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Received for review July 19, 1991 Revised manuscript received December 16, 1991 Accepted January 3,1992
Torque and Frictional Force Acting on a Slowly Rotating Sphere Arbitrarily Positioned in a Circular Cylinder GuangHong Zheng, Robert L. Powell, and Pieter Stroeve* Department of Chemical Engineering, University of California, Davis, California 95616
The method of reflections is used to derive expressions for the frictional force and torque experienced by a rotating sphere settling at an arbitrary position inside a cylinder filled with a viscous fluid. In limiting cases, the analytical results are shown to give the correct numerical predictions for the torque and frictional force. The expressions for the frictional force are in quantitative agreement with previous theoretical results. Other expressions published previously for the torque and frictional force are shown to be in error. Introduction Many theoretical studies have considered the effect of rotation of a settling sphere in an incompressible, viscous fluid using the Stokes approximation. Lamb (1945) gave the solution for the torque necessary to maintain the steady rotational motion of a single sphere in an unbounded, viscous fluid at small angular Reynolds numbers. Several authors have studied the rotation of a sphere near a planar wall. Dean and O'Neil(l963) examined the rotation of a sphere about an axis parallel to a plane. Goldman et al. (1967) gave solutions for the same problem using the method of reflections and lubrication theory. Malysa and Van de Ven (1986) compared experimental data with available theories. Brenner (1964) studied the slow rotation of a sphere about the axis of circular cylinders of finite and infinite lengths. Brenner and Sonshine (1964) supplemented this study by examining a sphere rotating slowly about the axis of an infinitely long cylinder. Haberman (1961) investigated the rotation of a sphere within a coaxially rotating cylinder. Tozeren (1982,1983) gave the perturbation solutions for the off-axis rotation of a sphere in a circular cylinder. On the basis of the expressions obtained by Greenatein (1967), Greenstein and Schiavina (1975) and Greenstein and Som (1976) investigated the general case where the sphere may occupy any position inside a cylinder and may rotate slowly with a constant angular velocity 0888-5885I 9 2 12631-1190S03.00,IO I
,
~~
about any axis. Hirschfeld et al. (1984) expanded the results of Hirschfeld (1972) for the frictional force acting on a sphere placed eccentrically inside a cylinder. Their results showed a significant difference from the earlier work of Greenstein and Som (1976). In this paper we revisit the work of Greenstein and Schiavina (1975) and Greenstein and Som (1976) to attempt to reconcile their results with those of Hirschfeld et al. (1984). We use the method of reflections and compare our results with available predictions (Greenstein and Schiavina, 1975; Greenstein and Som, 1976; Hirschfeld et al., 1984; Goldman et al., 1967; Tozeren, 1983).
Formulation of the Problem Consider a sphere, with radius a, rotating slowing with an arbitrary constant angular velocity 0 = in, + j02+ kO, through a viscous incompressible fluid. The fluid is bounded by an infinitely long circular cylinder and is at rest at infinity. The distance of sphere center to the axis of the cylinder is b, and the cylinder has a radius R,,. The coordinate systems utilized in our analysis are shown in Figure 1. The angular sphere Reynolds number, 4u21Ql/~, is assumed sufficiently small to neglect the inertial terms in the Navier-Stokes equations which become
v2v = ( l / p ) V P v.v = 0 0 1992 American Chemical Society
(1) (2)
Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992 1191 with the following boundary conditions: V=QXr atr=a
(34
V=O atR=Ro (3b) V=O atZ=fm (3c) where V is the fluid velocity with respect to a coordinate system having its origin at the sphere center and r is measured from the sphere center. As in the previous studies (Greenstein, 1967; Greenstein and Schiavina, 1975; Greenstein and Som, 1976), this boundary value problem can be solved by the method of reflections when a
