Chapter 5.1, Problem 1P

To determine

Use Equation 4.19 and obtain the continuity equation for steady or unsteady incompressible flow through a fixed control volume.

Answer to Problem 1P

The continuity equation for steady or unsteady incompressible flow through a fixed control volume is ${\displaystyle \underset{\text{cs}}{\int }\text{V}}\cdot \widehat{n}dA=0$

Explanation of Solution

Consider the Reynolds transport theorem.

$\frac{D{B}_{sys}}{Dt}=\frac{\partial }{\partial t}{\displaystyle \underset{\text{cv}}{\int }\rho bd}V+{\displaystyle \underset{\text{cs}}{\int }\rho b\text{V}\cdot \widehat{n}dA}$ (I)

Here, control surface is $\text{cs}$, control surface is $\text{cs}$, velocity is $\text{V}$, product of the component of velocity is $V\cdot \widehat{n}dA$ and density is $\rho$.

Substitute $V$ for $B$ and $\frac{1}{\rho }$ for $b$ in Equation (I).

$\begin{array}{c}\frac{D{V}_{sys}}{Dt}=\frac{\partial }{\partial t}{\displaystyle \underset{\text{cv}}{\int }\rho \frac{1}{\rho }d}V+{\displaystyle \underset{\text{cs}}{\int }\rho \frac{1}{\rho }\text{V}\cdot \widehat{n}dA}\\ =\frac{\partial }{\partial t}{\displaystyle \underset{\text{cv}}{\int }d}V+{\displaystyle \underset{\text{cs}}{\int }\text{V}\cdot \widehat{n}dA}\\ =\frac{\partial V_{\text{cv}}}{\partial t}+{\displaystyle \underset{\text{cs}}{\int }\text{V}\cdot \widehat{n}dA}\end{array}$ (II)

For incompressible flow, the volume of the system $V_{sys}$ is constant with time.

$\frac{D{V}_{sys}}{Dt}=0$ (III)

Also for a fixed control volume, $V_{\text{cv}}$ does not change and therefore

$\frac{\partial V_{\text{cv}}}{\partial t}=0$ (IV)

Combine Equations (II), (III) and (IV) to get,

${\displaystyle \underset{\text{cs}}{\int }\text{V}}\cdot \widehat{n}dA=0$

Hence, the continuity equation for steady or unsteady incompressible flow through a fixed control volume is ${\displaystyle \underset{\text{cs}}{\int }\text{V}}\cdot \widehat{n}dA=0$.