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Calculation of Hydrodynamic Forces for Unsteady Stokes Flows by Singularity Integral Equations Based on Fundamental Solutions

Published online by Cambridge University Press:  08 August 2013

C. H. Hsiao  and
D. L. Young
C. H. Hsiao
Affiliation:
Department of Civil Engineering and Hydrotech Research Institute, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
D. L. Young*
Affiliation:
Department of Civil Engineering and Hydrotech Research Institute, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
*
*dlyoung@ntu.edu.tw
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Abstract

The attractive feature of the singularity method for steady Stokes flows is that the hydrodynamic forces acting on the particle can be calculated by the total strength of distributed singularities. For unsteady Stokes flows, however we have to derive hydrodynamic forces acting on a solid body in terms of the strengths of both unsteady Stokeslets as well as unsteady potential dipoles if mass and force sources are both taken into consideration. Since the hydrodynamic force formulation results in a Volterra integral equation of the first kind, and the strengths are numerically approximated by means of the Lubich convolution quadrature method (CQM) in this study. As far as the numerical solutions of time-domain integral formulations of the unsteady Stokes equations are concerned, this paper requires only the Laplace-domain instead of the time- domain fundamental solutions of the governing equations. The stability and accuracy of the proposed method are verified through some well selected numerical examples. In total we include two examples presenting the accuracy of Lubich CQM, and another two examples for calculating general hydrody-namic forces of a sphere in oscillating or non-oscillating unsteady Stokes flows. It is concluded that this study is able to extend the unsteady Stokes flow theory to more general transient motions instead to limit to the oscillating flow assumption.

Keywords

Fundamental solutionsStokesletsDipoleVolterra integral equationConvolution quadrature methodSingularity methods
Type
Research Article
Information
Journal of Mechanics , Volume 30 , Issue 2 , April 2014 , pp. 129 - 136
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2013 

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References

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