Rotation

Considering the same element ABCD again, we notice that any rotation of AB or AD is brought about by a change in u velocity along y-direction and that of v velocity along x-direction. Let $d\alpha$ and $d\beta$ be the angles through which sides AB and AD rotate.

Figure 4.9: Rotation of a fluid Element

Now it is required that

$$= {\partial v \over \partial x} dx  dt$$

$$BB'$$

$$DD' = {\partial u \over \partial y} dy  dt$$ (4.35)

Since angles $d\alpha$ and $d\beta$ are small we have,

$$d\alpha ={{BB'} \over {AB}}= {{{{\partial v \over \partial x} dx  dt}} \over {dx} }  = {\partial v \over \partial x} dt$$

$$d\beta ={{DD'} \over {AD}}= {{{{\partial u \over \partial y} dy  dt}} \over {dy} }  = {\partial u \over \partial y} dt$$ (4.36)

Rotation or angular velocity of the element (about the z-axis) is defined as

$$\omega_z = {1 \over 2} \left( {d\alpha \over {dt}} - {d\beta \over {dt}}\right)$$ (4.37)

Combining eqns. 4.36 and 4.37 we have,

$$\omega_z = {1 \over 2} \left( {\partial v\over {\partial x}} - {\partial u \over {\partial y}}\right)$$ (4.38)

Similarly we have for rotation about the other axes,

$$\omega_x = {1 \over 2} \left( {\partial w\over {\partial y}} - {\... ...\left( {\partial u\over {\partial z}} - {\partial w \over {\partial x}}\right)$$ (4.39)

We see that $\omega$ is a vector given by $\omega = \hat{i}\omega_x + \hat{j}\omega_y + \hat{k}\omega_z$, which can be written in vector notation as,

$$\omega =  {1 \over 2} (curl \vec{V}) = {1 \over 2}\left\vert\begi... ...ial y} & \partial \over {\partial z} \ u & v & w \ \end{array}\right\vert$$ (4.40)

We now introduce another term Vorticity which is defined as twice rotation or

$$\zeta = 2\omega = curl \vec{V}$$ (4.41)

This brings us to a class of flows for which vorticity (i.e., rotation) is zero. These are known as Irrotational Flows. These are governed by the equation,

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney