The long-wave (lubrication) approximation governing the evolution of a thin film over a uniformly heated topographical substrate is solved numerically. We study the initial-value problem for a variety of governing dimensionless parameters and topographical substrates. We demonstrate that the dynamics is characterized by a slow relaxation process with continuous coarsening of drops up to a large time where coarsening is terminated and the interface organizes into a series of drops each of which is located in a trough in topography.

We analyze coherent-structures interaction and formation of bound states in active dispersivedissipative nonlinear media using a viscous film coating a vertical fiber as a prototype. The coherent structures in this case are drop-like pulses that dominate the evolution of the film. We study experimentally the interaction dynamics and show evidence for formation of bound states. A theoretical explanation is provided through a coherent structures theory of a simple model for the flow.

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.

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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.