IJPAM: Volume 75, No. 4 (2012)

ON THE PAPKOVICH-NEUBER FORMULATION FOR STOKES
FLOWS DRIVEN BY A TRANSLATING/ROTATING
PROLATE SPHEROID AT ARBITRARY ANGLES

Dali Kong$^1$, Zhen Cui$^2$, Yongxin Pan$^3$, Keke Zhang$^4$
$^{1,2,4}$Department of Mathematical Sciences
University of Exeter
Exeter, UK
$^3$Institute of Geology and Geophysics
Chinese Academy of Sciences
Beijing, P.R. CHINA


Abstract. We investigate, via the Papkovich-Neuber formulation using prolate spheroidal coordinates, a fully three-dimensional Stokes flow in the exterior of a prolate spheroid driven by its translation or rotation. The Stokes flow is primarily characterized by four parameters: the eccentricity $\e$ of the spheroid, the angle of attack $\gamma$ in the case of translation and two rotating angles $\alpha$ and $\beta$ in the case of rotation. Our mathematical analysis comprises the three parts:


(i) derive an analytical three-dimensional solution for the Stokes flow driven by a translating spheroid at an arbitrary angle $\gamma$;


(ii) derive an analytical three-dimensional solution for the Stokes flow driven by a rotating spheroid with arbitrary angles $\alpha$ and $\beta$; and


(iii) derive two analytical formulas for the corresponding drag and torque as a function of $\e$, $\alpha$, $\beta$ and $\gamma$.

Received: October 31, 2011

AMS Subject Classification: 76D07, 35Q35

Key Words and Phrases: Stokes flow, prolate spheroid, and spheroidal harmonics

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 75
Issue: 4