Gravity-driven dielectric thin film flow over corrugated substrate under a normal electric field: An inverse problem approach

Gravity-driven flow of a dielectric viscous liquid film down an inclined corrugated substrate subjected to a normal electric field generated by two capacitor plates is investigated. The direct problem of determining the free surface for a prescribed bottom topography has been well studied; here, attention is directed to the corresponding inverse problem: given a prescribed single frequency periodic free-surface profile, the bottom topography that sustains it in the presence of an electric field is sought. The weighted-residual integral boundary-layer method is employed as the theoretical framework, yielding two coupled evolution equations for the film thickness and the flow rate that incorporates the electrostatic contribution. The influence of the electric field strength, inertia, surface tension, and inclination angle on the reconstructed bottom topography is examined. The proposed analytical and numerical approaches are validated against benchmark results, demonstrating excellent agreement. For weakly undulated free surfaces, the bottom topography is recovered analytically via a first-order perturbation expansion, revealing that the electric field modifies the amplitude and phase of the fundamental mode through the parameters k1 and k2, and introduces a second harmonic through k3 that is absent in the non-electrified case. These findings establish the electric field as an effective control mechanism for substrate design in coating and film flow applications. Moreover, a linear stability analysis of the corresponding direct problem demonstrates the stabilizing effect of the applied electric field.