Chapter 7.1, Problem 1P

To determine

The units of velocity gradient.

Answer to Problem 1P

The unit of velocity gradient is $\frac{1}{s}$.

Explanation of Solution

Write the expression for the velocity gradient.

    $G=\sqrt{\frac{\dot{W}}{\mu \sout{V}}}$                                                              (I)

Here, the power input is $\dot{W}$, the fluid viscosity is $\mu$ and the basin volume is $\sout{V}$.

Conclusion:

Substitute $\text{ft}\cdot \text{lb}/\text{s}$ for $\dot{W}$, $\text{lb}\cdot \text{s}/{\text{ft}}^{\text{2}}$ for $\mu$ and ${\text{ft}}^{\text{3}}$ for $\sout{V}$ in Equation (I).

    $\begin{array}{c}G=\sqrt{\frac{\left(\text{ft}\cdot \text{lb}/\text{s}\right)}{\left(\text{lb}\cdot \text{s}/{\text{ft}}^{\text{2}}\right)\left({\text{ft}}^{\text{3}}\right)}}\\ =\sqrt{\frac{\left(\text{ft}\cdot \text{lb}/\text{s}\right)}{\left(\text{lb}\cdot \text{s}\cdot \text{ft}\right)}}\\ =\sqrt{\frac{1}{\left({\text{s}}^2\right)}}\\ =\frac{1}{\text{s}}\end{array}$

Thus, the unit of velocity gradient is $\frac{1}{s}$.