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A numerical investigation of non-rotational, rigid spherical particle sedimentation problem in viscous fluid

This paper can be considered as continuous part of [1], where the generalized diffusion theory of rigid spherical particle sedimentation in viscous fluid was investigated. Here a numerical solution of non-stationary sedimentation process is obtained by using the explicit finite difference method. The obtained results show that this model can be used for qualitative study of physical phenomenon of sedimentation problem. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. 24, 2002, No 1 ( 46 - 50) A NUMERICAL INVESTIGATION OF NON-ROTATIONAL, RIGID SPHERICAL PARTICLE SEDIMENTATION PROBLEM IN VISCOUS FLUID NGUYEN HONG PHAN, NGUYEN VAN DIEP Institute of Mechanics, NCST, 264 Doican, Hanoi, Vietnam ABSTRACT. This paper can be considered as continuous part of [1], where the generalized diffusion theory of rigid spherical particle sedimentation in viscous fluid was investigated. Here a numerical solution of non-stationary sedimentation process is obtained by using the explicit finite difference method. The obtained results show that this model can be used for qualitative study of physical phenomenon of sedimentation problem. 1. The Governing Equation System Let 's consider a sedimentation process ofrigid, spherical, non-rotational particles in a viscous suspension filled up the space between two horizontal parallel planes separated by the vert ical distance L (Fig. 1). It is assumed that the suspension is in a initial stationary state and being forced only by the gravity. In this case the movement of particles under the gravity force is considered only in the vertical z direction, and there is not a mean volume suspension velocity. Therefore r.p, J, pare functions of variables (t , z): r.p = r.p(t, z); J = (0, 0, J); J = J(t , z); P = P(t , z) (1.1) On basis of above mentioned assumptions , the equation system has the following form : fJr.p + ~ f)J = 0 fJt Pi fJz fJJ r.p(l - r.p) fJt +Ki 1 + r.p(kp - 1) J (1.2) + r.p(l - r.p) + K 2 1 + r.p(kp - 1) Dr.p fJr.p 1 + r.p(kp - 1) fJz = 0 And after determination of r.p and J we can define the pressure p by: fJP fJz + pig [kp fJJ - r.p(kp - 1)] - (kp - l)Bt = O (1.3) In addition, in (1.2) and (1.3) it was supposed that: 8 D = ( µ1) 8 r.p p,T _ P2 p1 = const; K 1 = Tpi = const; K2 = (Pi - P2)g = const; k p PI O'.n (1.4) 46 where: . P1, P?, P_ - densities of particle, fluid phase and suspension; J - .