Rotatory oscillations of arbitrary axi-symmetric bodies in an axi-symmetric viscous flow: Numerical solutions

Abstract

The problem of the rotatory oscillation of an axi-symmetric body in an axi-symmetric, viscous, incompressible flow at low Reynolds number has been studied. In contrast to the steady rotation of a body, which involves solving the Laplace equation, the study of an oscillating body requires solution of the Helmholtz equation which results from the simplification of the unsteady Stokes equations. In the present work, we have numerically evaluated the local stresses and torques on a selection of free, oscillating, axi-symmetric bodies in the continuum regime in an axi-symmetric viscous incompressible flow. The Helmholtz equation was solved by a Green’s function technique. The accuracy of the technique is tested against known solutions for a sphere, a prolate spheroid, a thin disk, and an infinitely long cylinder. Good agreements have been obtained. Finite cylinders have been studied and the edge correction factors for the circular disk geometry, that are basic to oscillating disk viscometers, have been calculated. It has been found that the calculated edge correction factors, based on the ratios of the real parts of the actual torques (calculated from this work) to the ideal torques, agree to within 1% to 10% with the reported values obtained by Clark et al. [Physica A 89, 539 (1977)] using the theory of Kestin and Wang [J. Appl. Mech. 24, 197 (1957)]. However, since the ratios of the real parts and the ratios of the imaginary parts of the torques do not coincide, the edge correction factors depend upon which ratio is used.

Slip at the surface of an oscillating spheroidal particle in a micropolar fluid

“…The accuracy of their technique is tested against known solutions for a sphere, a prolate spheroid, a thin disk and an infinitely long cylinder. Tekasakul and Loyalka [29] extended the work of Tekasakul et al [28] into the slip regime. An accurate numerical result for local stress and torque on spheres and spheroids as function of the frequency parameter and the slip coefficients have been obtained.…”

“…This value of the couple is needed in designing and calibrating viscometries and better predictions of the couple are essential in order to improve the accuracy of viscosity measurements. Numerical solutions for rotary oscillations of arbitrary axisymmetric bodies in an axisymmetric viscous flow has been investigated by Tekasakul et al [28]. They evaluated numerically the local stresses and torques on a selection of free, oscillating, axisymmetric bodies in the continuum regime in an axisymmetric viscous incompressible flow.…”

“…Applying the boundary conditions (29)- (31) to the general solution (27) and (28), for the hydrodynamically rectilinear oscillations of a slip spheroid in a micropolar fluid and using the perturbation method, we obtain the following system of algebraic equations:…”

The drag on a microcantilever oscillating near a wall

“…Feng, Ganatos & Weinbaum (1998a) treated the mathematically equivalent problem of a disk moving in a Brinkman medium (discussed further below). Tekasakul, Tompson, & Loyalka (1998) provide an overview of oscillatory flows generated by a variety of axisymmetric bodies.…”

Torsional oscillations of a sphere in a Stokes flow

“…Similarly, for small-amplitude oscillations and negligible secondary fluid motion, the flow resulting from the decaying oscillations of a torsion pendulum was determined to be circumferential in annuli around the axis of rotation [10,11]. Tekasul et al [12] numerically solved for the torque on a torsionally oscillating sphere in an unbound medium using a Green's function approach and found a less than 0.1% difference with the analytic solution of Lamb. Kanwal [13] investigated the oscillatory rotation of rigid axi-symmetric bodies about an axis of symmetry in a viscous, incompressible fluid using Stoke's stream function.…”