next up previous
Next: Mass Conservation Up: Mathematical Models of Fluid Previous: Viscosity

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

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

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

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

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


next up previous
Next: Mass Conservation Up: Mathematical Models of Fluid Previous: Viscosity
Richard Fitzpatrick 2016-03-31