer_1998, title={An elastoplasticity model for antiplane shearing with a non-associative flow rule: Genuine nonlinearity of plastic waves}, volume={219}, ISSN={["0022-247X"]}, DOI={10.1006/jmaa.1997.5817}, abstractNote={In elastoplasticity models, there is a stress threshold or yield condition that plays a role in determining whether the material is deforming elastically or plastically. If the stress is below the threshold, then the deformation is elastic, and is typically modeled by linear elasticity. If the stress reaches the threshold, it is said to be at yield, and the deformation is considered to be plastic. In models of plastic deformation in which the material hardens with increasing stress, the stress-strain constitutive law is Ž typically nonlinear. Since the equations are hyperbolic at least up to some . maximum stress , nonlinearities can in principle lead to the formation of}, number={2}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Garaizar, FX and Gordon, M and Shearer, M}, year={1998}, month={Mar}, pages={344–363} } @article{schwarz_horie_shearer_1998, title={Discrete element investigation of stress fluctuation in granular flow at high strain rates}, volume={57}, ISSN={["2470-0053"]}, DOI={10.1103/physreve.57.2053}, number={2}, journal={PHYSICAL REVIEW E}, author={Schwarz, OJ and Horie, Y and Shearer, M}, year={1998}, month={Feb}, pages={2053–2061} } @article{howle_schaeffer_shearer_zhong_1998, title={Lithotripsy: The treatment of kidney stones with shock waves}, volume={40}, ISSN={["1095-7200"]}, DOI={10.1137/S0036144597322630}, abstractNote={This paper discusses mathematical models for the response of a small air bubble in water to an ultrasound pulse, a context that arises in the modern treatment for kidney stones. The paper reviews Rayleigh's 1917 theory for bubble response, applies asymptotics to describe large-amplitude solutions of Rayleigh's equations, and briefly discusses effects neglected in the simple model. The style is expository, intended both to introduce this application to mathematicians and to illustrate the use of asymptotic methods to nonmathematicians.}, number={2}, journal={SIAM REVIEW}, author={Howle, L and Schaeffer, DG and Shearer, M and Zhong, P}, year={1998}, month={Jun}, pages={356–371} } @inbook{shearer_garaizar_gordon_1997, title={Formation of Shear Bands in Models of Granular Materials}, ISBN={9789401063241 9789401155205}, ISSN={0925-0042}, url={http://dx.doi.org/10.1007/978-94-011-5520-5_31}, DOI={10.1007/978-94-011-5520-5_31}, abstractNote={In this paper we analyze the behavior of a granular material just prior to the appearance of a shear band. Shear band formation has long been associated with stress and strain localization — the deformation develops abruptly and locally near the site of the shear band (Rudnicki and Rice, 1975). It is also recognized that the phenomenon is connected with a change of type in the governing partial differential equations resulting from a continuum theory (Schaeffer, 1990).}, booktitle={IUTAM Symposium on Mechanics of Granular and Porous Materials}, publisher={Springer Netherlands}, author={Shearer, M. and Garaizar, F. X. and Gordon, M. K.}, year={1997}, pages={343–352} } @inproceedings{schaeffer_shearer_1997, place={Rotterdam, Netherlands}, title={Models for stress fluctuations in plasticity}, booktitle={Powders & grains 97 : proceedings of the third International Conference on Powders & Grains, Durham, North Carolina, 18-23 May 1997}, publisher={Balkema}, author={Schaeffer, D.G. and Shearer, Michael}, editor={Behringer, R.P. and Jenkins, J.Editors}, year={1997}, pages={325–328} } @article{gordon_shearer_schaeffer_1997, title={Plane shear waves in a fully saturated granular medium with velocity and stress controlled boundary conditions}, volume={32}, ISSN={["0020-7462"]}, DOI={10.1016/S0020-7462(96)00080-7}, abstractNote={A one-dimensional system describing small shearing disturbances in a semi-infinite, fully saturated granular medium is studied. The system is fully non-linear as a result of the incrementally non-linear constitutive law for the material. In particular, there are two different wave speeds corresponding to loading or unloading of the material. Global solutions are constructed with boundary data consisting of a single pulse in either velocity or stress. In the case of velocity controlled boundary conditions, the solution is a traveling pulse of increasing kinetic energy which eventually unloads the material, regardless of whether the initial pulse loads or unloads. The solution with stress controlled boundary conditions has these features only if the initial stress pulse unloads the material. If the initial stress pulse is loading then the solution is a slowly traveling pulse of decreasing kinetic energy which is also loading.