B Conservation of Mass - The Continuity Equation

Once we adopt the continuum hypothesis and choose to describe fluid motions and heat transfer processes from a macroscopic point of view, we derive the governing equations by invoking the familiar conservation principles of classical continuum physics. These are conservation of mass and energy, plus Newton s second and third laws of classical mechanics. 

The simplest of the various conservation principles to apply is conservation of mass. It is instructive to consider its application relative to two different, but equivalent, descriptions of our fluid system. In both cases, we begin by identifying a specific macroscopic body of fluid that lies within an arbitrarily chosen volume element at some initial instant of time. Because we have adopted the continuum mechanics point of view, this volume element will be large enough that any flux of mass across its surface that is due to random molecular motions can be neglected completely. Indeed, in this continuum description of our system, we can resolve only the molecular average (or continuum point) velocities, and it is convenient to drop any reference to the averaging symbol (). The continuum point velocity vector is denoted as u.4 

In the first description of mass conservation for our system, we consider on arbitrarily chosen volume element (here called a control volume) of fixed position and shape as illustrated in Fig. 2-2. Thus, at each point on its surface, there is a mass flux of fluid pu n through the surface. With n chosen as the outer unit normal to the surface, this mass flux will be negative at points where fluid enters the volume element and positive where it exits. 

There is no reason, at this point, to assume that the fluid density p is necessarily constant. Indeed, conservation of mass requires the density inside the volume element to change with time in such a way that any imbalance in the mass flux in and out of the volume element is compensated for by an accumulation of mass inside. Expressing this statement in mathematical terms we obtain 

Then we note that this integral condition on p and u can be satisfied for an arbitrary volume element only if the integrand is identically zero, that is, 

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