University of Leeds

Abstract

A thin liquid mass of fixed volume spreading under the action of gravity on an inclined plane develops a fingering instability at the front. In this study we consider the motion of a viscous sheet down a pre-wetted plane with a large inclination angle. Previous theory has dealt mainly with the linear stage of the instability and hence only strictly valid for infinitesimal disturbances of a truly nonlinear system. It then seems pertinent to develop a theory for the nonlinear stage. A constant thickness precursor film is assumed ahead of the macroscopic front . We demonstrate that the instability is a phase instability due to the transitional symmetry of the system in the streamwise direction. Using methods from dynamical systems theory, a Kuramoto-Sivashinsky type partial differential equation for the evolution of fingers in the weakly nonlinear regime of the instability is derived. The equation is accurate up to third order in the amplitude of the disturbances. It is shown that the apparent contact line develops a saw-tooth pattern-before the equation blows-up in finite-time-qualitatively similar to that observed experimentally for completely wetting fluids on a dry surface. The developed fingers are characterized by sharp minima and flat maxima. Interestingly, the parallel-sided fingers for partially wetting fluids on dry surfaces were never observed. This implies that the precursor film model cannot be used to model spreading of partially wetting fluids on dry surfaces. Moreover, the presence of a thick film decreases the number of fingers and eventually suppresses fingering. In contrast, for spreading on a dry plane, existing experimental data indicate that the more wetting the fluid the larger the number of developed fingers. Hence, spreading on a pre-wetted surface seems to be different from spreading on dry surfaces. Our theoretical findings are in good agreement with a numerical solution of the two-dimensional free-surface evolution equation as an initial value problem.