}, number={3}, journal={INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS}, author={Gordon, MS and Shearer, M and Schaeffer, D}, year={1997}, month={May}, pages={489–503} } @inproceedings{schaeffer_shearer_1997, place={New York}, title={Stress fluctuations in granular materials}, booktitle={Mechanics of deformation and flow of particulate materials : proceedings of a symposium, Evanston, Illinois, June 29-July 2, 1997}, publisher={ASCE Publications}, author={Schaeffer, D.G. and Shearer, Michael}, editor={Chang, C.S. and Misra, A. and Liang, R.Y. and Babic, M.Editors}, year={1997} } @article{schaeffer_shearer_1997, title={The influence of material non-uniformity preceding shear-band formation in a model for granular flow}, volume={8}, ISSN={["0956-7925"]}, DOI={10.1017/S0956792597003070}, abstractNote={The onset of shear-banding in a deforming elastoplastic solid has been linked to change of type of the governing partial differential equations. If uniform material properties are assumed, then (i) deformations prior to shear-banding are uniform, and (ii) the onset of shear-banding occurs simultaneously at all points in the sample. In this paper we study, in the context of a model for anti-plane shearing of a granular material, the effect of a small variation in material properties (e.g. in yield strength) within the sample. Using matched asymptotic expansions, we find that (i) the deformation is extremely non-uniform in a short time period immediately preceding the formation of shear-bands; and (ii) generically, a shear-band forms at a single location in the sample.}, number={1997 Oct.}, journal={EUROPEAN JOURNAL OF APPLIED MATHEMATICS}, author={Schaeffer, DG and Shearer, M}, year={1997}, month={Oct}, pages={457–483} } @inproceedings{shearer_1996, place={River Edge, New Jersey}, title={Fully nonlinear hyperbolic systems related to hypoplasticity}, booktitle={Proceedings of the Fifth International Conference on Hyperbolic problems: Theory, numerics, applications}, publisher={World Scientific}, author={Shearer, Michael}, editor={Glimm, J. and Graham, M.J. and Grove, J.W. and Plohr, B.J.Editors}, year={1996}, pages={440–446} } @article{shearer_schaeffer_1996, title={Riemann Problems for 5×5 Systems of Fully Non-linear Equations Related to Hypoplasticity}, volume={19}, ISSN={0170-4214 1099-1476}, url={http://dx.doi.org/10.1002/(sici)1099-1476(199612)19:18<1433::aid-mma824>3.0.co;2-x}, DOI={10.1002/(sici)1099-1476(199612)19:18<1433::aid-mma824>3.0.co;2-x}, abstractNote={The equations of motion for two-dimensional deformations of an incompressible elastoplastic material involve five equations, two equations expressing conservation of momentum, and three constitutive laws, which we take in the rate form, i.e. relating the stress rate to the strain rate. In hypoplasticity, the constitutive laws are homogeneous of degree one in the stress and strain rates. This property has the consequence that although the equations are not in conservation form, there is nonetheless a natural way to characterize planar shock waves. The Riemann problem is the initial value problem for plane waves, in which the initial data for stress and velocity consist of two constant vectors separated by a single discontinuity. The main result is that, under appropriate assumptions, the Riemann problem has a scale invariant piecewise constant solution. The issue of uniqueness is left unresolved. Indeed, we give an example satisfying the conditions for existence, for which there are many solutions. Using asymptotics, we show how solutions of the Riemann problem are approximated by smooth solutions of a system regularized by the addition of viscous terms that preserve the property of scale invariance.}, number={18}, journal={Mathematical Methods in the Applied Sciences}, publisher={Wiley}, author={Shearer, Michael and Schaeffer, David G.}, year={1996}, month={Dec}, pages={1433–1444} } @article{shearer_schaeffer_1995, title={A class of fully nonlinear 2×2 systems of partial differential equations}, volume={20}, ISSN={0360-5302 1532-4133}, url={http://dx.doi.org/10.1080/03605309508821126}, DOI={10.1080/03605309508821126}, abstractNote={This paper is a study of certain fully nonlinear 2x2 systems of partial differential equations in one space variable and time. The nonlinearity contains a term proportional to {vert_bar}{partial_derivative}U/{partial_derivative}x{vert_bar} where U - U(x,t) {element_of} {Re}{sup 2} is the unknown function and {vert_bar}.{vert_bar} is the Euclidean norm on {Re}{sup 2}; i.e., a term homogeneous of degree 1 in {partial_derivative}U/{partial_derivative}x and singular at the origin. Such equations are motivated by hypoplasticity. The paper introduces a notion of hyperbolicity for such equations and, in the hyperbolic case, proves existence of solutions for two initial value problems admitting similarity solutions: the Riemann problem and the scale-invariant problem. Uniqueness is addressed in a companion paper. 10 refs., 10 figs.}, number={7-8}, journal={Communications in Partial Differential Equations}, publisher={Informa UK Limited}, author={Shearer, Michael and Schaeffer, David G.}, year={1995}, month={Jan}, pages={1105–1131} } @article{shearer_schaeffer_1995, title={Fully nonlinear hyperbolic systems of partial differential equations related to plasticity}, volume={20}, ISSN={0360-5302 1532-4133}, url={http://dx.doi.org/10.1080/03605309508821127}, DOI={10.1080/03605309508821127}, abstractNote={In this paper, we study systems of two fully nonlinear partial differential equations in one space dimension, of the form U{sub t}+F(U{sub x})+O, where U=U(x,t){element_of}{Re}{sup 2}. The nonlinearity F : {Re}{sup 2}{r_arrow} {Re}{sup 2} is homogeneous of degree one, i.e. F({alpha}V)={alpha}F(V), for all {alpha}>O, and is smooth away from the origin. A model example is provided by F(V)+AV+b{vert_bar}V{vert_bar}, where A is a 2x2 matrix and b{var_epsilon}{Re}, which was studied. 