Thin liquid film formation on hemispherical and conical substrate

The deposition and coating of thin films onto curved rigid substrate, involving displacement of air by a liquid, has numerous applications within the technology sectors but faces two major challenges: (i) control of the local film thickness; (ii) ensuring that the coating remains stable. The work reported here investigates the full coverage of three‐dimensional curved geometries, of hemispherical and conical shape, by a continuously fed, gravity‐driven, thin liquid layer. The modelling approach adopted utilises a first integral formulation [1,2] of the Navier‐Stokes equations leading to a variational formulation in the case of steady flow and an advantageous re‐formulation of the dynamic boundary condition at the free surface [3]. Asymptotic analysis, underpinned by the long‐wave approximation, enables analytic solutions for the local film thickness to be obtained.