13. Incompressible Viscous Fluid Flow

The flow of incompressible and viscous fluid is considered in this chapter. The continuity condition and the equations of motion, the Navier-Stokes equations for the Newtonian fluid, are derived. The Stokes equation for the low Reynolds number flow is discussed and its associated functional expression is presented. The governing equations are then made dimensionless and employing the Galerkin method, the finite element equation is derived, using two different shape functions for the velocity components and pressure. The general finite element equations are then reduced to the case of two-dimensional fluid flow and the elements of mass, stiffness, and force matrices are calculated. The general form of boundary conditions are discussed and the selection/limitation of the shape functions for the velocity components and pressure is explained. As another method of fluid flow problems, the vorticity transport technique is presented and the finite element modeling of the fluid flow in terms of the vorticity and stream function is presented and the element matrices are derived. Since the governing finite element equations are nonlinear, the method of linearization technique is discussed. The finite element matrices for two-dimensional flow of an incompressible viscous fluid employing two-dimensional simplex elements are derived.