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A boundary integral method for the numerical computation of the forces exerted on a sphere in viscous incompressible flows near a plane wall. (English) Zbl 0624.76033

This paper is based on the theoretical results [the first author, Math. Methods Appl. Sci 8, 23-40 (1986; Zbl 0616.76032) and: Über die langsame Bewegung eines starren Körpers in einer zähen inkompressiblen Flüssigkeit längs einer ebenen Wand. Thesis, Techn. Hochschule Darmstadt (1983)] on the existence and construction of asymptotic expansions for the solution of the Navier Stokes problem with small Reynolds numbers. Examples on slow uniform flows of viscous, incompressible fluid past a rigid sphere near a plane wall are considered. The drag and lateral forces exerted on the sphere by the fluid are completed. The numerical results are compared with existing theoretical and experimental data. Boundary element method was used and proved to be useful.
Reviewer: P.K.Mahanti

MSC:

76D10 Boundary-layer theory, separation and reattachment, higher-order effects
76M99 Basic methods in fluid mechanics
35Q30 Navier-Stokes equations

Citations:

Zbl 0616.76032
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References:

[1] H. Brenner,Hydrodynamic resistance of particles at small Reynolds numbers. Advances in Chemical Engineering (T. B. Drew et al. ed.), Vol. 6, 287. Academic Press, New York 1966.
[2] H. Buggisch,Langsame Relativbewegung von festen Partikeln in Strömungsfeldern-Anwendungsbeispiele aus der mechanischen Verfahrenstechnik. ZAMM64, T3 (1984). · doi:10.1002/zamm.19840640403
[3] R. G. Cox and H. Brenner,The lateral migration of solid particles in Poiseuille flow-I Theory. Chem. Engng. Sci.23, 147 (1968). · doi:10.1016/0009-2509(68)87059-9
[4] R. G. Cox and S. K. Hsu,The lateral migration of solid particles in a laminar flow near a plane. Int. J. Multiphase Flow3, 201 (1977). · Zbl 0366.76020 · doi:10.1016/0301-9322(77)90001-5
[5] W. R. Dean and M. E. O’Neill,A slow motion of viscous liquid caused by the rotation of a solid sphere. Mathematika10, 13 (1963). · Zbl 0117.20301 · doi:10.1112/S0025579300003314
[6] H. Faxén,Einwirkung der Gefä?wände auf den Widerstand gegen die Bewegung einer kleinen Kugel in einer zähen Flüssigkeit. Thesis, Uppsala 1921. · JFM 48.0946.02
[7] T. M. Fischer,An integral equation procedure for the exterior three-dimensional slow viscous flow. Integral Equations Operator Theory5, 490 (1982). · Zbl 0483.76033 · doi:10.1007/BF01694049
[8] T. M. Fischer,Über die langsame Bewegung eines starren Körpers in einer zähen, inkompressiblen Flüssigkeit längs einer ebenen Wand. Thesis, Techn. Hochschule Darmstadt 1983. · Zbl 0614.76022
[9] T. M. Fischer,Wall effects on the slow steady motion of a particle in a viscous incompressible fluid. Math. Meth. Appl. Sci.8, 23 (1986). · Zbl 0616.76032 · doi:10.1002/mma.1670080103
[10] T. M. Fischer, G. C. Hsiao and W. L. Wendland,On the exterior three-dimensional slow viscous flow problem. ZAMM64, T 276 (1984).
[11] T. M. Fischer, G. C. Hsiao and W. L. Wendland,Singular perturbations for the exterior three dimensional slow viscous flow problem. J. Math. Anal. Appl.110, 583 (1985). · Zbl 0582.76026 · doi:10.1016/0022-247X(85)90318-X
[12] A. J. Goldman, R. G. Cox and H. Brenner,Slow viscous motion of a sphere parallel to a plane wall-I Motion through a quiescent fluid. Chem. Engng. Sci.22, 637 (1967). · doi:10.1016/0009-2509(67)80047-2
[13] A. J. Goldman, R. G. Cox and H. Brenner,Slow viscous motion of a sphere parallel to a plane wall-II Couette flow. Chem. Engng. Sci.22, 653 (1967). · doi:10.1016/0009-2509(67)80048-4
[14] H. L. Goldsmith and S. G. Mason,The microrheology of dispersions. Rheology, Theory and Applications (F. R. Eirich ed.), Vol. 4, 85. Academic Press, New York 1967.
[15] J. Happel and H. Brenner,Low Reynolds Number Hydrodynamics. Prentice-Hall, Englewood Cliffs 1965. · Zbl 0612.76032
[16] F.-K. Hebeker,A theorem of Faxén and the boundary integral method for three-dimensional viscous incompressible fluid flows. Math. Meth. Appl. Sci., to appear.
[17] A. Heertsch,Experimente zur Entmischung kugelförmiger Teilchen in Mikrokanälen. Max-Planck-Institut für Strömungsforschung, Göttingen, Bericht 20/1980.
[18] B. P. Ho and L. G. Leal,Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech.65, 365 (1974). · Zbl 0284.76076 · doi:10.1017/S0022112074001431
