Bilayer Couette flow over a profiled plate

Abstract

The shear–driven, bilayer flow of two immiscible liquids sandwiched between a moving planar and a stationary corrugated plate, in parallel alignment and under Stokes flow conditions, is explored. The motivation, partly inspired by biomimetic considerations, is the use of surface texturing/topography to reduce friction and wear while increasing load carrying capacity in lubrication problems. The starting point of the associated mathematical formulation is a variational principle which, in addition to the classical fluid mechanical field quantities, includes potential fields enabling an elegant reformulation of the dynamic condition at the interface separating the two layers. Two methods of solution are utilised: (i) an analytic one of the field equations, boundary and interface conditions by invoking the long–wave approximation; (ii) an exact semi–analytic one, obtained after transformation to complex variables and application of a spectral method. For validation and comparison purposes a FE formulation of the classical continuity and Navier–Stokes equations is solved for. Resulting streamline patterns, interface shapes and wall shear stresses are discussed.