Modeling, simulation, and nonlinear analysis for film flow over inclined wavy bottoms

The gravity-driven free surface flow of a viscous liquid down an inclined plane has various engineering applications, for instance in cooling and coating processes. However, in many applications the bottom is not flat but has a wavy profile. This may be due to natural irregularities or by design, e.g., to increase the contact area in heat conductors. Thus, studying the stability of stationary solutions over wavy inclines is of great interest. If perturbations of the free surface decay to zero, we call the stationary solution stable, otherwise unstable. From linear analysis it is well known that stability is mainly determined by the dimensionless Reynolds number R, which is a measure for the ratio of inertial forces to viscous forces. More precisely, there exists a critical Reynolds number Rcrit depending on the bottom waviness and the inclination angle such that the free surface becomes unstable for R > Rcrit. In this thesis, we first derive model equations for the evolution of the film thickness F and the local flow rate Q. In case of a thin film over a weakly undulated bottom, we can introduce a small perturbation parameter which allows to solve the underlying NavierStokes equations by an asymptotic expansion approach. Using this solution as ansatz and test function in a Galerkin method for the downstream momentum equation, we obtain a system of parabolic partial differential equations for F and Q. According to the used methods, this is called weighted residual integral boundary layer (WRIBL) equation. Comparing numerical simulations of the WRIBL equation with available experimental data and full Navier-Stokes numerics, we can justify its validity for a large range of parameters. Since we used a second-order velocity profile in the Galerkin method, we can even simulate parameter regimes for which eddies occur in the troughs of the bottom. Moreover, by reducing the inclination angle we find a new phenomenon, namely a short wave instability for laminar flows, which does not exist over flat bottoms. Finally, we prove nonlinear stability of stationary solutions in the spectrally stable situation, which corresponds to Reynolds numbers smaller than Rcrit. To be more precise, we show that small perturbations decay in a universal manner determined by the Burgers equation. Since the WRIBL equation has a whole family of stationary solutions, the corresponding linear differential operator always has essential spectrum up to zero. Thus, stability cannot be shown by considering the linear system alone. Instead, we have to take into account the full nonlinearity, where we encounter the following difficulties. In contrast to a flat bottom, where the linearized WRIBL equation can be analyzed by Fourier transform, here we have to use Bloch analysis to generalize the spectral theory from spatially homogeneous stationary solutions to spatially periodic ones. Furthermore, since the WRIBL equation is quasilinear, we cannot show local existence and uniqueness of solutions by applying the variation-of-constants formula but rather have to use the method of maximal regularity. The asymptotic decay behavior of perturbations follows then by a renormalization process.