Generalized eigenfunctions and complete semiseparable solutions for Stokes flow in spheroidal coordinates
Authors:
G. Dassios, M. Hadjinicolaou and A. C. Payatakes
Journal:
Quart. Appl. Math. 52 (1994), 157-191
MSC:
Primary 76D07; Secondary 33C90, 35Q30
DOI:
https://doi.org/10.1090/qam/1262325
MathSciNet review:
MR1262325
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Abstract: The stream function $\psi$ for axisymmetric Stokes flow satisfies the well-known equation ${E^4}\psi = 0$. In spheroidal coordinates the equation ${E^2}\psi = 0$ admits separable solutions in the form of products of Gegenbauer functions of the first and second kind, and the general solution is then represented as a series expansion in terms of these eigenfunctions. Unfortunately, this property of separability is not preserved when one seeks solutions of the equation ${E^4}\psi = 0$. The nonseparability of ${E^4}\psi = 0$ in spheroidal coordinates has impeded considerably the development of theoretical models involving particle-fluid interactions around spheroidal objects. In the present work the complete solution for $\psi$ in spheroidal coordinates is obtained as follows. First, the generalized 0-eigenspace of the operator ${E^2}$ is investigated and a complete set of generalized eigenfunctions is given in closed form, in terms of products of Gegenbauer functions with mixed order. The general Stokes stream function is then represented as the sum of two functions: one from the 0-eigenspace and one from the generalized 0-eigenspace of the operator ${E^2}$. A rearrangement of the complete expansion, in such a way that the angular-type dependence enters through the Gegenbauer functions of successive order, leads to some kind of semiseparable solutions, which are given in terms of full series expansions. The proper solution subspace that provides velocity and vorticity fields, which are regular on the axis, is given explicitly. Finally, it is shown how these simple and generalized eigenfunctions reduce to the corresponding spherical eigenfunctions as the focal distance of the spheroidal system tends to zero, in which case the separability is regained. The usefulness of the method is demonstrated by solving the problem of the flow in a fluid cell contained between two confocal spheroidal surfaces with Kuwabara-type boundary conditions.
A. T. Chwang, Hydromechanics of Low-Reynolds-Number Flow. Part 3, Motion of Spheroidal Particle in Quadratic Flows, J. Fluid Mech. 72, 17–34 (1975)
A. T. Chwang and T. Y. Wu, Hydromechanics of Low-Reynolds-Number Flow. Part 1. Rotation of Axisymmetric Prolate Bodies, J. Fluid Mech. 63, 607–622 (1974)
A. T. Chwang and T. Y. Wu, Hydromechanics of Low-Reynolds-Number Flow. Part 2. Singularity Method for Stokes Flows, J. Fluid Mech. 67, 787–815 (1975)
A. T. Chwang and T. Y. Wu, Hydromechanics of Low-Reynolds-Number Flow. Part 4. Translation of Spheroids, J. Fluid Mech. 75, 677–689 (1976)
T. Dabros, A Singularity Method for Calculating Hydrodynamic Forces and Particle Velocities in Low-Reynolds-Number Flows, J. Fluid Mech 156, 1–21 (1985)
G. Dassios, M. Hadjinicolaou, and A. C. Payatakes, On Stokes flow through a swarm of spheroidal particles, submitted (1991)
S. Goldstein, Modern Developments in Fluid Dynamics, Clarendon Press, Oxford, 1938, pp. 114–115
J. Happel, Viscous Flow in Multiparticle Systems: Slow Motion of Fluids Relative to Beds of Spherical Particles, A. I. Ch. E. J. 4, 197–201 (1958)
J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Prentice Hall, Englewood Cliffs, NJ, 1965; and Martinus Nijholl Publishers, Dordrecht, 1986
D. J. Jeffrey and Y. Onishi, Calculation of the Resistance and Mobility Functions for Two Unequal Rigid Spheres in Low-Reynolds Number Flow, J. Fluid Mech 139, 261–290 (1985)
S. Kuwabara, The Forces Experienced by Randomly Distributed Parallel Circular Cylinders or Spheres in a Viscous Flow at Small Reynolds Numbers, J. Phys. Soc. Japan 14, 527–532 (1959)
H. Lamb, Hydrodynamics, Dover, New York, 1932, pp. 604–605
