Stokes flow between sinusoidal walls

In this paper, we study two-dimensional Stokes flow between sinusoidal walls. A stream function is introduced, thus transforming the Stokes equation into a biharmonic one, whose solution is then derived for a single periodic cell of length equal to the wall fluctuation wavelength, and for a given pressure drop. Relevant boundary conditions are the no-slip and no-flow conditions on the boundary, as well as those deriving from the periodicity and an auxiliary condition based on an energy argument. For such a mathematical problem, an approximate solution is possible via a series expansion in terms of a small parameter equal to the ratio between the mean channel width and the wavelength. We present closed-form second-order expressions for stream function, flow rate, and velocity components, and discuss the implications of the zero-order solution (lubrication approximation) for different values of two dimensionless parameters. Expressions derived for the velocity components show flow reversal for strong channel sinuosity; they will be useful for several purposes, such as study of solute transport in rough-walled fractures or of heat and mass transfer in conduits with wavy walls.