Volumetric Dilatation Rate

The volumetric dilatation rate is a measure of how much a fluid element is shrinking or expanding per unit time. Consider the following rectangular two-dimensional fluid element:

The deformation of each side is given by (see the Linear Strain Rate derivation):

$\frac{1}{dx}\frac{d(dx)}{dt}=\frac{\partial u}{\partial x}$

$\frac{1}{dy}\frac{d(dy)}{dt}=\frac{\partial v}{\partial y}$

$\frac{1}{dz}\frac{d(dz)}{dt}=\frac{\partial w}{\partial z}$

The volume of a fluid element is just the product of its sides, thus

$dV=dxdydz$

We apply the chain rule to take the derivative of the above expression to obtain the volume change

$d(dV)=d(dx)dydz+dxd(dy)dz+dxdyd(dz)$

Divide the above expression by a $dV$

$\frac{1}{dV}d(dV)=\frac{d(dx)}{dx}+ \frac{d(dy)}{dy}+\frac{d(dz)}{dz}$

And using the linear strain rate relations

$\frac{1}{dV}\frac{d(dV)}{dt} =\frac{\partial u}{\partial x}+ \frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}$

This can also be expressed as:

$\frac{1}{dV}\frac{d(dV)}{dt} = \triangledown \bullet \vec{u}$

The volumetric dilatation rate is directly related to the divergence of the velocity field. If the divergence of the velocity field is 0, the volume of the fluid element remains constant as it travels from one position to another.