P. Moon and D. E. Spencer, Field Theory Handbook, Springer-Verlag, New York, 1961, pp. 28–29
P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vols. I, II, McGraw-Hill, New York, 1953, pp. 547–549, 600–604
A. Oberbeck, Ueber Stationäre Flüssigkeitsbewegungen mit Berücksichtigung der inneren Reibung, J. Reine Angew. Math. 81, 62–80 (1876)
L. E. Payne and W. H. Pell, The Stokes Flow Problem for a Class of Axially Symmetric Bodies, J. Fluid Mech. 7, 529–549 (1960)
E. R. Rainville, Special Functions, Chelsea, New York, 1971, p. 173
J. M. Rallison and A. Acrivos, A Numerical Study of the Deformation and Burst of a Viscous Drop in an Extensional Flow, J. Fluid Mech. 89, 191–200 (1978)
R. A. Sampson, On Stoke’s Current Function, Philos. Trans. Roy. Soc. London Ser. A 182, 449–518 (1891)
M. Stimson and G. B. Jeffery, The Motion of Two-Spheres in a Viscous Fluid, Proc. Roy. Soc. A 111, 110–116 (1926)
B. J. Yoon and S. Kim, A Boundary Collocation Method for the Motion of Two Spheroids in Stokes Flows: Hydrodynamic and Colloidal Interactions, Internat. J. Multiphase Flow 16, 639–649 (1990)
A. T. Chwang, Hydromechanics of Low-Reynolds-Number Flow. Part 3, Motion of Spheroidal Particle in Quadratic Flows, J. Fluid Mech. 72, 17–34 (1975)
A. T. Chwang and T. Y. Wu, Hydromechanics of Low-Reynolds-Number Flow. Part 1. Rotation of Axisymmetric Prolate Bodies, J. Fluid Mech. 63, 607–622 (1974)
A. T. Chwang and T. Y. Wu, Hydromechanics of Low-Reynolds-Number Flow. Part 2. Singularity Method for Stokes Flows, J. Fluid Mech. 67, 787–815 (1975)
A. T. Chwang and T. Y. Wu, Hydromechanics of Low-Reynolds-Number Flow. Part 4. Translation of Spheroids, J. Fluid Mech. 75, 677–689 (1976)
T. Dabros, A Singularity Method for Calculating Hydrodynamic Forces and Particle Velocities in Low-Reynolds-Number Flows, J. Fluid Mech 156, 1–21 (1985)
G. Dassios, M. Hadjinicolaou, and A. C. Payatakes, On Stokes flow through a swarm of spheroidal particles, submitted (1991)
S. Goldstein, Modern Developments in Fluid Dynamics, Clarendon Press, Oxford, 1938, pp. 114–115
J. Happel, Viscous Flow in Multiparticle Systems: Slow Motion of Fluids Relative to Beds of Spherical Particles, A. I. Ch. E. J. 4, 197–201 (1958)
J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Prentice Hall, Englewood Cliffs, NJ, 1965; and Martinus Nijholl Publishers, Dordrecht, 1986
D. J. Jeffrey and Y. Onishi, Calculation of the Resistance and Mobility Functions for Two Unequal Rigid Spheres in Low-Reynolds Number Flow, J. Fluid Mech 139, 261–290 (1985)
S. Kuwabara, The Forces Experienced by Randomly Distributed Parallel Circular Cylinders or Spheres in a Viscous Flow at Small Reynolds Numbers, J. Phys. Soc. Japan 14, 527–532 (1959)
H. Lamb, Hydrodynamics, Dover, New York, 1932, pp. 604–605
P. Moon and D. E. Spencer, Field Theory Handbook, Springer-Verlag, New York, 1961, pp. 28–29
P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vols. I, II, McGraw-Hill, New York, 1953, pp. 547–549, 600–604
A. Oberbeck, Ueber Stationäre Flüssigkeitsbewegungen mit Berücksichtigung der inneren Reibung, J. Reine Angew. Math. 81, 62–80 (1876)
L. E. Payne and W. H. Pell, The Stokes Flow Problem for a Class of Axially Symmetric Bodies, J. Fluid Mech. 7, 529–549 (1960)
E. R. Rainville, Special Functions, Chelsea, New York, 1971, p. 173
J. M. Rallison and A. Acrivos, A Numerical Study of the Deformation and Burst of a Viscous Drop in an Extensional Flow, J. Fluid Mech. 89, 191–200 (1978)
R. A. Sampson, On Stoke’s Current Function, Philos. Trans. Roy. Soc. London Ser. A 182, 449–518 (1891)
M. Stimson and G. B. Jeffery, The Motion of Two-Spheres in a Viscous Fluid, Proc. Roy. Soc. A 111, 110–116 (1926)
B. J. Yoon and S. Kim, A Boundary Collocation Method for the Motion of Two Spheroids in Stokes Flows: Hydrodynamic and Colloidal Interactions, Internat. J. Multiphase Flow 16, 639–649 (1990)
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Article copyright:
© Copyright 1994
American Mathematical Society