4 refs., 5 figs.}, number={7-8}, journal={Communications in Partial Differential Equations}, publisher={Informa UK Limited}, author={Shearer, Michael and Schaeffer, David G.}, year={1995}, month={Jan}, pages={1133–1153} } @article{shearer_yang_1995, title={The Riemann problem for a system of conservation laws of mixed type with a cubic nonlinearity}, volume={125}, ISSN={0308-2105 1473-7124}, url={http://dx.doi.org/10.1017/s0308210500030298}, DOI={10.1017/s0308210500030298}, abstractNote={Using the viscosity-capillarity admissibility criterion for shock waves, we solve the Riemann problem for the system of conservation laws}, number={4}, journal={Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, publisher={Cambridge University Press (CUP)}, author={Shearer, Michael and Yang, Yadong}, year={1995}, pages={675–699} } @article{jacobs_mckinney_shearer_1995, title={Traveling Wave Solutions of the Modified Korteweg-deVries-Burgers Equation}, volume={116}, ISSN={0022-0396}, url={http://dx.doi.org/10.1006/jdeq.1995.1043}, DOI={10.1006/jdeq.1995.1043}, number={2}, journal={Journal of Differential Equations}, publisher={Elsevier BV}, author={Jacobs, D. and Mckinney, B. and Shearer, M.}, year={1995}, month={Mar}, pages={448–467} } @inproceedings{shearer_garaizar_schaeffer_trangenstein_1994, place={Research Triangle, North Carolina}, title={Formation and development of shear bands in granular material}, booktitle={Transactions of the 11th Army Conference on Applied Mathematics and Computing}, publisher={Army Research Office}, author={Shearer, Michael and Garaizar, F.X. and Schaeffer, D.G. and Trangenstein, J.}, year={1994}, pages={15–28} } @article{shearer_schaeffer_1994, title={Unloading near a shear band in granular material}, volume={52}, ISSN={0033-569X 1552-4485}, url={http://dx.doi.org/10.1090/qam/1292207}, DOI={10.1090/qam/1292207}, abstractNote={We consider a model for dynamic deformations of granular materials which allows for the localisation of flow and the consequent development of shear bands. We construct solutions, asymptotic in small time, of Riemann initial value problems in one space dimension that include a shear band. An unusual feature of the solutions is that they are not scale invariant}, number={3}, journal={Quarterly of Applied Mathematics}, publisher={American Mathematical Society (AMS)}, author={Shearer, Michael and Schaeffer, David G.}, year={1994}, month={Sep}, pages={579–600} } @article{schaeffer_schecter_shearer_1993, title={Nonstrictly Hyperbolic Conservation Laws with a Parabolic Line}, volume={103}, ISSN={0022-0396}, url={http://dx.doi.org/10.1006/jdeq.1993.1043}, DOI={10.1006/jdeq.1993.1043}, number={1}, journal={Journal of Differential Equations}, publisher={Elsevier BV}, author={Schaeffer, D.G. and Schecter, S. and Shearer, M.}, year={1993}, month={May}, pages={94–126} } @article{shearer_schaeffer_1993, title={The Initial Value Problem for a System Modelling Unidirectional Longitudinal Elastic-Plastic Waves}, volume={24}, ISSN={0036-1410 1095-7154}, url={http://dx.doi.org/10.1137/0524065}, DOI={10.1137/0524065}, abstractNote={The authors analyze initial value problems for a hyperbolic system of equations that is a simplification of models of dynamic longitudinal elastoplastic deformations in a rod with hardening. The simplified system has a positive characteristic speed associated with stress waves, and a zero speed associated with the time independence of hardening during elastic deformation. The equations are piecewise linear in stress derivatives, and thus fully nonlinear. The main result is that for bounded uniformly continuous initial data, the Cauchy problem has a unique continuous solution that can be approximated by piecewise linear solutions of the equations.}, number={5}, journal={SIAM Journal on Mathematical Analysis}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Shearer, Michael and Schaeffer, David G.}, year={1993}, month={Sep}, pages={1111–1144} } @article{schaeffer_shearer_1993, title={Unloading near a shear band: a free boundary problem for the wave equation}, volume={18}, ISSN={0360-5302 1532-4133}, url={http://dx.doi.org/10.1080/03605309308820974}, DOI={10.1080/03605309308820974}, number={7-8}, journal={Communications in Partial Differential Equations}, publisher={Informa UK Limited}, author={Schaeffer, David G. and Shearer, Michael}, year={1993}, month={Jan}, pages={1271–1298} } @article{schaeffer_shearer_1992, title={Scale-invariant initial value problems in one-dimensional dynamic elastoplasticity, with consequences for multidimensional nonassociative plasticity}, volume={3}, ISSN={0956-7925 1469-4425}, url={http://dx.doi.org/10.1017/S0956792500000814}, DOI={10.1017/S0956792500000814}, abstractNote={This paper solves a class of one-dimensional, dynamic elastoplasticity problems for equations which describe the longitudinal motion of a rod. The initial conditionsU(x, 0)are continuous and piecewise linear, the derivative ∂U/∂x(x, 0) having just one jump atx= 0. Both the equations and the initial data are invariant under the scalingŨ(x, t) = α−1U(αx, αt), where α > 0; hence the termscale-invariant. Both in underlying motivation and in solution, this problem is highly analogous to the Riemann problem from gas dynamics. These ideas are applied to the Sandler–Rubin example of non-unique solutions in dynamic plasticity with a nonassociative flow rule. We introduce an entropy condition that re-establishes uniqueness, but we also exhibit problems regarding existence.