The velocity potential is defined as a scalar function of space and time φ(x, y ,z, t) such that its negative derivative with respect to any direction is the fluid velocity in that direction:

The velocity potential function only exists for irrotational flows.

Velocity potential functions are defined for two-dimensional and three-dimensional, steady and unsteady flows. The velocity potential φ is always a scalar.

If for any reason w = ∇ x v = 0 everywhere, then a flow is said to be irrotational or Velocity potential function φ

\(u = - \frac{{\partial \phi }}{{\partial x}};v = - \frac{{\partial \phi }}{{\partial y}};w = - \frac{{\partial \phi }}{{\partial z}}\)

• Flow always occurs in the direction of decreasing potential function.
• Velocity potential function exists only for irrotational flow.

Laplace equation is given by ;

\(\frac{{{\partial ^2}\phi }}{{\partial {x^2}}} + \frac{{{\partial ^2}\phi }}{{\partial {y^2}}} = 0\)

If the velocity potential function satisfies Laplace equation then it also satisfies the continuity equation and hence the flow is possible.