Evidence of an inertialess Kapitza instability due to viscosity stratification

Abstract

The classical Kapitza instability of a gravity-driven falling film requires finite inertia to operate. We show that a surface-mode instability can arise in the complete absence of inertia when the film possesses a continuous viscosity stratification, a feature relevant to particle-laden films with shear-induced migration, thermally stratified coatings, and concentration-graded flows. The viscosity field, prescribed as a linear profile across the film thickness, evolves through an advection-diffusion equation characterized by a P$\'{e}$clet number. Using long-wave asymptotics and Chebyshev spectral computations, we solve the coupled eigenvalue problem for the perturbation streamfunction and viscosity fields and demonstrate that viscosity stratification destabilizes the surface mode in the zero-inertia (Stokes) limit. The instability is confined to a finite window of P$\'{e}$clet numbers. Increasing the stratification parameter lowers the critical P$\'{e}$clet number, broadens the range of unstable wavenumbers, and increases the growth rate. The instability mechanism is traced to the phase relationship between perturbation vorticity and the interface displacement: viscosity stratification shifts the vorticity to a lagging configuration, which reinforces interface deformation, following the framework of Hinch (1984). The mechanism bears a structural resemblance to the surfactant-driven Marangoni instability in creeping two-layer flows, extending this class of scalar-mediated, inertialess instabilities to bulk viscosity stratification.