}, number={3}, journal={European Journal of Applied Mathematics}, publisher={Cambridge University Press (CUP)}, author={Schaeffer, David G. and Shearer, Michael}, year={1992}, month={Sep}, pages={225–254} } @article{schecter_shearer_1991, title={Undercompressive shocks for nonstrictly hyperbolic conservation laws}, volume={3}, ISSN={1040-7294 1572-9222}, url={http://dx.doi.org/10.1007/bf01047709}, DOI={10.1007/bf01047709}, abstractNote={We study 2×2 systems of hyperbolic conservation laws near an umbilic point. These systems have Undercompressive shock wave solutions, i.e., solutions whose viscous profiles are represented by saddle connections in an associated family of planar vector fields. Previous studies near umbilic points have assumed that the flux function is a quadratic polynomial, in which case saddle connections lie on invariant lines. We drop this assumption and study saddle connections using Golubitsky-Schaeffer equilibrium bifurcation theory and the Melnikov integral, which detects the breaking of heteroclinic orbits. The resulting information is used to construct solutions of Riemann problems.}, number={2}, journal={Journal of Dynamics and Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Schecter, Stephen and Shearer, Michael}, year={1991}, month={Apr}, pages={199–271} } @book{shearer_1991, title={Viscous Profiles and Numerical Methods for Shock Waves}, url={http://dx.doi.org/10.21236/ada246110}, DOI={10.21236/ada246110}, abstractNote={Abstract : A workshop on shock waves was held at North Caroline State University, May 23-25, 1990. The workshop brought together mathematicians interested in the following areas related to shock waves: the theory of hyperbolic conservation laws, numerical methods for hyperbolic and parabolic systems of equations, the theory of travelling waves (viscous profiles) for parabolic systems, and applications.}, institution={Defense Technical Information Center}, author={Shearer, Michael}, year={1991}, month={Jun} } @article{schaeffer_shearer_pitman_1990, title={Instability in Critical State Theories of Granular Flow}, volume={50}, ISSN={0036-1399 1095-712X}, url={http://dx.doi.org/10.1137/0150003}, DOI={10.1137/0150003}, abstractNote={Conditions characterizing the occurrence of instability in the partial differential equations arising in critical state theories for granular flow are derived. This continues earlier work studying ill-posedness in these equations. Implications of the stability results for the formation of shear bands are discussed.}, number={1}, journal={SIAM Journal on Applied Mathematics}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Schaeffer, David G. and Shearer, Michael and Pitman, E. Bruce}, year={1990}, month={Feb}, pages={33–47} } @inbook{schaeffer_shearer_1990, title={Loss of Hyperbolicity in Yield Vertex Plasticity Models under Nonproportional Loading}, ISBN={9781461390510 9781461390497}, ISSN={0940-6573}, url={http://dx.doi.org/10.1007/978-1-4613-9049-7_15}, DOI={10.1007/978-1-4613-9049-7_15}, abstractNote={Several authors [5,8,10,12] have shown that the dynamic partial differential equations arising from continuum models for granular flow may be linearly ill-posed. In a typical theory with shear-strain hardening, the equations are linearly well-posed for small deformations but become linearly ill-posed at some critical deformation which occurs before the maximum shear stress is achieved. Before this critical deformation the dynamic equations are hyperbolic, but after this point the equations are of no definite type; rather they resemble u tt = u xx - u yy , the wave equation with a rotated time axis, in that the possible uncontrolled growth of a plane wave depends on its direction of propagation.}, booktitle={The IMA Volumes in Mathematics and Its Applications}, publisher={Springer New York}, author={Schaeffer, David G. and Shearer, Michael}, year={1990}, pages={192–217} } @inbook{shearer_schecter_1990, title={Undercompressive Shocks in Systems of Conservation Laws}, ISBN={9781461390510 9781461390497}, ISSN={0940-6573}, url={http://dx.doi.org/10.1007/978-1-4613-9049-7_16}, DOI={10.1007/978-1-4613-9049-7_16}, abstractNote={In this paper, we describe recent progress in our understanding of Riemann problems that involve undercompressive shock waves for 2 × 2 systems of nonstrictly hyperbolic conservation laws. A 2 × 2 system of conservation laws (1.1) $$ {U_t} + F{(U)_x} = 0 $$ , U = U(x,t) ∈ R 2, F: R 2 → R 2, is nonstrictly hyperbolic if the eigenvalues λ1(U) ≤ λ2(U) of dF(U) are real, but not distinct for every U. As defined in [4], system (1.1) has an umbilic point at U = U* if dF(U*) is a multiple of the identity. Hyperbolic equations with an isolated umbilic point can be classified locally according to properties of the quadratic map d 2 F(U*). Since linear changes of coordinates do not affect the shocks or rarefaction waves for quadratic nonlinearities F, the general family of quadratic nonlinearities F with a unique umbilic point can be reduced to a two parameter family, which we write as (1.2) $$ Q(u,v) = d(a{u^3}/3 + b{u^2}v + u{v^2}),a \ne 1 + {b^2} $$ , where d denotes gradient with respect to U = (u,v).