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A spectral boundary element approach to three-dimensional Stokes flow

Published online by Cambridge University Press:  26 April 2006

G. P. Muldowney  and
J. J. L. Higdon
G. P. Muldowney
Affiliation:
Department of Chemical Engineering, University of Illinois, Urbana, IL 61801, USA Present address: Mobil Research and Development Co, Billingsport Rd, Paulsboro, NJ 08066, USA.
J. J. L. Higdon
Affiliation:
Department of Chemical Engineering, University of Illinois, Urbana, IL 61801, USA
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Abstract

A novel method is introduced for solving the three-dimensional Stokes equations via a spectral element approach to the boundary integral method. The accuracy and convergence of the method are illustrated through applications involving rigid particles, deformable droplets and interacting particles. New physical results are obtained for two applications in low Reynolds number flow: the permeability of periodic models of a porous membrane and the instability of a toroidal droplet subject to non-axisymmetric perturbations. Further applications are described in the companion paper (Higdon & Muldowney 1995).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 298 , 10 September 1995 , pp. 167 - 192
Copyright
© 1995 Cambridge University Press

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References

Alarcon, E. & Reverter, A. 1986 p-adaptive boundary element. Intl J. Numer. Meth. Engng 23, 801–829.Google Scholar
Braß, H. & Hammerlin, G. 1982 Numerical Integration III. Birkhauser.
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.
Canuto, C. & Quarteroni, A. 1988 Spectral and Higher Order Methods for Partial Differential Equations. North-Holland: Elsevier.
Cerrolaza, M. & Alarcon, E. 1987 P-adaptive boundary elements for three dimensional potential problems. Commun. Appl. Numer. Meth. 3, 335–344.Google Scholar
Cerrolaza, M. & Alarcon, E. 1989 Further applications of P-adaptive boundary elements. In Boundary Elements X.
Chapman, A. M. & Higdon, J. J. L. 1992 Oscillatory Stokes flow in periodic porous media. Phys. Fluids A 4, 2099–2116.Google Scholar
Chan, C. Y., Beris, A. N. & Advani, S. G. 1992 Second order boundary element method calculations of hydrodynamic interactions between particles in close proximity. Intl J. Numer. Meth. Fluids 14, 1063–1086.Google Scholar
Claeys, I. L. & Brady, J. F. 1993a Suspensions of prolate spheroids in Stokes flow. Part 1. Dynamics of a finite number of particles in an unbounded fluid. J. Fluid Mech. 251, 411–442.Google Scholar
Claeys, I. L. & Brady, J. F. 1993b Suspensions of prolate spheroids in Stokes flow. Part 2. Statistically homogeneous dispersions. J. Fluid Mech. 251, 443–477.Google Scholar
Claeys, I. L. & Brady, J. F. 1993c Suspensions of prolate spheroids in Stokes flow. Part 3. Hydrodynamic transport properties of crystalline dispersions. J. Fluid Mech. 251, 479–500.Google Scholar
Duffy, M. G. 1982 Quadrature over a pyramid or cube of integrands with a singularity at a vertex. J. Numer. Anal. 19, 1260–1262.Google Scholar
Espelid, T. O. & Genz, A. 1992 Numerical Integration: Recent Developments, Software and Applications. Kluwer.
Gottlieb, D. & Orszag, S. A. 1984 Numerical Analysis of Spectral Methods: Theory and Applications. SIAM.
Guo, B. Q., Petersdorff, T. von & Stephan, E. P. 1990 An h-p version of BEM for plane mixed boundary value problems. In Advances in Boundary Elements.
Hammerlin, G. 1979 Numerische Integration. Birkhauser.
Hammerlin, G. 1982 Numerical Integration. Birkhauser.
Happel, J. & Brenner, H. 1973 Low Reynolds Number Hydrodynamics. Noordhoff.
Hayami, K. 1992 A Projection Transformation Method for Nearly Singular Surface Boundary Element Integrals. Springer.
Hebeker, F. K. 1986 Efficient boundary element methods for three dimensional exterior viscous flows. Num. Meth. Partial Diff. Equats. 2, 273–297.Google Scholar
Hebeker, F. K. 1988 On the numerical treatment of viscous flows against bodies with corners and edges by boundary element and multigrid methods. Numer. Meth. 52, 81–99.Google Scholar
Higdon, J. J. L. 1985 Stokes flow in arbitrary two-dimensional domains: shear flow over ridges and cavities. J. Fluid Mech. 159, 195–226.Google Scholar
Higdon, J. J. L. 1990 Effects of pressure gradients on Stokes flow over cavities. Phys. Fluids A 2, 112–114.Google Scholar
