tailieunhanh - Generalized diffusion theory of non - rotational, rigid spherical particle sedimentation in viscous fluid

This paper is devoted to application of Generalized Diffusion Theory for solving a sedimentation problem of rigid spherical particles in viscous fluid. The governing equations have been obtained. It is shown that, in this case the governing equation system is a hyperbolic one, and the equations in the characteristic form have been derived. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. 23, 2001, No 3 (159 - 166) GENERALIZED DIFFUSION THEORY OF NON-ROTATIONAL, RIGID SPHERICAL PARTICLE SEDIMENTATION IN VISCOUS FLUID NGUYEN HONG PHAN, NGUYEN VAN DIEP Institute of Mechanics, NCNST, 264 Doican, Hanoi, Vietnam ABSTRACT. This paper is devoted to application of Generalized Diffusion Theory for solving a sedimentation problem of rigid spherical particles in viscous fluid. The governing equations have been obtained. It is shown that, in this case the governing equation system is a hyperbolic one, and the equations in the characteristic form have been derived. The mathemetical properties of the obtained equation system and the solution for stationary sedimentation where investigated numericaly. 1. The General Motion Equation System of Two-Phase ParticulesFluid Flows In [1-3] the Generalized Diffusion Theory of Multiphase Flows of fluids with microstructural particles (microdeformation, rotation, and arbitrary shape) has been developed. In the simplest case, the movement of viscous fluid carrying non-rotational, rigid spherical particles without external forces, except the force of gravity, is described by the following general equations system: \l · U=O dcp 1- -=--\l·J dt PI ' dU - p - = pg - \J p dt D] T = + P2 ( 1 - 2µ_ -e, T T 8c 8 C1 d Dt t PI - P2 D] PI Dt {J + [a1- an (h1 - p h2)] v;. 2 PI - ( . ) = -d (. ) T P1 PT]} P2) dU - _l_(v µi) PI dt 1 - . 1' 2 = ± D P2[Kp + cp(l - Kp)] () And they are equations of characteristic curves 0 1,2 of the system () . Due to p2[Kp D + cp(l - the system () is hyperbolic. 161 >O Kp)] - . The variation of characteristic directions In figures 1 - 4 the variation of characteristic directions depending on coefficient D, density ratio KP = pif p2 and concentration ----+----; 3000 4000 D 5000 _ _ + u I. I 05 + - - - + - - - + - - - t - - - r - - - ; 0 D =1000; Kp= 0 L 0