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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

REFERENCES

1.Chwang, A. T. and Wu, T. Y., “Hydrodynamics of low-Reynolds-number Flow. Part 2. Singularity method for Stokes Flows,” Journal of Fluid Mechanics, 67, pp. 787–815 (1975).CrossRefGoogle Scholar
2.Dabros, T., “A Singularity Method for Calculating Hydrodynamic Forces and Particle Velocities in Low-Reynolds-Number Flows,” Journal of Fluid Mechanics, 156, pp. 1–21 (1985).Google Scholar
3.Zhou, H. and Pozrikidis, C., “An Adaptive Singularity Method for Stokes Flow Past Particles,” Journal of Computational Physics, 117, pp. 79–89 (1995).Google Scholar
4.Young, D. L., Jane, S. J., Fan, C. M., Murugesan, K. and Tsai, C. C., “The Method of Fundamental Solutions for 2D and 3D Stokes Problems,” Journal of Computational Physics, 211, pp. 1–8 (2006).CrossRefGoogle Scholar
5.Feng, J. and Joseph, D. D., “The Unsteady Motion of Solid Bodies in Creeping Flows,” Journal of Fluid Mechanics, 303, pp. 83–102 (1995).Google Scholar
6.Brunner, H., Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, New York (2004).Google Scholar
7.Muhammad, M., Nurmuhammad, A., Mori, M. and Sugihara, M., “Numerical Solution of Integral Equations by Means of the Sinc Collocation Method Based on the Double Exponential Transformation,” Journal of Computational and Applied Mathematics, 177, pp. 269–286 (2005).Google Scholar
8.Lubich, C. and Schadle, A., “Fast Convolution for Non- Reflecting Boundary Conditions,” SIAM Journal on Scientific Computing, 24, pp. 161–182 (2002)Google Scholar
9.Bedivan, D. M. and Fix, G. J., “Analysis of Finite Element Approximation and Quadrature of Volterra Integral Equations,” Numerical Methods in Partial Differential Equations, 13, pp. 663–672 (1997).Google Scholar
10.Pozrikidis, C., “A Singularity Method for Unsteady Linearized Flow,” Physics of Fluids A1, pp. 1508–1520 (1989).Google Scholar
11.Banerjee, P. K., The Boundary Element Methods in Engineering, McGraw-Hill, New York (1994).Google Scholar
12.Tsai, C. C., Young, D. L., Fan, C. M. and Chen, C. W., “Time-Dependent Fundamental Solutions for Unsteady Stokes Problems,” Engineering Analysis with Boundary Elements, 30, pp. 897–908 (2006).Google Scholar
13.Stokes, G. G., “On the Effect of the Internal Friction of Fluids on the Motion of Pendulums,” Transactions of the Cambridge Philosophical Society, 9, pp. 8–106 (1851).Google Scholar
14.Basset, A. B., A Treatise on Hydrodynamics, 2, Cambridge: Deighton Bell, Ch. 21–22 (1888).Google Scholar