Higdon, J. J. L. 1993 The kinematics of the four roll mill. Phys. Fluids A 5, 274–276.Google Scholar
Higdon, J. J. L. & Muldowney, G. P. 1995 Resistance functions for spherical particles, droplets and bubbles in cylindrical tubes. J. Fluid Mech. 298, 193–210.Google Scholar
Higdon, J. J. L. & Schnepper, C. A. 1994 High order boundary integral methods for viscous free surface flows. In Free Boundaries in Viscous Flows. Springer.
Hinch, E. J. 1972 Note on the symmetries of certain material tensors for a particle in Stokes flow. J. Fluid Mech. 54, 423–425.Google Scholar
Hsiao, G. C. 1988 On the stability of boundary element methods for integral equations of the first kind. In Boundary Elements IX, pp. 177–192.
Ingber, M. S. & Mondy, L. A. 1993 Direct second kind boundary integral formulation for Stokes flow problems. Comput. Mech. 11, 11–27.Google Scholar
Isaacson, E. & Keller, H. B. 1966 Analysis of Numerical Methods. Wiley.
Jackson, G. W. & James, D. F. 1986 The permeability of fibrous porous media. Can. J. Chem. Engrs 64, 364–374.Google Scholar
Kim, S. & Karrila, S. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.
Kojima, M., Hinch, E. J. & Acrivos, A. 1984 The formation and expansion of a toroidal drop moving in a viscous fluid. Phys. Fluids 27, 19–32.Google Scholar
Larson, R. E. & Higdon, J. J. L. 1986 Microscopic flow near the surface of two-dimensional porous media. Part 1. Axial flow. J. Fluid Mech. 166, 449–472.Google Scholar
Larson, R. E. & Higdon, J. J. L. 1987 Microscopic flow near the surface of two-dimensional porous media. Part 2. Transverse flow. J. Fluid mech. 178, 119–136.Google Scholar
Li, X. Z., Barthes-Biesel, D. & Helmy, A. 1988 Large deformations and burst of a capsule freely suspended in an elongational flow. J. Fluid Mech. 187, 179–196.Google Scholar
Lorentz, H. A. 1907 Ein allgemeiner satz, die bewegung einer reibenden flussigkeit betreffend, nebst einigen andwendungendesselben. Abhand. Theor. Phys. 1, 23–42.Google Scholar
Lyness, J. N. 1992 On handling singularities in finite elements. In Numerical Integration: Recent Developments, Software and Applications (ed. T. O. Espelid & A. Genz). Kluwer.
Maday, Y. & Patera, A. T. 1989 Spectral element methods for the incompressible Navier-Stokes equations. In State of the Art Surveys in Computational Mechanics (ed. A. K. Noor & J. T. Oden). ASME.
Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–18.Google Scholar
Muldowney, G. P. 1988 PhD dissertation, University of Illinois.
Occhialini, J. M. & Higdon, J. J. L. 1992 Convective mass transport from rectangular cavities in viscous flow. J. Electrochem. Soc. 10, 2845–2855.Google Scholar
Occhialini, J. M., Muldowney, G. P. & Higdon, J. J. L. 1992 Boundary integral/spectral element approaches to the Navier-Stokes equations. Intl J. Numer. Meth. Fluids. 15, 1361–1381.Google Scholar
Parriera, P. 1988 Self-adaptive hierarchical boundary elements in elastostatics. In Boundary Elements IX.
Phan-Thien, N., Tran-Cong, T. & Ramia, M. 1987 A boundary element analysis of flagellar propulsion. J. Fluid Mech. 184, 533–549.Google Scholar
Power, H. & Miranda, G. 1987 Second kind integral equation formulation of Stokes flow past a particle of arbitrary shape. SIAM J. Appl. Maths 47, 689–698.Google Scholar
Pozrikidis, C. 1990 The axisymmetric deformation of a red blood cell in a uniaxial strining Stokes flow. J. Fluid Mech. 216, 231.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singuarlity Methods for Linearized Viscous Flow. Cambridge University Press.
Rallison, J. M. 1984 The deformation of small viscous drops and bubbles in shear flows. Ann. Rev. Fluid Mech. 16, 45–66.Google Scholar
Rank, E. 1988 Adaptive boundary element methods. In Boundary Elements IX.
Schnepper, C. A. 1988 Application of the boundary integral method to the deformation of droplets in viscous flow. 1 thesis, University of Illinois.
Stone, H. A. 1994 Dynamics of drop deformation and breakup in viscous fluids. Ann. Rev. Fluid Mech. 26, 65–102.Google Scholar
Youngren, G. K. & Acrivos, A. 1975 Stokes flow past a particle of arbitrary shape: a numerical method of solution. J. Fluid Mech. 69, 377–403.Google Scholar
Youngrhn, G. K. & Acrivos, A. 1976 On the shape of a gas bubble in a viscous extensional flow. J. Fluid Mech. 76, 433–442.Google Scholar
Zick, A. A. & Homsy, G. M. 1982 Stokes flow through periodic arrays. J. Fluid Mech. 115, 13–26.Google Scholar
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