Chapter 19 - Viscous and Non-Newtonian Flows

Abstract

The solutions to viscous and non-Newtonian fluid flow problems are presented. The basic equations governing two-dimensional steady incompressible Newtonian flow and its boundary conditions are stated in terms of the pressure, velocity, and velocity gradient, along with possible solution methods. The solution methods are based on the use of the stream function formulation, which treats the stream function as an unknown, the velocity–pressure formulation with velocity components and the pressure as unknowns, and the stream function–vorticity formulation with the stream function and vorticity as unknowns. The stream function formulation is outlined using a variational approach. Because no universally accepted variational principle is available for solving Navier–Stokes equations, the pseudovariational principle, given by Olson, is used to present the finite element solution, based on an 18 degrees of freedom–conforming triangular element. The velocity–pressure formulation, based on the Galerkin method, is outlined by developing the finite element equations. An iterative solution is given for solving full Navier–Stokes equations. The stream function–vorticity formulation, coupled with the variational approach, is used to derive the finite element equations. Finally, the finite element solution to non-Newtonian fluid flow problems is presented using the Galerkin method along with an iterative method for the solution.