Chapter 3, Problem 3.1P

For an irrotational flow. show that Bernoulli’s equation holds between any points in the flow, not just along a streamline.

To determine

To Show: Bernoulli’s equation for irrotational flow.

Answer to Problem 3.1P

$\text{ p+}\frac{1}{2}\rho V^2=\text{constant}$

Explanation of Solution

Given Information:

Flow type is irrotational.

Calculation:

Considering steady and inviscid flow:

$\begin{array}{l}x\text{momentum:}\rho u\frac{\partial u}{\partial x}+\rho v\frac{\partial u}{\partial y}+\rho w\frac{\partial u}{\partial z}=-\frac{\partial p}{\partial x}\left(1\right)\\ y\text{momentum:}\rho u\frac{\partial v}{\partial x}+\rho v\frac{\partial v}{\partial y}+\rho w\frac{\partial v}{\partial z}=-\frac{\partial p}{\partial y}\left(2\right)\\ z\text{momentum}:\rho u\frac{\partial w}{\partial x}+\rho v\frac{\partial w}{\partial y}+\rho w\frac{\partial w}{\partial z}=-\frac{\partial p}{\partial z}\left(3\right)\end{array}$ ss

Multiplying (1), (2) and (3) by $dx,dy\text{and}dz\text{respectively}$

$\begin{array}{l}u\frac{\partial u}{\partial x}dx+v\frac{\partial u}{\partial y}dx+w\frac{\partial u}{\partial z}dx=-\frac{1}{\rho }\frac{\partial p}{\partial x}dx\left(4\right)\\ u\frac{\partial v}{\partial x}dy+v\frac{\partial v}{\partial y}dy+w\frac{\partial v}{\partial z}dy=-\frac{1}{\rho }\frac{\partial p}{\partial y}dy\left(5\right)\\ u\frac{\partial w}{\partial x}dz+v\frac{\partial w}{\partial y}dz+w\frac{\partial w}{\partial z}dz=-\frac{1}{\rho }\frac{\partial p}{\partial z}dz\left(6\right)\end{array}$

Now, adding equation 4,5 and 6

$u\left(\frac{\partial u}{\partial x}dx+\frac{\partial v}{\partial x}dy+\frac{\partial w}{\partial x}dz\right)+v\left(\frac{\partial u}{\partial y}dx+\frac{\partial v}{\partial y}dy+\frac{\partial v}{\partial x}dy\right)+w\left(\frac{\partial u}{\partial z}dx+\frac{\partial v}{\partial z}dy+\frac{\partial w}{\partial x}dz\right)=-\frac{1}{\rho }\left(\frac{\partial p}{\partial x}dx+\frac{\partial p}{\partial y}dy+\frac{\partial p}{\partial z}dz\right)\left(7\right)$

We know that for irrotational flow $\nabla \times V=0$

$hence\frac{\partial w}{\partial y}=\frac{\partial v}{\partial z};\frac{\partial u}{\partial z}=\frac{\partial w}{\partial x};\frac{\partial v}{\partial x}=\frac{\partial u}{\partial y}\left(8\right)$

Substituting Equation (8) into (7) we get

$udu+vdv+wdw=-\frac{1}{\rho }dp$

$\begin{array}{l}\frac{1}{2}d\left(u^2+v^2+w^2\right)=-\frac{1}{2}d\left(V^2\right)=VdV=-\frac{1}{\rho }d\rho \\ dp=-\rho VdV\text{which}\text{integrates}\text{to }\\ \text{ p+}\frac{1}{2}\rho V^2=\text{constant}\end{array}$