}, booktitle={The IMA Volumes in Mathematics and Its Applications}, publisher={Springer New York}, author={Shearer, Michael and Schecter, Stephen}, year={1990}, pages={218–231} } @article{fehribach_shearer_1989, title={Approximately periodic solutions of the elastic string equations}, volume={32}, ISSN={0003-6811 1563-504X}, url={http://dx.doi.org/10.1080/00036818908839835}, DOI={10.1080/00036818908839835}, abstractNote={Glimm's method is used to compute solutions of initial-boundary value problems for the nonlinear system of equations describing the motion of an elastic string. For certain initial data, numerical results suggest the existence of a stable, approximately periodic solution containing no shocks. This solution consists of two new exact solutions which are patched together by the numerical algorithm}, number={1}, journal={Applicable Analysis}, publisher={Informa UK Limited}, author={Fehribach, Joseph D. and Shearer, Michael}, year={1989}, month={Jan}, pages={1–14} } @article{shearer_trangenstein_1989, title={Loss of real characteristics for models of three-phase flow in a porous medium}, volume={4}, ISSN={0169-3913 1573-1634}, url={http://dx.doi.org/10.1007/bf00179533}, DOI={10.1007/bf00179533}, number={5}, journal={Transport in Porous Media}, publisher={Springer Science and Business Media LLC}, author={Shearer, Michael and Trangenstein, JohnA.}, year={1989}, month={Oct} } @inbook{shearer_schecter_1989, place={Berlin}, series={Springer Lecture Notes in Physics}, title={Riemann problems involving undercompressive shocks}, ISBN={3540516174}, url={http://dx.doi.org/10.1007/bfb0024943}, DOI={10.1007/bfb0024943}, booktitle={PDEs and Continuum Models of Phase Transitions}, publisher={Springer-Verlag}, author={Shearer, Michael and Schecter, Stephen}, editor={Rascle, M. and Serre, D. and Slemrod), M.Editors}, year={1989}, pages={187–200}, collection={Springer Lecture Notes in Physics} } @article{shearer_1989, title={The Riemann problem for 2 × 2 systems of hyperbolic conservation laws with case I quadratic nonlinearities}, volume={80}, ISSN={0022-0396}, url={http://dx.doi.org/10.1016/0022-0396(89)90088-0}, DOI={10.1016/0022-0396(89)90088-0}, abstractNote={On etudie les problemes de Riemann pour des systemes 2×2 de lois de conservation non strictement hyperboliques avec des non-linearites quadratiques}, number={2}, journal={Journal of Differential Equations}, publisher={Elsevier BV}, author={Shearer, Michael}, year={1989}, month={Aug}, pages={343–363} } @article{shearer_schaeffer_1989, title={The quasidynamic approximation in critical state plasticity}, volume={108}, ISSN={0019-9710 1432-0673}, url={http://dx.doi.org/10.1007/bf01052974}, DOI={10.1007/bf01052974}, number={4}, journal={Archive for Rational Mechanics and Analysis}, publisher={Springer Science and Business Media LLC}, author={Shearer, Michael and Schaeffer, David G.}, year={1989}, pages={267–280} } @article{shearer_1988, title={Dynamic phase transitions in a van der Waals gas}, volume={46}, ISSN={0033-569X 1552-4485}, url={http://dx.doi.org/10.1090/qam/973380}, DOI={10.1090/qam/973380}, abstractNote={is of mixed hyperbolic and elliptic type when a is a monotonically increasing function except in an interval (a,/?) (see Fig. 1). System (1.1) has been used to describe dynamic changes of phase in a van der Waals gas [ 10] and to model elastic deformations in a rod under tension [2], Mathematically, changes of phase for Eq. (1.1) are associated with jump discontinuities (shocks) in weak solutions (u,v) of system (1.1), in which u jumps across the interval (a, /?). There is a continuing problem of how to distinguish the phase jumps that are physically relevant. Mathematically, one would like to impose an entropy admissibility condition on all jump discontinuities that selects the physically relevant shocks, including the correct phase jumps, while giving well-posedness of the Cauchy problem. For systems of mixed type, it is not known in general what the appropriate admissibility condition should be, even if the initial data are restricted to lie entirely in the hyperbolic regions. This is largely due to the presence of noncompressive shocks, which fail to satisfy the classical entropy conditions of the theory of conservation laws [3, 4] . In the context of (1.1), noncompressive shocks are phase jumps that are typically nearly stationary (i.e., with nearly zero shock speed). When the shock speed is exactly zero, the phase jump is referred to as the Maxwell line. A requirement of an admissibility condition is that the Maxwell line should