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The Derivation and Application of Fundamental Solutions for Unsteady Stokes Equations

Published online by Cambridge University Press:  18 September 2015

C-H. Hsiao  and
D.-L. Young
C-H. Hsiao
Affiliation:
Department of Civil Engineering and Hydrotech Research Institute, National Taiwan University, Taipei, Taiwan
D.-L. Young*
Affiliation:
Department of Civil Engineering and Hydrotech Research Institute, National Taiwan University, Taipei, Taiwan
*
* Corresponding author ([email protected])
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Abstract

In this paper, two formulations in explicit form to derive the fundamental solutions for two and three dimensional unsteady unbounded Stokes flows due to a mass source and a point force are presented, based on the vector calculus method and also the Hörmander’s method. The mathematical derivation process for the fundamental solutions is detailed. The steady fundamental solutions of Stokes equations can be obtained from the unsteady fundamental solutions by the integral process. As an application, we adopt fundamental solutions: an unsteady Stokeslet and an unsteady potential dipole to validate a simple case that a sphere translates in Stokes or low-Reynolds-number flow by using the singularity method instead by the traditional method which in general limits to the assumption of oscillating flow. It is concluded that this study is able to extend the unsteady Stokes flow theory to more general transient motions by making use of the fundamental solutions of the linearly unsteady Stokes equations.

Keywords

Fundamental solutionsUnsteady Stokes flowsGreen’s functionTranslating sphere
Type
Research Article
Information
Journal of Mechanics , Volume 31 , Issue 6 , December 2015 , pp. 683 - 691
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2015 

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