Stability of Traveling Waves in Thin Liquid Films Driven by Gravity and Surfactant

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.