0.3 The kinematics of fluid motion  (Page 3/9)

where the parameter $t^{\text{'}}$ along it lies in the interval $0\le t^{\text{'}}\le t$ . If we regard the motion as having been proceeding for all time, then the origin of time is arbitrary and t' can take negative values $-\infty \le t^{\text{'}}\le t$ .

The flow field illustrated in 4.13 by Aris is assigned as an exercise.

Dilatation

We noticed earlier that if the coordinate system is changed from coordinates $\xi$ to coordinates $x$ , then the element of volume changes by the formula

If we think of $\xi$ as the material coordinates, they are the Cartesian coordinates at $t=0$ , so that $d{\xi }_1$ $d{\xi }_2$ $d{\xi }_3$ is the volume $d{V}_o$ of an elementary rectangular parallelepiped. Consider this elementary parallelepiped about a given point $\xi$ at the initial instant. By the motion this parallelepiped is moved and distorted but because the motion is continuous it cannot break up and so at some time $t$ is some neighborhood of the point $\mathbf{x}=\mathbf{x}(\xi ,t)$ . By the above equation, its volume is $dV=Jd{V}_o$ and hence

It is called the dilation or expansion . The assumption that $\mathbf{x}=\mathbf{x}(\xi ,t)$ can be inverted to give $\xi =\xi (\mathbf{x},t)$ , and vice versa, is equivalent to requiring that neither $J$ nor $J^{-}1$ vanish. Thus,

We can now ask how the dilation changes as we follow the motion. To answer this we calculate the material derivative $DJ/Dt$ . However,

Now

for $D/Dt$ is differentiation with $\xi$ constant so that the order can be interchanged. Now if we regard $v_i$ as a function of $x_1$ , $x_2$ , $x_3$ ,

The above relation can now be applied to differentiation of the Jacobian.

where we made use of the property of the determinant that the determinant of a matrix with repeated rows is zero. Thus,

We thus have an important physical meaning for the divergence of the velocity field. It is the relative rate of dilation following a particle path. It is evident that for an incompressible fluid motion,

Use of a stream function to satisfy the mass-conservation equation (batchelor, 1967)

In the cases of flow of an incompressible fluid, and of steady flow of a compressible fluid, the mass-conservation equation reduces to the statement that a vector divergence is zero, the divergences being of $\mathbf{u}$ and $\rho \mathbf{u}$ respectively. If we impose the further restriction that the flow field either is two-dimensional or has axial symmetry, this vector divergence is the sum of only two derivatives, and the mass-conservation equation can then be regarded as defining a scalar function from which the components of $\mathbf{u}$ or $\rho \mathbf{u}$ are obtained by differentiation. The procedure will be described here for the case of an incompressible fluid.

Assume first that the motion is two-dimensional, so that $\mathbf{u}=(u,v,0)$ and $u$ and $v$ are independent of $z$ . The mass-conservation equation for an incompressible fluid then has the form