Stream function - vorticity formulation of mixture mass flow

Abstract

Employing a generalized quasi two-phase bulk mixture mass flow model derived from a general two-phase model (Pudasaini, 2012), here, we formulate a stream function - vorticity and vorticity-transport equation for a rapid flow of mixture of viscous fluid and solid particles down a channel. The original system of partial differential equations (PDEs) in velocity and pressure is converted into the reduced stream function - vorticity form as a close system of equations that is free of pressure which replaces the original system of three equations by a set of just two PDEs as an advantage of the new model. A novel pressure Poisson equation in terms of stream function, vorticity and the rate-dependent mixture viscosity is derived. For given mixture viscosity, our pressure Poisson equation can provide mixture pressure. The two equations are coupled through the Poisson equation to provide the full system in stream function, vorticity and mixture pressure to describe the dynamics of mixture flow. However, the pressure is decoupled and can also be computed separately using stream function and vorticity. We further reduce the new system to obtain exact expressions for stream function. One of the most important advancements here is the construction of a new pressure Poisson equation for shear mixture flows that includes yield strength of the mixture. We also discuss the importance of pressure Poisson equation induced by the flow field intensity, the yield strength, and free surface geometry. Our results show that the pressure Poisson equation is mainly characterized by the non-linear diffusion of the free surface. Furthermore, mixture pressures are derived analytically for thin and thick flows. Similarly, different flow scenarios such as pressure dominated flow; thick, low yield strength flow; and thin, high yield strength flows are analyzed. Several exact/analytical solutions are constructed for the pressure and flow depth distributions for incipient flow, shearing flow, free surface flow, propagating bore front and mass deposition. The novel models developed here and analytical results are consistent with the observed phenomena indicating their application potential in detailed description of the mixture flow dynamics more efficiently than the existing complex models.