Convective Time Derivative

(1.38) |

where we have Taylor expanded up to first order in , and where

Clearly, the so-called

The continuity equation (1.37) can be rewritten in the form

because . [See Equation (A.174).] Consider a volume element that is co-moving with the fluid. In general, as the element is convected by the fluid its volume changes. In fact, it is easily seen that

(1.41) |

where is the bounding surface of the element, and use has been made of the divergence theorem. In the limit that , and is approximately constant across the element, we obtain

Hence, we conclude that the divergence of the fluid velocity at a given point in space specifies the fractional rate of increase in the volume of an infinitesimal co-moving fluid element at that point.