On the use of Green’s formula for the solution of the Navier-Stokes equations


A general iterative procedure for solving Navier-Stokes system in stream function-vorticity variables  is considered. Using the fundamental solution for Laplace operator we find a boundary condition for vorticity. We define an application such that the vorticity is its fixed point. In order to construct this application we solve a Dirichlet problem for Laplace equation and a boundary value problem for vorticity equation in divergence form. A FEM and a classical BEM are used to accomplish this task. Some numerical experiments are carried out.


C.I. Gheorghiu
Tiberiu Popoviciu Institute of Numerical Analysis


Navier-Stokes system; stream function; vorticity; Laplace operator; fundamental solution; iterative solver; FEM; BEM;


See the expanding block below.

Cite this paper as

C.I. Gheorghiu, On the use of Green’s formula for the solution of the Navier-Stokes equations, Adv. Water Resources, 14 (1991) 113-117
doi: 10.1016/0309-1708(91)90002-6



About this paper

Print ISSN


Online ISSN






Google Scholar


1 Roache, P. J. Computational Fluid Dynamics, Hermosa PubI., Albuquerque. USA, 1972
2 Orszag, S. A. and Israeli. M. Numerical simulation of viscous incompressible flows. Ann. Rev. Fluid Mech.. 6, 281-318
3 Gupta, M. M. and Manohar, R. P. Boundary approximations and accuracy in viscous flow computations, d. Comp. Phys.. 31. 265 288
4 Cea, J.. Lhomme, B. and Peyret, R. The Use of Green’r Formula for Vorticity Boundary Values. Publications ,Mathematiques Appliquees. Univ. de Nice. 1981
5 Wu, J. C. and Thompson, J. F. Numerical solution of time-dependent incompressible Navier-Stokes equations using an integro-differential formulation, Computer and Fluids I. 197-215
6 Wu, J. C. and Wahbah. M. M. Numerical solution of viscou:, flow equations using integral representations, Lecture Notes in Physics. 1976, 59, 448453
7 Glowinski. R. and Pironneau. O. Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. SIA.!,[ Reciew, 1979, 21, 167-2t2
8 Petrila, T. and Gheorghiu, C. I. Finite Element Methods and Applications,  Ed. Academiei, Bucarest, Romania, 1987 (in Romanian)
9 Johnson, C. Numerical Solutions Partial Differential Equations by the Finite Element Method, Cambridge Uniersity Press. Cambridge. UK, 1987
10 Iacob, C. Determination de la second approximation de I’ecoulement compressible subsonique autour d’un profil donne. Archivum Mechaniki Stosowanej, 1964, 2{16)
11 Gheorghiu, C. I. The FEM in Some Problems Viscous Fluid Mechanics  Ph. Dissertation, Univ. of Bucharest. Faculty of Mathematics, Romania, 1984
12 lngham, D. B. and Kelmanson. M. A. Boundari Integral Equation analysis of Singular, Potential and Biharmonic Problems, Springer-Verlag, 1984

Related Posts