Conservation Laws

Suppose that $\theta({\bf r}, t)$ is the density of some bulk fluid property (e.g., mass, momentum, or energy) at position ${\bf r}$ and time $t$ . In other words, suppose that, at time $t$ , an infinitesimal fluid element of volume $d V$ , located at position ${\bf r}$ , contains an amount $\theta({\bf r},t)\,dV$ of the property in question. Note, incidentally, that $\theta$ can be either a scalar, a component of a vector, or even a component of a tensor. The total amount of the property contained within some fixed volume $V$ is

$${\mit\Theta} = \int_V \theta\,dV,$$ (1.28)

where the integral is taken over all elements of $V$ . Let $d{\bf S}$ be an outward directed element of the bounding surface of $V$ . Suppose that this element is located at point ${\bf r}$ . The volume of fluid that flows per second across the element, and so out of $V$ , is ${\bf v}({\bf r}, t)\cdot d{\bf S}$ . Thus, the amount of the fluid property under consideration that is convected across the element per second is $\theta({\bf r},t)\,{\bf v}({\bf r}, t)\cdot d{\bf S}$ . It follows that the net amount of the property that is convected out of volume $V$ by fluid flow across its bounding surface $S$ is

$${\mit\Phi}_{\mit\Theta} = \oint_S \theta\,{\bf v}\cdot d{\bf S},$$ (1.29)

where the integral is taken over all outward directed elements of $S$ . Suppose, finally, that the property in question is created within the volume $V$ at the rate $S_{\mit\Theta}$ per second. The conservation equation for the fluid property takes the form

$$\frac{d{\mit\Theta}}{dt} = S_{\mit\Theta}-{\mit\Phi}_{\mit\Theta}.$$ (1.30)

In other words, the rate of increase in the amount of the property contained within $V$ is the difference between the creation rate of the property inside $V$ , and the rate at which the property is convected out of $V$ by fluid flow. The previous conservation law can also be written

$$\frac{d{\mit\Theta}}{dt} +{\mit\Phi}_{\mit\Theta}= S_{\mit\Theta}.$$ (1.31)

Here, ${\mit\Phi}_{\mit\Theta}$ is termed the flux of the property out of $V$ , whereas $S_{\mit\Theta}$ is called the net generation rate of the property within $V$ .