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Transverse motion of a disk through a rotating

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3. Fluid Mech. (1995), vol. 301, pp. 295-324 Copyright @ 1995 Cambridge University Press 295 Transverse motion of a disk through a rotating viscous fluid By JOHN P. TANZOSH AND H. A. STONE Division of Applied Sciences, Harvard University, Cambridge, MA 02138, USA (Received 27 June 1994 and in revised form 19 June 1995) A thin rigid disk translates edgewise perpendicular to the rotation axis of an un- bounded fluid undergoing solid-body rotation with angular velocity R. The disk face, with radius a, is perpendicular to the rotation axis. For arbitrary values of the Taylor number, F = Qa2/v, and in the limit of zero Reynolds number We, the linearized viscous equations reduce to a complex-valued set of dual integral equations. The solution of these dual equations yields an exact representation for the velocity and pressure fields generated by the translating disk. For large rotation rates S >> 1, the O(1) disturbance velocity field is confined to a thin O(S-'/2) boundary layer adjacent to the disk. Within this boundary layer, the flow field near the disk centre undergoes an Ekman spiral similar to that created by a nearly geostrophic flow adjacent to an infinite rigid plate. Additionally, flow within the boundary layer drives a weak O(S-'/*) secondary flow which extends parallel to the rotation axis and into the far field. This flow consists of two counter-rotating columnar eddies, centred over the edge of the disk, which create a net in-plane flow at an angle of 45" to the translation direction of the disk. Fluid is transported axially toward/away from the disk within the core of these eddies. The hydrodynamic force (drag and lift) varies as O(S'/2) for S >> 1; this scaling is consistent with the viscous stresses created in the Ekman boundary layer. Additionally, an approximate expression, suitable for all Taylor numbers, is given for the hydrodynamic force on a disk translating broadside along the rotation axis and edgewise transverse to the rotation axis. 1. Introduction There exist few analytical solutions describing particle motion transverse to the rotation axis of a fluid in solid-body rotation. Such motions occur in centrifugal separations and swirling flows of particle-laden fluids. The analysis of such problems is difficult owing to the three-dimensional character of the disturbance flow field created by a moving particle. To further the theoretical understanding in this problem area, we present an exact solution for the velocity and pressure fields created by the edgewise translation of a circular disk transverse to the rotation axis. Consider a thin disk of radius a translating with velocity Up through a fluid in solid-body rotation with angular velocity s2. The plane of the disk is perpendicular to the rotation axis, and the disk translates in the edgewise direction (figure 1). The fluid motion is characterized by two dimensionless parameters : the Reynolds number Be = lUpla/v and the Taylor number 9 = s2a2/v, where v denotes the kinematic viscosity of the fluid. We study the limit where the Reynolds number We << 1, so that296 J. P. Tanzosh and H. A. Stone +* Y FIGURE 1. Disk translating edgewise through an unbounded fluid undergoing solid-body rotation with angular velocity Q. The z-axis is aligned parallel to the rotation axis, and the disk translates along the x-direction. Investigation Disk motion Ray (1936) Davis (1991; 1993) Moore & Saffman (1969a,b) Hocking et al. (1979) Vedensky & Ungarish (1994) Ungarish & Vedensky (1995) Here broadside/edgewise, in-plane rotation; F = 0 broadside/edgewise translation near walls; F = 0 broadside/edgewise; rotating fluids Y >> 1 broadside; in a long container F >> 1 broadside; bounded/unbounded; all F edgewise motion; unbounded; all F TABLE 1. Research related to the edgewise translation of a disk in a Stokes (F = 0) or rotating viscous flow and formulated using dual integral equations. the convective acceleration effects may be neglected, and obtain an exact solution, valid for any Taylor number, for the linearized equations governing rotating viscous flows. To our knowledge, this is the first solution to describe transverse particle motion at arbitrary rotation rates. The mathematical procedures employed here reduce the linearized governing equa- tions to a set of dual integral equations. Table l summarizes related research in which dual integral equation formulations describe the motion of a disk either in a Stokes flow or through a fluid in solid-body rotation. In particular, Ray (1936) and Davis (1991) analysed the edgewise translation of a circular disk in an unbounded Stokes flow (F = 0); Moore & Saffman (1969a,b) considered broadside and edgewise translation of a disk through a rotating fluid in the rapid rotation limit (F >> 1) by examining the appropriate boundary layer equations ; and Vedensky & Ungarish (1994) and Ungarish & Vedensky (1995) investigated broadside motion along the rotation axis through unbounded and bounded geometries at arbitrary Taylor num- ber. Tanzosh (1994) and Tanzosh & Stone (1995) present a general approach for analysing a variety of disk motions including the Stokes flow problem, the transverse translation of a disk perpendicular to the rotation axis, the broadside motion of a disk parallel to the rotation axis, and the in-plane rotation of a disk along an axis parallel to the rotation axis. Solid-body rotation produces a number of interesting and often surprising effects on the structure of the flow field caused by transverse particle motion. For fluids in rapid rotation so that F >> 1 but = IU,(/SZa << 1, the flow field tends toward having a two-dimensional structure. The mathematical statement describing this inviscid flow limit is the Taylor-Proudman theorem, which states that the fluid velocity II satisfies s2 - Vu = 0. Thus, disturbances generated by particle motion, or perhaps by topological disturbances to an imposed flow, have long-range effects inTransverse motion of a disk 297 t” * D- F~GURE 2. Geometric length scales of a particle translating between rigid walls in a fluid undergoing solid-body rotation. directions parallel to the rotation axis. The classic example of this effect is the Taylor column in which a circumscribing column of fluid accompanies a


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