Computation of the flow of a two-layer fluid near a rotating wall under spin-up

Abstract

Consider a disk having a circular cavity of rectangular cross-section filled with two immiscible viscous fluids and rotating about the z-axis with angular velocity ~0 (Fig. 1).Let pl and P2 be the densities and Vl and u2 the kinematic viscosities of each of the phases , with pl -< p2. During an extended uniform rotation the interface between the fluids becomes vertical f(z) = r*, and all components of the velocity vector in a cylindrical coordinate system (r, z, 0) attached to the spinning disk are zero: v~ --vo = v~ = 0. As the angular velocity of the disk ~ = ~0 + w increases the particles of fluid immediately adjacent to the horizontal walls of the cavity acquire an additional angular velocity and initiate a radial flow under the action of the increased centrifugal force. The latter leads to a deformation of the interphase boundary f ~d the appearance of centrifugal forces that stabilize its position.This class of flows has become widespread in engineering and has been studied fairly completeiy [1-3] for a homogeneous fluid (Pl --P2, Ul --u2). :At the same time no theoretical study has yet been made of the dynamics of a two-layer fluid, which is of interest in optimization of mass exchange modes in centrifuges.We have studied in detail in [4] a mathematical model of the evolution of the flow in a boundary layer of a rotating horizontal wall. For the case when the relative increment of angular velocity is small we obtained the characteristic scales of the process and tests for similarity. Their values were computed from the typical modal parameters ~0 = 100sec -1, w = 10sec -1, pl = 900kg/m 3, p2 = 100kg/m 3, r* = 10 cm, ul = u2 = 10 -s m2/sec, and are as follows:time of evolution of the flow thickness of the boundary layer radial scale radial and angular velocities vertical velocity Rossby number Ekman number Richardson number the r0 = 1/120, 10 -2 sec; z0 = ~/'~-1/~0, 0.1 ram; /~o----~r*, lcm; o v~=v ~ lm/sec; v.~ ~ lcm/sec; e =w/~o, 0.1; 6 = zo/Ro, 0.01; R = (P2 -Pl)/~P1, 1.1. It has been shown that up to terms of order 0(62 + e) in dimensionless quantities X = (r-r*)lRo, Z = ZlZo, t = r/~'o, F = (f-r*)/Ro, u = v,lv~, ~ = ~o1,~, ~ = ~=1~7 hydrodynamics of this process at the spin-up stage can be described by the boundary-layer equations Ou Ou Ou 0 / Ou x -~ + ~ + w~ = 2. + ~[kv~) + R~k.,Ov ~v Ov 0 / Ov \ (4) z=0: u=w=0, v=l,z=~: u=v=0,I~,1 = oo: ,,, = 0, OulOx = OvlOx = o,t =0 :u = w = v = 0,