Viscous Fluid Mechanics

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.