Inverse Temperature Reconstruction in Thermocapillary-driven Thin Liquid Films

A thin liquid film subject to a temperature gradient undergoes thermocapillary convection because of the non-uniform surface tension at the free surface. This induced flow perturbs the film free surface and generate a free surface velocity field. These observable consequences can be thought of as the “signature” of the imposed temperature field and this work investigates whether the temperature field can be reconstructed from this signature for general three-dimensional flows. Using a model based on the lubrication approximation, we show that one can explicitly formulate the partial differential equation which governs this inverse problem. This equation is solved using finite differences. We illustrates the feasibility of this reconstruction exercise on a set of “artificial” experimental data obtained by first solving the direct problem which consists in computing the free surface deformation and free surface velocity field for a given applied temperature field.