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where the parameter along it lies in the interval . 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 .
The flow field illustrated in 4.13 by Aris is assigned as an exercise.
We noticed earlier that if the coordinate system is changed from coordinates to coordinates , then the element of volume changes by the formula
If we think of as the material coordinates, they are the Cartesian coordinates at , so that is the volume of an elementary rectangular parallelepiped. Consider this elementary parallelepiped about a given point 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 is some neighborhood of the point . By the above equation, its volume is and hence
It is called the dilation or expansion . The assumption that can be inverted to give , and vice versa, is equivalent to requiring that neither nor vanish. Thus,
We can now ask how the dilation changes as we follow the motion. To answer this we calculate the material derivative . However,
Now
for is differentiation with constant so that the order can be interchanged. Now if we regard as a function of , , ,
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,
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 and 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 or 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 and and are independent of . The mass-conservation equation for an incompressible fluid then has the form
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