be admissible. This paper is a continuation of the study of the viscosity-capillarity criterion for shocks, introduced by Slemrod [11], In particular, I discuss solutions of the Riemann initial value problem for (1.1). This is the Cauchy problem with piecewise constant initial data having a single jump. The main result is that for initial data near the Maxwell line, the Riemann problem has a solution consisting of two weak shock or rarefaction waves, separated by a slowly moving phase jump. All the jump discontinuities are required to satisfy the viscosity-capillarity criterion. Alternative admissibility criteria for shocks in solutions of the Riemann problem for (1.1) are discussed in [1, 5, 9],}, number={4}, journal={Quarterly of Applied Mathematics}, publisher={American Mathematical Society (AMS)}, author={Shearer, Michael}, year={1988}, month={Dec}, pages={631–636} } @inbook{shearer_1988, place={New York}, series={The IMA volumes in mathematics and its applications}, title={Loss of strict hyperbolicity for the Buckley-Leverett equations of three phase flow in a porous medium}, booktitle={Numerical simulation in oil recovery}, publisher={Springer}, author={Shearer, Michael}, editor={Wheeler, M.Editor}, year={1988}, collection={The IMA volumes in mathematics and its applications} } @misc{shearer_1987, title={Phase jumps near the Maxwell line}, ISBN={9780821850695 9780821876503}, ISSN={1098-3627 0271-4132}, url={http://dx.doi.org/10.1090/conm/060/873536}, DOI={10.1090/conm/060/873536}, journal={Contemporary Mathematics}, publisher={American Mathematical Society}, author={Shearer, Michael}, year={1987}, pages={111–114} } @article{schaeffer_shearer_1987, title={Riemann problems for nonstrictly hyperbolic 2x2 systems of conservation laws}, volume={304}, ISSN={0002-9947}, url={http://dx.doi.org/10.1090/s0002-9947-1987-0906816-5}, DOI={10.1090/s0002-9947-1987-0906816-5}, abstractNote={The Riemann problem is solved for 2 × 2 2 \times 2 systems of hyperbolic conservation laws having quadratic flux functions. Equations with quadratic flux functions arise from neglecting higher order nonlinear terms in hyperbolic systems that fail to be strictly hyperbolic everywhere. Such equations divide into four classes, three of which are considered in this paper. The solution of the Riemann problem is complicated, with new types of shock waves, and new singularities in the dependence of the solution on the initial data. Several ideas are introduced to help organize and clarify the new phenomena.}, number={1}, journal={Transactions of the American Mathematical Society}, publisher={American Mathematical Society (AMS)}, author={Schaeffer, David G. and Shearer, Michael}, year={1987}, month={Jan}, pages={267–306} } @article{shearer_schaeffer_marchesin_paes-leme_1987, title={Solution of the riemann problem for a prototype 2x2 system of non-strictly hyperbolic conservation laws}, volume={97}, ISSN={0003-9527 1432-0673}, url={http://dx.doi.org/10.1007/bf00280409}, DOI={10.1007/bf00280409}, number={4}, journal={Archive for Rational Mechanics and Analysis}, publisher={Springer Science and Business Media LLC}, author={Shearer, M. and Schaeffer, D. G. and Marchesin, D. and Paes-Leme, P. L.}, year={1987}, pages={299–320} } @article{schaeffer_shearer_1987, title={The classification of 2 × 2 systems of non-strictly hyperbolic conservation laws, with application to oil recovery}, volume={40}, ISSN={0010-3640 1097-0312}, url={http://dx.doi.org/10.1002/cpa.3160400202}, DOI={10.1002/cpa.3160400202}, abstractNote={On considere un systeme 2×2 a une variable d'espace: U + +F(U) x =0, −∞ 0, ou U=U(x,t)∈R 2 et F:R 2 →R 2 . On montre que, pour des equations hyperboliques, des points ombilicaux sont typiquement isoles et que en un point ombilical isole dF est typiquement diagonalisable. On donne une classification des equations hyperboliques a points ombilicaux isoles}, number={2}, journal={Communications on Pure and Applied Mathematics}, publisher={Wiley}, author={Schaeffer, David G. and Shearer, Michael}, year={1987}, month={Mar}, pages={141–178} } @book{shearer_fehribach_1987, place={Raleigh, North Carolina}, title={The elastic string equations: numerical results using Glimm’s method, and two new exact solutions}, institution={NC State University}, author={Shearer, Michael and Fehribach, J.}, year={1987} } @article{shearer_1986, title={Nonuniqueness of admissible solutions of Riemann initial value problems for a system of conservation laws of mixed type}, volume={93}, ISSN={0003-9527 1432-0673}, url={http://dx.doi.org/10.1007/bf00250844}, DOI={10.1007/bf00250844}, number={1}, journal={Archive for Rational Mechanics and Analysis}, publisher={Springer Science and Business Media LLC}, author={Shearer, Michael}, year={1986}, month={Mar}, pages={45–59} } @inproceedings{shearer_schaeffer_1986, place={Research Triangle, North Carolina}, title={Recent developments in nonstrictly hyperbolic conservation laws}, booktitle={Transactions of the Fourth Army Conference on Applied Mathematics and Computing}, publisher={U.S. Army Research Office}, author={Shearer, Michael and Schaeffer, D.G.}, year={1986}, month={May} } @inproceedings{shearer_1986, title={Shock waves and bifurcation}, booktitle={Proceedings of the Brazilian National Colloquium}, author={Shearer, Michael}, editor={Gama, L.Editor}, year={1986} } @article{shearer_1986, title={The Riemann problem for the planar motion of an elastic string}, volume={61}, ISSN={0022-0396}, url={http://dx.doi.org/10.1016/0022-0396(86)90116-6}, DOI={10.1016/0022-0396(86)90116-6}, abstractNote={where T= 7’( Ir,l) is the tension in the string, taken here to be a given smooth monotonically increasing function of the stretch 1~~~1 alone. In (l.l), we have also taken the density of the material of the string to be constant. The derivation of Eq. (1.1) (and of more general equations of motion for the string) is explained in [l]. In a previous paper [5], the Riemann problem was solved for system ( 1.1) under two assumptions: that the graph of T has exactly one inflection point, at t,, with}, number={2}, journal={Journal of Differential Equations}, publisher={Elsevier BV}, author={Shearer, Michael}, year={1986}, month={Feb}, pages={149–163} } @inbook{shearer_schaeffer_1986, place={Philadelphia, Pennsylvania}, title={Three phase flow in porous media-recent developments in nonstrictly hyperbolic conservation laws}, booktitle={Advances in Multiphase Flow and Related Problems}, publisher={SIAM}, author={Shearer, Michael and Schaeffer, D.G.}, editor={Papanicolaou, G.Editor}, year={1986}, pages={210–218} } @article{shearer_1985, title={Elementary Wave Solutions of the Equations Describing the Motion of an Elastic String}, volume={16}, ISSN={0036-1410 1095-7154}, url={http://dx.doi.org/10.1137/0516032}, DOI={10.1137/0516032}, abstractNote={The equations of planar motion of an elastic string form a $4 \times 4$ system of first order conservation laws. Two of the characteristic fields correspond to genuinely nonlinear longitudinal shocks and rarefaction waves, involving changes in the tension in the string, but not the slope. The other two fields correspond to contact discontinuities, across which the slope of the string jumps, reflecting the absence of any resistance to bending.Here, the tension T is related to the local elongation $\xi > 1$ in such a way as to ensure strict hyperbolicity: $T''(\xi ) > T{{(\xi )} / {\xi \geq 0}}$. The other main assumption is chosen to reflect properties of typical materials such as nylon and rubber. That is, $T'(\xi) $ is negative for $\xi > \xi _I $ and positive for $\xi > \xi _I $ for some $\xi _I > 1 $. The principal result of the paper is that the Riemann problem has a unique solution among combinations of centered waves, with a natural entropy condition placed on shocks. That is, any initial jump disco...}, number={3}, journal={SIAM Journal on Mathematical Analysis}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Shearer, Michael}, year={1985}, month={May}, pages={447–459} } @inbook{shearer_1985, place={New York}, series={Lecture Notes in Pure and Applied Mathematics}, title={The interaction of transverse waves for the vibrating string}, booktitle={Physical Mathematics and Nonlinear Partial Differential Equations}, publisher={Marcel Dekker}, author={Shearer, Michael}, editor={Lightbourne, J.H. and Rankin, SMEditors}, year={1985}, pages={229–238}, collection={Lecture Notes in Pure and Applied Mathematics} } @article{shearer_1985, title={The nonlinear interaction of smooth travelling waves in an elastic string}, volume={7}, ISSN={0165-2125}, url={http://dx.doi.org/10.1016/0165-2125(85)90044-7}, DOI={10.1016/0165-2125(85)90044-7}, abstractNote={It is shown that the collision of approaching weak travelling waves in a nonlinearly elastic leads to shock formation in finite time. Whereas the travelling waves are transverse, the shock is formed in the longitudinal waves generated by the collision.}, number={2}, journal={Wave Motion}, publisher={Elsevier BV}, author={Shearer, Michael}, year={1985}, month={Mar}, pages={169–175} } @inproceedings{shearer_schaeffer_1985, place={Research Triangle, North Carolina}, title={Three phase flow in a porous medium and the classification of non-strictly hyperbolic conservation laws}, booktitle={Transactions of the third Army Conference on Applied Mathematics and Computing}, publisher={U.S. Army Research Office}, author={Shearer, Michael and Schaeffer, D.G.}, year={1985}, month={May}, pages={509–518} } @article{shearer_1983, title={Admissibility criteria for shock wave solutions of a system of conservation laws of mixed type}, volume={93}, ISSN={0308-2105 1473-7124}, url={http://dx.doi.org/10.1017/s0308210500015948}, DOI={10.1017/s0308210500015948}, abstractNote={Synopsis}, number={3-4}, journal={Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, publisher={Cambridge University Press (CUP)}, author={Shearer, Michael}, year={1983}, pages={233–244} } @article{shearer_1982, title={The Riemann problem for a class of conservation laws of mixed type}, volume={46}, ISSN={0022-0396}, url={http://dx.doi.org/10.1016/0022-0396(82)90103-6}, DOI={10.1016/0022-0396(82)90103-6}, number={3}, journal={Journal of Differential Equations}, publisher={Elsevier BV}, author={Shearer, Michael}, year={1982}, month={Dec}, pages={426–443} } @article{shearer_1981, title={Coincident bifurcation of equilibrium and