Question

Why the velocity potential Φ(x,y,z,t) exists only for irrotational flow

Answer 1

`\omega_y,\omega_x,\omega_Z` all are zero.

Step-by-step explanation:

We know that if flow is possible then it will satisfy the below equation

`(\partial u)/(\partial x)+(\partial v)/(\partial y)+(\partial w)/(\partial z)=0`

Where u is the velocity of flow in the x-direction ,v is the velocity of flow in the y-direction and w is the velocity of flow in z-direction.

And velocity potential function
`\phi` given as follows

`u=-(\partial \phi )/(\partial x),v=-(\partial \phi )/(\partial y),w=-(\partial \phi )/(\partial z)`

Rotationality of fluid is given by
`\omega`

`(\partial v)/(\partial x)-(\partial u)/(\partial y)=\omega_Z`

`(\partial v)/(\partial z)-(\partial w)/(\partial y)=\omega_x`

`(\partial w)/(\partial x)-(\partial u)/(\partial z)=\omega_y`

So now putting value in the above equations ,we will find

`\omega =(\partial \phi )/(\partial x),u=(\partial \phi )/(\partial x),`

`\omega_y=(\partial^2 \phi )/(\partial z\partial x)-(\partial^2 \phi )/(\partial z\partial x)`

So
`\omega_y`=0

Like this all
`\omega_y,\omega_x,\omega_Z` all are zero.

That is why velocity potential flow is irroational flow.