Rotation of a fluid element

Consider the following two-dimensional fluid element:

After some time $dt$, the angle sides $AB$ and $AD$ will have rotated a small angle $d\alpha$ and $d\beta$, respectively, due to the velocity difference that exists between points $A-B$ and point $A-D$. The differential angles are:

$tan(d\alpha) \approx d\alpha=\frac{\frac{\partial v}{\partial x}dxdt}{dx}$

$tan(d\beta) \approx d\beta=\frac{\frac{\partial u}{\partial y}dydt}{dy}$

Rearranging the previous relations

$\frac{d\alpha}{dt}=\frac{\partial v}{\partial x}$

$\frac{d\alpha}{dt}=\frac{\partial u}{\partial y}$

Given that the rotation is defined positive if it's counterclockwise, the angle $d\beta$ is actually:

$\frac{d\beta}{dt}=-\frac{\partial u}{\partial y}$

The angular velocity of the element, along the $z$ axis, is defined as the average angular velocity of sides $A-B$ and $A-C$. Thus

$\Omega_{z}=\frac{1}{2}(\frac{d\alpha}{dt}+\frac{d\beta}{dt})=\frac{1}{2}(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y})$

Similarly, for the other two directions:

$\Omega_{x}=\frac{1}{2}(\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z})$

$\Omega_{y}=\frac{1}{2}(\frac{\partial u}{\partial z}-\frac{\partial w}{\partial x})$

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.

Linear Strain Rate

The linear strain rate is a measure of how much a fluid element is shrinking or expanding per unit time. We use a Taylor Series Expansion on the velocities to approximate the velocity on the sides of the fluid elements. Note that the fluid element might have an arbitrary shape. Consider the following rectangular two-dimensional fluid element:

The velocity difference between the sides of the fluid element is the cause of its expansion or shrinking. See that the size of sides of the object have changed when a differential time $dt$ has passed.

The linear strain rate is defined as the rate of change of a side of the fluid elements divided by the original size of the side. Thus, for the $x$ component, the change on the size of the side is:

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

Bringing the $dxdt$ to the other side of the equation:

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

This is exactly the linear strain rate.

If we do the same for the other directions we get that:

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

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

where $u,v,w$ are the velocity components in the $x,y,z$ directions, respectively.

It is common to find the linear strain rate written in the following form:

$\epsilon_{xx}=\frac{\partial u}{\partial x}$

$\epsilon_{yy}=\frac{\partial v}{\partial y}$

$\epsilon_{zz}=\frac{\partial w}{\partial z}$