Flow domain identification from free surface velocity in thin inertial films

We consider the flow of a viscous liquid along an unknown topography. A new strategy is presented to reconstruct the topography and the free surface shape from one component of the free surface velocity only. In contrast to the classical approach in inverse problems based on optimization theory we derive an ordinary differential equation which can be solved directly to obtain the inverse solution. This is achieved by averaging the Navier-Stokes equation and coupling the function parameterizing the flow domain with the free surface velocity. Even though we consider nonlinear systems including inertia and surface tension, the inverse problem can be solved analytically with a Fourier series approach. We test our method on a variety of benchmark problems and show that the analytical solution can be applied to reconstruct the flow domain from noisy input data. It is also demonstrated that the asymptotic approach agrees very well with numerical results of the Navier-Stokes equation. The results are finally confirmed with an experimental study where we measure the free surface velocity for the film flow over a trench and compare the reconstructed topography with the measured one.