Abstract
Solutions of the biharmonic equation governing steady two dimensional viscous flow of an incompressible newtonian fluid are obtained by employing a direct biharmonic boundary integral equation (BBIE) method in which Green’s Theorem is used to reformulate the differential equation as a pair of coupled integral equations.
The classical BBIE gives poor convergence in the presence of singularities arising in the solution domain. The rate of convergence is improved dramatically by including the analytic behaviour of the flow in the neighbourhood of the singularities. The modified BBIE (MBBIE) effectively ‘subtracts out’ this analytic behaviour in terms of a series representation whose coefficients are initially unknown. In this way the modified flow variables are regular throughout the entire solution domain.
Also presented is a method for including the asymptotic nature of the flow when the solution domain is unbounded.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M.A.JASWON and G.T.SYMM, in Integral Equation Methods in Potential Theory and Elastostatics, Academic Press, London (1977).
C.A.BREBBIA, J.C.F.TELLES and L.C.WROBEL, Boundary Element Techniques: Theory and Applications in Engineering, Springer Verlag, Berlin and New York, 1984.
S.K.LASKAR, Ph.D Thesis, Solution of Certain Boundary Integral Equations in Potential Theory, University College London (1971).
M.R.SPIEGEL, Vector Analysis, McGraw-Hill, London (1974).
C.W.MILLER, Ph.D Thesis, Numerical Solution of 2-D Potential Theory Problems using Integral Equation Techniques, University of Iowa (1979).
T.BRATANOW, I.SPEHERT and C.A.BREBBIA, 3-D analysis of flows around ship hulls using boundary elements. In Recent Advances in Boundary Element Methods (ed. C.A.Brebbia), Pentech Press, London (1978).
F.J.RIZZO and D.J.SHIPPY, A boundary integral approach to potential and elasticity problems for axisymmetric bodies with arbitrary boundary conditions, Mech. Research Comm., 6(2), 99, (1979).
C.J.COLEMAN, A contour integral formulation of plane creeping newtonian flow, Q. J. Mech. Appl. Math., 34(4), 453, (1981).
F.PAN and A.ACRIVOS, Steady flows in rectangular cavities, J. Fluid Mech., 28(4), 643, (1967).
O.BURGGRAF, Analytical and numerical studies on the structure of steady separated flows, J. Fluid Mech., 24(1), 113, (1966).
R.P.GUPTA and M.M.MANOHAR, Boundary approximations and accuracy in viscous flow computations, J. Comput. Phys., 31, 265, (1979).
R.P.GUPTA, M.M.MANOHAR and B.NOBLE, Nature of viscous flows near sharp corners, Computers and Fluids, 9(4), 379, (1981).
D.H.MICHAEL, The separation of viscous liquid at a straight edge, Mathematika, 5, 82, (1958).
W.R.DEAN and P.E.MONTAGNON, On the steady motion of viscous liquid in a corner, Proc. Camb. Phil. Soc, 45, 389, (1949).
S.D.R.WILSON, The development of Poiseuille flow, J. Fluid Mech., 38(4), 793, (1969).
A.C.HURD and A.R.PETERS, Analysis of flow separation in a confined 2-D channel, Trans. ASME J. Basic Engng., D92, 908, (1970).
M.VAN DYKE, Entry flow in a channel, J. Fluid Mech., 44(4), 813, (1970).
S.D.R.WILSON, Entry flow in a channel; Part 2, J. Fluid Mech., 46(4), 787, (1971).
S.C.R.DENNIS and F.T.SMITH, Steady flow through a channel with a symmetrical constriction in the form of a step, Proc. R. Soc. Lond., A372, 393, (1980).
H.HOLSTEIN Jr. and D.J.PADDON, A singular finite difference treatment of re-entrant corner flows. Part 1: newtonian fluids, J. Non-newtonian Fluid Mech., 8, 81, (1981).
J.S.BRAMLEY and S.C.R.DENNIS, The calculation of eigenvalues for the stationary perturbation of Poiseuille flow, J. Comput. Phys., 47(2), 179, (1982).
M.MAITI and S.K.CHAKRABARTY, Integral equation solution for simply supported polygonal plates, Int. J. Engng. Sci., 12, 793, (1974).
H.K.MOFFATT, Viscous and resistive eddies near a sharp corner, J. Fluid Mech., 18(1), 1, (1964).
G.D.SMITH, in Numerical Solution of Partial Differential Equations, Finite Difference Methods, Clarendon Press, Oxford (1978).
D.B.INGHAM and M.A.KELMANSON, Modified integral equation solution of viscous flow in a bifurcating channel. In Proc. 4th Int. Conf. on Boundary Element Methods, Springer-Verlag (1982).
G.T.SYMM, Treatment of singularities in the solution of Laplace’s equation by an integral equation method, National Physics Laboratory Report NAC31, (1973).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1984 Springer-Verlag Berlin, Heidelberg
About this chapter
Cite this chapter
Ingham, D.B., Kelmanson, M.A. (1984). Modified Integral Equation Solution of Viscous Flows Near Sharp Corners. In: Boundary Integral Equation Analyses of Singular, Potential, and Biharmonic Problems. Lecture Notes in Engineering, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-82330-5_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-82330-5_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-13646-0
Online ISBN: 978-3-642-82330-5
eBook Packages: Springer Book Archive