Sergey Saprykin

Free-surface thin-film flows over topography : influence of inertia and viscoelasticity

We consider viscoelastic flows over topography in the presence of inertia. Such flows are modelled by an integral-boundary-layer approximation of the equations of motion and wall/free-surface boundary conditions. Steady states for flows over a step-down in topography are characterized by a capillary ridge immediately before the entrance to the step. A similar capillary ridge has also been observed for non-inertial Newtonian flows over topography. The height of the ridge is found to be a monotonically decreasing function of the Deborah number. Further, we examine the interaction between capillary ridges and excited non-equilibrium inertia/viscoelasticity-driven solitary pulses. We demonstrate that ridges have a profound influence on the drainage dynamics of such pulses: they accelerate the drainage process so that once the pulses pass the topographical feature they become equilibrium ones and are no longer excited.

Two-dimensional wave dynamics in thin films. I. Stationary solitary pulses

We consider two-dimensional stationary solitary pulses in a falling film by using the two-dimensional generalized Kuramoto-Sivashinsky equation as a model system. We numerically construct solitary wave solutions of this equation as a function of the dispersion parameter. We obtain an analytical estimate for the speed of these waves in the strongly dispersive case by using a perturbation from the Korteweg-de Vries limit. An impulse response analysis in which the nonlinearity is replaced with a delta function leads to an approximate analytical solution for the shape of two-dimensional solitary waves. The analytical predictions are in excellent agreement with numerical results for the speed and shape of these waves.

Two-dimensional wave dynamics in thin films. II. Formation of lattices of interacting stationary solitary pulses

We develop an inelastic coherent structure theory that describes the weak interaction of the two-dimensional (2D) solitary pulses constructed in Part I. We focus on the interaction between two equilibrium pulses. We project the interaction dynamics on to the zero modes associated with the translational invariances of the system in the streamwise and spanwise directions and we obtain a dynamical system for the locations of the two solitons. We show that in the strongly dispersive case, 2D pulses self-organize into V shapes. Our theoretical findings are in excellent agreement with our time-dependent computations of the fully nonlinear system.

Coherent structures theory for the generalized Kuramoto-Sivashinsky equation

We examine coherent structures interaction and formation of bound states in active–dispersive–dissipative nonlinear media. A prototype for such media is a simple weakly nonlinear model, the generalized Kuramoto-Sivashinsky (gKS) equation, that retains the fundamental mechanisms of any nonlinear process involving wave evolution, namely, a dominant nonlinearity, instability, stability and dispersion. We develop a weak interaction theory for the solitary pulses of the gKS equation by representing the solution as a superposition of the pulses and an overlap function. We derive a linearized equation for the overlap function in the vicinity of each pulse and project the dynamics of this function onto the discrete part of the spectrum of the linearized interaction operator. This leads to a coupled system of ordinary differential equations describing the evolution of the locations of the pulses. By analyzing this system, we prove a criterion for the existence of a countable infinite or finite number of bound states, depending on the strength of the dispersive term in the equation. The theoretical findings are corroborated by computations of the full equation.