Chapter 2, Problem 2.1P

Consider a body of arbitrary shape. If the pressure distribution over the surface of the body is constant, prove that the resultant pressure force on the body is zero. [Recall that this fact was used in Equation (2.77).]

To determine

To prove:

The resultant force of the constant pressure on the arbitrary body is zero.

Explanation of Solution

Assume the following arbitrary shaped body.

Consider an elemental area ds on the control surface, the pressure acting on the surface is given as Pds.

The net force acting on the body is given as follows:

$\begin{array}{l}\overrightarrow{F}=-{\displaystyle \underset{s}{∯}Pd\overrightarrow{s}}\\ \text{where }\overrightarrow{F}\text{and}\overrightarrow{s}\text{ are the elemental surfaces}\text{.}\end{array}$

The pressure distribution is given as the constant.

$P=P_{\infty }$

Substitute the value of constant pressure $P=P_{\infty }$.

$\begin{array}{l}\overrightarrow{F}=-{\displaystyle \underset{s}{∯}{P}_{\infty }d\overrightarrow{s}}\\ \overrightarrow{F}=-P_{\infty }{\displaystyle \underset{s}{∯}d\overrightarrow{s}}\end{array}$

The surface integral on any closed surface is always zero.

$\begin{array}{l}\overrightarrow{F}=-P_{\infty }\times 0\\ \overrightarrow{F}=0\end{array}$

Therefore, the pressure force acting the arbitrary body under constant pressure is zero.