Stokes flow before plane boundary with mixed stick-slip boundary conditions

doi:10.1007/s10483-011-1459-8

Applied Mathematics and Mechanics (English Edition) ›› 2011, Vol. 32 ›› Issue (6): 795-804.doi: https://doi.org/10.1007/s10483-011-1459-8

Stokes flow before plane boundary with mixed stick-slip boundary conditions

N. AKHTAR¹, G. A. H. CHOWDHURY¹, S.K.SEN²

1. Department of Mathematics, Shahjalal University of Science and Technology, ylhet 3114, Bangladesh;
2. Research Centre for Mathematical and Physical Sciences, University of Chittagong, Chittagong 4331, Bangladesh

收稿日期:2010-07-09 修回日期:2011-02-19 出版日期:2011-06-01 发布日期:2011-06-01

Stokes flow before plane boundary with mixed stick-slip boundary conditions

N. AKHTAR¹, G. A. H. CHOWDHURY¹, S.K.SEN²

1. Department of Mathematics, Shahjalal University of Science and Technology, ylhet 3114, Bangladesh;
2. Research Centre for Mathematical and Physical Sciences, University of Chittagong, Chittagong 4331, Bangladesh

Received:2010-07-09 Revised:2011-02-19 Online:2011-06-01 Published:2011-06-01

摘要/Abstract

摘要： A general theorem for the Stokes flow over a plane boundary with mixed tick-slip boundary conditions is established. This is done by using a representation for the velocity and pressure fields in the three-dimensional Stokes flow in terms of a biharmonic function and a harmonic function. The earlier theorem for the Stokes flow due to fundamental singularities before a no-slip plane boundary is shown to be a special case of the present theorem. Furthermore, in terms of the Stokes stream function, a corollary of the theorem is also derived, providing a solution to the problem of the axisymmetric Stokes flow along a rigid plane with stick-slip boundary conditions. The formulae for the drag and torque exerted by the fluid on the boundary are established. An illustrative example is given.

关键词: Stokes flow, Stokes stream function, stick-slip boundary conditions, harmonic function, biharmonic function

Abstract: A general theorem for the Stokes flow over a plane boundary with mixed tick-slip boundary conditions is established. This is done by using a representation for the velocity and pressure fields in the three-dimensional Stokes flow in terms of a biharmonic function and a harmonic function. The earlier theorem for the Stokes flow due to fundamental singularities before a no-slip plane boundary is shown to be a special case of the present theorem. Furthermore, in terms of the Stokes stream function, a corollary of the theorem is also derived, providing a solution to the problem of the axisymmetric Stokes flow along a rigid plane with stick-slip boundary conditions. The formulae for the drag and torque exerted by the fluid on the boundary are established. An illustrative example is given.

Key words: Stokes flow, Stokes stream function, stick-slip boundary conditions, harmonic function, biharmonic function

中图分类号:

76D07

N. AKHTAR G. A. H. CHOWDHURY S.K.SEN. Stokes flow before plane boundary with mixed stick-slip boundary conditions[J]. Applied Mathematics and Mechanics (English Edition), 2011, 32(6): 795-804.

N. AKHTAR;G. A. H. CHOWDHURY;S.K.SEN. Stokes flow before plane boundary with mixed stick-slip boundary conditions[J]. Applied Mathematics and Mechanics (English Edition), 2011, 32(6): 795-804.

参考文献

[1] Akhtar, N., Rahman, F., and Sen, S. K. Stokes flow due to fundamental singularities before a
plane boundary. Applied Mathematics and Mechanics (English Edition), 25(4), 799–805 (2004)
DOI 10.1007/BF02437572

[2] Happel, J. and Brenner, H. Low Reynolds Number Hydrodynamics, Martinus Nijhoff Publishers,
Dordrecht (1986)

[3] Collins, W. D. Note on a sphere theorem for the axisymmetric Stokes flow of a viscous fluid.
Mathematika, 5, 118–121 (1958)

[4] Aderogba, K. On Stokeslets in a two-fluid space. Journal of Engineering Mathematics, 10(2),
143–151 (1976)

[5] Palaniappan, D., Nigam, S. D., Amaranath, T., and Usha, R. Lamb’s solution of Stokes’s equations:
a sphere theorem. The Quarterly Journal of Mechanics and Applied Mathematics, 45(1),
47–56 (1992)

[6] Padmavathi, B. S., Amaranath, T., and Nigam, S. D. Stokes flow past a sphere with mixed
slip-stick boundary conditions. Fluid Dynamics Research, 11, 229–234 (1993)

[7] Schmitz, R. and Felderhof, B. U. Creeping flow about a sphere. Physica, 92A, 423–437 (1978)

[8] Raja, S. G. P., Tejeswara, R. K., Padmavathi, B. S., and Amaranath, T. Two-dimensional Stokes
flows with slip-stick boundary conditions. Mechanics Research Communications, 22(5), 491–501
(1995)

[9] Palaniappan, D. and Daripa, P. Interior Stokes flows with stick-slip boundary conditions. Physica,
297A, 37–63 (2001)

[10] Palaniappan, D. and Daripa, P. Exterior Stokes flows with stick-slip boundary conditions.
Zeitschrift f ¨ur Angewandte Mathematik und Physik, 53, 281–307 (2002)

[11] Basset, A. B. A Treatise on Hydrodynamics, Dover Publications, New York (1961)

[12] Lamb, H. Hydrodynamics, Dover Publications, New York (1945)

[13] Batchelor, G. K. An Introduction to Fluid Dynamics, Cambridge University Press, London (1967)

[14] Sneddon, I. N. Elements of Partial Differential Equations, McGraw Hill, Singapore (1985)

[15] Stimson, M. and Jeffery, G. B. The motion of two spheres in a viscous fluid. Proceedings of the
Royal Society, A111, 110–116 (1926)

[16] Collins, W. D. A note on Stokes’s stream function for the slow steady motion of viscous fluid
before plane and spherical boundaries. Mathematika, 1, 125–130 (1954)

[17] Spiegel, M. R. Theory and Problems of Advanced Calculus, McGraw Hill, Singapore (1963)

Stokes flow before plane boundary with mixed stick-slip boundary conditions

Stokes flow before plane boundary with mixed stick-slip boundary conditions

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