Viscous Fluid Mechanics

  • D. Bonneau
  • G. Bezine
Part of the Topics in Boundary Element Research book series (TBOU, volume 5)


First we give the governing equations for an incompressible viscous newtonian fluid completed with boundary conditions.

An integral equation method for two-dimensional Stokes flows is presented which consists in solving the biharmonic equation.

A direct boundary integral formulation is developed for the biharmonic equation. The representation of the stream function and its derivative obtained involves all the quantities defined on the boundary.

In the case of Stokes flow the discretization of these representations leads to a linear system of equations.

When the inertia effects are taken into account, the evaluation of these terms is necessary. In this latter case four internal parameters are defined: the two components of the velocity and the two gradients of the vorticity. By discretizing the domain we obtain nonlinear algebraic equations which can be solved by classical method for small Reynold’s numbers, but much elaborated methods are necessary when the inertia effects are important.

Finally we present some examples which prove the numerical efficiency of this formulation compared with results given by other methods.


Stream Function Boundary Element Method Inertia Effect Internal Parameter Nonlinear Algebraic Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Bourot, J.M., “Sur un procédé de résolution approchée du problème aux limites, pour certains écoulements de fluide parfait ou visqueux, en présence d’obstacles et de parois”, C.R. Acad. Sci., 266, 470–473 (1968).Google Scholar
  2. 2.
    Smith, G.D., “Numerical Solution of Partial Differential Equations”, Oxford University Press, London 1969.Google Scholar
  3. 3.
    Chung, T.J., “Finite Element Analysis in Fluid Dynamics”, McGraw-Hill 1978.Google Scholar
  4. 4.
    Bezine, G. and Gamby, D., “A New Integral Equation Formulation for Plate Bending Problems”, Recent Advances in Boundary Element Methods, Pentech Press, London 1978.Google Scholar
  5. 5.
    Bezine, G., “Boundary Integral Formulation for Plate Flexure with Arbitrary Boundary Conditions”, Mech. Res. Comm., 5, 197–206 (1978).MATHCrossRefGoogle Scholar
  6. 6.
    Bergman, S. and Schiffer, M., “Kernel Functions and Elliptic Differential Equations in Mathematical Physics”, Academic Press, New York, 1953.MATHGoogle Scholar
  7. 7.
    Batchelor, G.K., “An Introduction to Fluid Dynamics”, Cambridge University Press, 1967.Google Scholar
  8. 8.
    Bonneau, D., “Formation du film lubrifiant dans les contacts à alimentation non surabondante. Aspects expérimentaux et théoriques”, Thèse de Doctorat des Sciences Physiques. Poitiers, 1986.Google Scholar
  9. 9.
    Bezine, G. and Bonneau, D., “On a Procedure for Numerical Evaluation of Surface Integrals in Two-Dimensional Boundary Integral Equation Method”, Engineering Analysis, 2, 2–8 (1985).CrossRefGoogle Scholar
  10. 10.
    Demidovitch, B. and Maron, I., “Elements de calcul numérique”, Editions de Moscou, 1973.MATHGoogle Scholar
  11. 11.
    Coutanceau, M. and Bouard, R., “On the Computation of the Plane and Axisymmetric Creeping Flows Around Body in Duct”, European Mechanics Colloquium 129. Computation of Flow Around Systems of Airfoils, Bulgarie, 1980.Google Scholar
  12. 12.
    Curtis, A.R. and Man, C.R., “The Viscous Drag on Cylinders Falling Symmetrically Between Parallel Walls”. J. Phys. D. Appl. Phys. 11, 1173–1178 (1978).ADSCrossRefGoogle Scholar
  13. 13.
    Takaisi, Y., “The Drag on a Circular Cylinder Moving with Low Speeds in a Viscous Liquid Between two Parallel Walls”, J. Phys. Soc. Japan, 10, 685–693 (1955).MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Richardson, S., “A Stick-Slip Problem Related to the Motion of a Free Jet Low Reynolds Numbers”, Proc. Cambridge Phil. Soc., 67, 477–489 (1970).ADSMATHCrossRefGoogle Scholar
  15. 15.
    Michael, D.H., “The Separation of a Viscous Liquid at a Straight Edge”, Mathematika, 5, 82–84 (1958).MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Ruschak, K.J., “A method for Incorporating Free Boundaries with surface Tension in Finite Element Fluid-Flow Simulators”, Int. J. Numerical Methods Eng., 15, 639–648 (1980).ADSMATHCrossRefGoogle Scholar
  17. 17.
    Zienkiewicz, O.C., “The Finite Element Method”, McGraw-Hill, London, 1977.MATHGoogle Scholar
  18. 18.
    Haisler, W.E. and Stricklin, J.A., “Computational Methods for Solving Non-Linear Structural Mechanics Problems”, Proc. Int. Conf. on Computational Meth. in Non-Linear Mech., Université du Texas, USA, pp. 393–403, 1974.Google Scholar
  19. 19.
    Stoer, J. and Bulirsch, R., “Introduction to numerical Analysis”. Springer-Verlag, Berlin, 1976.Google Scholar
  20. 20.
    Golub, G.H. and Van Loan, C.F., “Matrix Computations”, North Oxford Academic, Oxford, 1983.MATHGoogle Scholar
  21. 21.
    Polak, E., “Computational Methods in Optimization”. Academic Press, New York 1971.Google Scholar
  22. 22.
    Kamat, M.P., Watson, L.T. and Junkins, J.L., “A Robust and Efficient Hybrid Method for Finding Multiple Equilibrium Solutions”, C.R. 3e Congrès Int. Methods Num. Ing., Paris, pp. 799–809, 1983.Google Scholar
  23. 23.
    Bouard, R., “Etude de l’écoulement autour d’un cylindre soumis à une translation uniforme après un départ impulsif pour des nombres de Reynolds allant de 0 à 104”, Thèse de Doctorat es Sciences, Poitiers, Juin 1986.Google Scholar
  24. 24.
    Keller, H.B. and Takami, H., “Numerical Studies of Steady Viscous Flow about Cylinders”, Proc. Symp. Math. Université du Wisconsin, USA, pp. 115–135, 1966.Google Scholar
  25. 25.
    Gartling, D. and Becker, E.B., “Computationally Efficient Finite Element Analysis of Viscous Flow ProblemsCh. in Non-Linear Mech., Université du Texas, USA, pp. 603–614, 1974.Google Scholar
  26. 26.
    Dennis, S.C.R. and Chang, G.Z., “Numerical Solutions for Steady Flow Past a Circular Cylinder at Reynolds Numbers up to 100”, J. of Fluid Mechanics, 42, 471 (1970).ADSMATHCrossRefGoogle Scholar
  27. 27.
    Ta Phuoc Loc, “Etude numérique de l’écoulement d’un fluide visqueux incompressible autour d’un cylindre fixe ou en rotation”. J. de Mécanique, 14, 109 (1975).ADSMATHGoogle Scholar
  28. 28.
    Dennis, S.C.R., “A Numerical Method for Calculating Steady Flow Past a Cylinder”, Lecture Notes Phys., Vol. 165, 1976.Google Scholar
  29. 29.
    Tuann, S.Y. and Olson, M.D., “Numerical Studies of the Flow Around a Circular Cylinder by a Finite Element Method”. Computers and Fluids, 6, 219 (1978).ADSMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin, Heidelberg 1989

Authors and Affiliations

  • D. Bonneau
  • G. Bezine

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