Flow of suspension of solid particles in a channel with porous walls

A boundary value problem for a nonstationary two-dimensional suspension flow is formulated based on the basis of the momentum and mass balance equations using the Darcy law and the equation for the settling rate of heavy particles. The proposed formulation takes into account the effect of sedimentation and leakage of the dispersion medium through porous walls. The solution of the problem is developed in the framework of the proposed finite element model describing the suspension flow and the evolution of the solid particle concentration distribution in a channel with porous walls. The system of differential equations of the boundary value problem is written in the Galerkin form using the Crank-Nicolson scheme, and discretization of the computational domain is carried using triangular elements. The resulting system of algebraic equations written in band form is solved by the Gauss method. An iterative procedure is introduced to correlate the velocities of the main flow of the suspension and the seepage rate of the liquid fraction through the walls at each time step. As an example, the calculation of the process of transporting a suspension in a flat channel with an efflux of the dispersion medium through porous walls and a non-stationary inhomogeneous distribution of the solid particle concentration is presented. The calculations were carried out using the original FEM FLOW package; the results obtained were then presented in a graphical form. It is shown that in the course of time the leakage of the dispersion medium through the porous walls causes solid particles to occupy an increasing volume, the concentration of particles in the channel grows higher, the effective viscosity increases, the movement of particles slows down. The flow rate of the suspension at a given constant pressure at the inlet decreases, and after some time becomes equal to the total rate of liquid phase leakage through the porous walls. The movement of particles stops, and the time it happens corresponds to the maximum length of the channel filled with particles.