Chapter 3 - Navier-Stokes equations for incompressible flows: Finite-element methods

Abstract

This chapter is concerned with the finite-element approximation of viscous, incompressible, laminar flows. The governing partial differential equations are the continuity equation and the Navier–Stokes equations. The first of these is the mathematical realization of the incompressibility of the flow. The Navier–Stokes equations are the mathematical realization of Newton's second law of motion along with a linear constitutive law relating stresses to rates of strains. Derivations of these equations may be found in numerous texts on fluid mechanics. The specific form of the governing equations depends on the choice of dependent variables. At first, the chapter concentrates on the “primitive variable” formulation. Subsequently, it discusses other commonly used formulations of the equations of viscous, incompressible, and laminar flow. It should be noted that only homogeneous incompressible flows are considered here. Also, for the most part, the chapter focuses on issues that are specific to finite-element methods, and do not dwell too much on material that is common to all discretization schemes for the simulation of incompressible flows.