Inverse determination of body force to achieve uniform film thickness on curved surfaces

This study presents a systematic inverse approach for determining the body force required to achieve uniform thin liquid film coatings on curved substrates. The methodology applies in the lubrication limit, where the film thickness is much smaller than the substrate length scale and the Reynolds number is small. By imposing uniform thickness as a target solution of the governing thin-film equation, we analytically derive the tangential forcing that admits this uniform state. Linearisation about the uniform solution reveals the critical role of the normal forcing component in governing the small perturbation dynamics and coating stability. The framework is demonstrated on geometries with constant curvature (flat plates, cylinders, spheres) and variable curvature (spheroids). Numerical experiments on axisymmetric curved substrates validate the theory, showing that the film remains close to uniform thickness over extended time periods when the prescribed forcing is applied. Physical realisability is demonstrated through multi-axial rotation of a spherical substrate. The framework serves as a diagnostic tool for assessing whether a desired uniform coating can be achieved under a given body-force configuration, with the methodology adaptable to other forcing mechanisms including electromagnetic and thermocapillary actuation.