Chapter 7, Problem 1P

To determine

The rate of rotation of a fluid element and the shear strain rate of the element located at y.

Explanation of Solution

Draw the schematic representation of velocity distribution as in Figure (1).

Calculation:

Consider the steady flow of an ideal fluid. According to the velocity profile in Figure 1,

    $\begin{array}{c}\frac{u}{y}=\frac{U}{h}\\ u=\left(\frac{U}{h}\right)y\end{array}$

Here, h is the depth of water below the top plate and U is the constant velocity of the top plate.

The velocity, v, perpendicular to the direction of flow is zero.

  $v=0$

Write the expression for the rate of rotation of a fluid element.

  ${\omega }_z=\frac{1}{2}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)$

Here, ${\omega }_z$ is the average angular velocity of the fluid elements, u and v are the velocity of fluid along the x-direction (along the flow) and y-direction (perpendicular to the direction of flow), respectively, and x and y are the reference axis along the flow and perpendicular to the flow, respectively.

Substitute $\left[\left(Uh\right)/y\right]$ for u and 0 for v.

  $\begin{array}{c}{\omega }_z=\frac{1}{2}\times \left[\frac{\partial }{\partial x}\left(0\right)-\frac{\partial }{\partial y}\left(\frac{U}{h}y\right)\right]\\ =\frac{1}{2}\times \left[0-\left(\frac{U}{h}\times 1\times y^{1-1}\right)\right]\\ =-\frac{U}{2h}\end{array}$

The negative value of the rate of rotation represents an anticlockwise rotation.

  ${\omega }_z=\frac{U}{2h}$

Write the mathematical expression for the rate of shear strain.

  ${\dot{\gamma }}_{xy}=\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}$

  $\begin{array}{c}{\dot{\gamma }}_{xy}=\frac{\partial }{\partial x}\left(0\right)+\frac{\partial }{\partial y}\left(\frac{U}{h}y\right)\\ =0+\left(\frac{U}{h}\times 1\times y^{1-1}\right)\\ =\frac{U}{h}\end{array}$

Therefore, the rate of rotation of a fluid element is $\underset{\_}{U/2h}$ and the shear strain rate of the element located at y is $\underset{\_}{U/h}$.