@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{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{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} }