The double von Mises transformation in the study of two-phase fluid flow over curved boundaries: Theory and analysis

A numerical method to handle the flow of a two-phase fluid over curved boundaries is proposed. The method is based on the double von Mises transformation which is derived in this work and is expected to be applicable to a variety of flow situations while utilizing the finite difference technique. In order to illustrate the numerical implementation of the method, dusty fluid flow through a porous channel possessing curved boundaries and the flow through a semi-infinite porous layer overlying a curved lower boundary are considered. The flow is assumed to be governed by model equations based on Brinkman's equation and reflecting boundary conditions are employed in the study based on a uniform dust particle distribution. Results indicate that an increase in the permeability results in decreasing the tangential velocity component in regions close to the curved boundary, and increasing the dust parameters decreases this component. The effects of the grid size and the extent of the computational domain are discussed. The results also shed some light on the applicability of the dusty fluid flow model and suggest that the model is best employed when the permeability is high, a conclusion that is consistent with the validity of Brinkman's equation.