The Equations of Fluid Dynamics

Abstract

The equations of fluid mechanics are derived from first principles here, in order to point out clearly all the underlying assumptions. The equations can take various different forms and in numerical work we will find that it often makes a difference what form we use for a particular problem. We will work solely with the continuum theory of fluids, and thus use conser-vation principles, supplemented by constitutive assumptions about the nature of the fluids. The conservation principles are common to any material where the continuum hypothesis is valid but different constitutive hypothesizes apply to different materials. Expressing the basic principles of conservation of mass, momentum, and energy in mathematical form leads to the governing equations for fluid flow. Here we derive the equations for fluid motion, with particular emphasize on incompressible flows. 1.1 General Flows The principle of conservation of mass states that mass can not be created nor destroyed. Therefore, if we consider a volume fixed in space, V , then the change of mass inside this volume can only take place if mass flows in or out through the boundary of this volume, S. 1 Stated more precisely d dt V ρdv = − S ρu · nds, (1) 1 In standard text books the fundamental laws are often stated for a volume of fluid moving with the fluid. In computational work the elementary volumes are usually stationary, therefore it is simpler to start with a stationary volume. 1 2 CHAPTER 1. THE EQUATIONS OF FLUID DYNAMICS—DRAFT where n is the outward normal, ρ the density and u the velocity. Here, the left hand side is the rate of change of mass in the volume V and the right hand side represents in and out flow through the boundaries of V . Since the volume is fixed in space we can take the derivative inside the integral, and by applying the divergence theorem (V