Convective Time Derivative

The quantity $\partial \rho({\bf r}, t)/\partial t$ , appearing in Equation (1.37), represents the time derivative of the fluid mass density at the fixed point ${\bf r}$ . Suppose that ${\bf v}({\bf r},t)$ is the instantaneous fluid velocity at the same point. It follows that the time derivative of the density, as seen in a frame of reference which is instantaneously co-moving with the fluid at point ${\bf r}$ , is

$$\lim_{\delta t\rightarrow 0} \frac{\rho({\bf r}+{\bf v}\,\delta t... ...\frac{\partial \rho}{\partial t} + {\bf v}\cdot\nabla \rho = \frac{D \rho}{Dt},$$ (1.38)

where we have Taylor expanded $\rho({\bf r}+{\bf v}\,\delta t, t+\delta t)$ up to first order in $\delta t$ , and where

$$\frac{D}{Dt}= \frac{\partial}{\partial t} + {\bf v}\cdot\nabla = \frac{\partial}{\partial t} + v_i\,\frac{\partial}{\partial x_i}.$$ (1.39)

Clearly, the so-called convective time derivative, $D/Dt$ , represents the time derivative seen in the local rest frame of the fluid.

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

$$\frac{1}{\rho}\,\frac{D\rho}{Dt}= \frac{D\ln\rho}{Dt}=-\nabla\cdot{\bf v},$$ (1.40)

because $\nabla \cdot(\rho\,{\bf v}) = {\bf v}\cdot\nabla \rho + \rho\,\nabla \cdot{\bf v}$ . [See Equation (A.174).] Consider a volume element $V$ 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

$$\frac{DV}{Dt} = \oint_S{\bf v}\cdot d{\bf S}=\oint_S v_i\,dS_i = ... ..._V \frac{\partial v_i}{\partial x_i}\,dV = \int_V \nabla\!\cdot \! {\bf v}\,dV,$$ (1.41)

where $S$ is the bounding surface of the element, and use has been made of the divergence theorem. In the limit that $V\rightarrow 0$ , and $\nabla\cdot{\bf v}$ is approximately constant across the element, we obtain

$$\frac{1}{V}\,\frac{DV}{Dt} = \frac{D\ln V}{Dt} = \nabla\cdot{\bf v}.$$ (1.42)

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.