periodic solutions of evolution equations}, volume={84}, ISSN={0022-247X}, url={http://dx.doi.org/10.1016/0022-247x(81)90154-2}, DOI={10.1016/0022-247x(81)90154-2}, abstractNote={Abstract : Bifurcation of equilibrium and periodic solutions of nonlinear evolution equations is considered in the neighbourhood of an equilibrium solution for which the corresponding linear problem admits both non-zero equilibrium and non-constant periodic solutions. These solutions of the linear problem are related to those of the nonlinear equation by deriving bifurcation equations possessing a simple symmetry property. This results in a simplification of the bifurcation analysis, illustrated by a discussion of two important special cases exhibiting secondary bifurcation of periodic solutions. (Author)}, number={1}, journal={Journal of Mathematical Analysis and Applications}, publisher={Elsevier BV}, author={Shearer, Michael}, year={1981}, month={Nov}, pages={113–132} } @article{shearer_walton_1981, title={On Bifurcation and Symmetry in Benard Convection and Taylor Vortices}, volume={65}, ISSN={0022-2526}, url={http://dx.doi.org/10.1002/sapm198165185}, DOI={10.1002/sapm198165185}, abstractNote={Symmetries of the nonlinear boundary value problems governing Bénard convection and Taylor vortices are described. Their effect on the corresponding bifurcation equation is deduced from the general analysis of Shearer (1978).}, number={1}, journal={Studies in Applied Mathematics}, publisher={Wiley}, author={Shearer, M. and Walton, I. C.}, year={1981}, month={Aug}, pages={85–93} } @article{shearer_1980, title={Bifurcation of axisymmetric buckled states of a thin spherical shell}, volume={4}, ISSN={0362-546X}, url={http://dx.doi.org/10.1016/0362-546x(80)90070-x}, DOI={10.1016/0362-546x(80)90070-x}, number={4}, journal={Nonlinear Analysis: Theory, Methods & Applications}, publisher={Elsevier BV}, author={Shearer, M.}, year={1980}, month={Jan}, pages={699–713} } @article{shearer_1980, title={One-parameter perturbations of bifurcation from a simple eigenvalue}, volume={88}, ISSN={0305-0041 1469-8064}, url={http://dx.doi.org/10.1017/s0305004100057388}, DOI={10.1017/s0305004100057388}, abstractNote={Abstract}, number={1}, journal={Mathematical Proceedings of the Cambridge Philosophical Society}, publisher={Cambridge University Press (CUP)}, author={Shearer, M.}, year={1980}, month={Jul}, pages={111–123} } @article{shearer_1980, title={Secondary Bifurcation Near a Double Eigenvalue}, volume={11}, ISSN={0036-1410 1095-7154}, url={http://dx.doi.org/10.1137/0511034}, DOI={10.1137/0511034}, abstractNote={General conditions are formulated under which secondary bifurcation is rigorously established for a family of bifurcation problems depending continuously on a real auxiliary parameter. With more specific conditions, it is shown that, although the presence of secondary bifurcation renders the problem a priori degenerate, a full local bifurcation analysis is still possible.The results of this paper demonstrate the prime importance of symmetry (or more generally, invariance) to the mechanism by which secondary bifurcation points are created as the auxiliary parameter is varied.}, number={2}, journal={SIAM Journal on Mathematical Analysis}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Shearer, M.}, year={1980}, month={Mar}, pages={365–389} } @article{shearer_1978, title={Bifurcation in the neighbourhood of a non-isolated singular point}, volume={30}, ISSN={0021-2172 1565-8511}, url={http://dx.doi.org/10.1007/bf02762000}, DOI={10.1007/bf02762000}, number={4}, journal={Israel Journal of Mathematics}, publisher={Springer Science and Business Media LLC}, author={Shearer, M.}, year={1978}, month={Dec}, pages={363–381} } @article{shearer_1978, title={On the null spaces of linear Fredholm operators depending on several parameters}, volume={84}, ISSN={0305-0041 1469-8064}, url={http://dx.doi.org/10.1017/s0305004100054979}, DOI={10.1017/s0305004100054979}, abstractNote={Abstract}, number={1}, journal={Mathematical Proceedings of the Cambridge Philosophical Society}, publisher={Cambridge University Press (CUP)}, author={Shearer, M.}, year={1978}, month={Jul}, pages={131–142} } @book{shearer_1978, place={Colchester, England}, title={Secondary bifurcation for one-parameter families of bifurcation problems}, number={97}, institution={University of Essex}, author={Shearer, Michael}, year={1978} } @article{shearer_1977, title={Small solutions of a non-linear equation in Banach space for a degenerate case}, volume={79}, ISSN={0308-2105 1473-7124}, url={http://dx.doi.org/10.1017/s0308210500016802}, DOI={10.1017/s0308210500016802}, abstractNote={Synopsis}, number={1-2}, journal={Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, publisher={Cambridge University Press (CUP)}, author={Shearer, M.}, year={1977}, pages={35–49} } @inbook{shearer_1976, title={Bifurcation from a multiple eigenvalue}, ISBN={9783540080589 9783540375173}, ISSN={0075-8434 1617-9692}, url={http://dx.doi.org/10.1007/bfb0087360}, DOI={10.1007/bfb0087360}, booktitle={Ordinary and Partial Differential Equations}, publisher={Springer Berlin Heidelberg}, author={Shearer, M.}, year={1976}, pages={417–424} }