[19] G. C. Hsiao, P. Kopp and W. L. Wendland,Some applications of a Galerkin-collocation method for integral equations of the first kind. Math. Meth. Appl. Sci.6, 280 (1984). · Zbl 0546.65091 · doi:10.1002/mma.1670060119
[20] G. C. Hsiao and R. C. MacCamy,Solution of boundary value problems by integral equations of the first kind. SIAM Rev.15, 687 (1973). · Zbl 0265.45009 · doi:10.1137/1015093
[21] G. C. Hsiao and W. L. Wendland,A finite element method for some integral equations of the first kind. J. Math. Anal. Appl.58, 449 (1977). · Zbl 0352.45016 · doi:10.1016/0022-247X(77)90186-X
[22] R. C. Jeffrey and J. R. A. Pearson,Particle motion in laminar vertical tube flow. J. Fluid Mech.22, 721 (1965). · doi:10.1017/S0022112065001106
[23] S. Kaplun and P. A. Lagerstrom,Asymptotic expansions of Navier-Stokes solutions for small Reynolds numbers. J. Math. Mech.6, 585 (1957). · Zbl 0080.18501
[24] H. A. Lorentz,Ein allgemeiner Satz, die Bewegung einer reibenden Flüssigkeit betreffend, nebst einigen Anwendungen desselben. Abhandlungen über Theoretische Physik I, 23. Teubner, Leipzig 1907.
[25] F. K. G. Odqvist,Über die Randwertaufgaben der Hydrodynamik zäher Flüssigkeiten. Math. Z.32, 329 (1930). · JFM 56.0713.04 · doi:10.1007/BF01194638
[26] D. R. Oliver,Influence of particle rotation on radial migration in the Poiseuille flow of suspensions. Nature194, 1269 (1962). · doi:10.1038/1941269b0
[27] M. E. O’Neill,A slow motion of viscous liquid caused by a slowly moving solid sphere. Mathematika11, 67 (1964). · Zbl 0142.43305 · doi:10.1112/S0025579300003508
[28] C. W. Oseen,Neuere Methoden und Ergebnisse in der Hydrodynamik. Akad. Verlagsgesellschaft, Leipzig 1927. · JFM 53.0773.02
[29] I. Proudman and J. R. A. Pearson,Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder. J. Fluid Mech.2, 237 (1957). · Zbl 0077.39103 · doi:10.1017/S0022112057000105
[30] S. I. Rubinow and J. B. Keller,The transverse force on a spinning sphere moving in a viscous fluid. J. Fluid Mech.11, 447 (1961). · Zbl 0103.19503 · doi:10.1017/S0022112061000640
[31] P. G. Saffman,The lift on a small sphere in a slow shear flow. J. Fluid Mech.22, 385 (1965). · Zbl 0218.76043 · doi:10.1017/S0022112065000824
[32] G. Segré and A. Silberberg,Behaviour of macroscopic rigid spheres in Poiseuille flow. Part 1 (Determination of local concentration by statistical analysis of particle passages through crossed light beams). J. Fluid Mech.14, 115 (1962). · Zbl 0118.43203 · doi:10.1017/S002211206200110X
[33] G. Segré and A. Silberberg,Behaviour of macroscopic rigid spheres in Poiseuille flow. Part 2 (Experimental results and interpretation). J. Fluid Mech.14, 136 (1962). · doi:10.1017/S0022112062001111
[34] A. H. Stroud,Approximate Calculation of Multiple Integrals. Prentice-Hall, Englewood Cliffs 1971. · Zbl 0379.65013
[35] M. Tachibana,On the behaviour of a sphere in the laminar tube flows. Rheol. Acta12, 58 (1973). · doi:10.1007/BF01526901
[36] P. Vasseur and R. G. Cox,The lateral migration of a spherical particle in two-dimensional shear flows. J. Fluid Mech.78, 385 (1976). · Zbl 0342.76039 · doi:10.1017/S0022112076002498
[37] P. Vasseur and R. G. Cox,The lateral migration of spherical particles sedimenting in a stagnant bounded fluid. J. Fluid Mech.80, 561 (1977). · Zbl 0351.76121 · doi:10.1017/S0022112077001840
[38] W. L. Wendland,On applications and the convergence of boundary integral methods. Treatment of Integral Equations by Numerical Methods (C. Baker and G. Miller ed.), 463. Academic Press, London 1982. · Zbl 0561.65085
[39] G. K. Youngren and A. Acrivos,Stokes flow past a particle of arbitrary shape: a numerical method of solution. J. Fluid Mech.69, 377 (1975). · Zbl 0314.76031 · doi:10.1017/S0022112075001486
[40] J. Zhu,A boundary integral equation method for the stationary Stokes problem in 3D. Boundary Elements, 5th Int. Con. Hiroshima (C. A. Brebbia et al. ed.), 283. Springer-Verlag, Berlin 1983.
[41] M. B. Bush and R. I. Tanner,Numerical solution of viscous flows using integral equation methods. Int. J. Num. Meth. Fluids3, 71 (1983). · Zbl 0515.76030 · doi:10.1002/fld.1650030107
[42] P. J. Davis and P. Rabinowitz,Numerical Integration. Blaisdell, Waltham 1967. · Zbl 0154.17802
[43] D. Leighton and A. Acrivos,The lift on a small sphere touching a plane in the presence of a simple shear flow. Z. angew. Math. Phys.36, 174 (1985). · doi:10.1007/BF00949042
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