It is defined as a scalar function of space and time such that its negative derivative with respect to any direction gives the fluid velocity in that direction. It is denoted by Φ (Phi).

For steady flow velocity potential Φ = f (x, y, z).

u  = - ∂Φ / ∂x         v  = - ∂Φ / ∂y        w  = - ∂Φ / ∂z

Negative sign indicate that flow take place in the direction in which ( ) decreases

Continuity equation for steady flow,

∂u / ∂x + ∂v / ∂y + ∂w / ∂z  =  0

∂ / ∂x  (- ∂Φ / ∂x) + ∂ / ∂y (∂Φ / ∂y) + ∂ / ∂z  (∂Φ / ∂z)  = 0

∂²Φ / ∂x² + ∂²Φ / ∂y² + ∂²Φ / ∂z²  =  0 is a Laplace equation,

For two-dimensional,

∂²Φ / ∂x² + ∂²Φ / ∂y²  =  0

1.    If velocity potential (Φ ) exist, the flow is irrotational.

2.    If the velocity potential (Φ ) satisfies the Laplace equation, it represents the possible steady incompressible irrotational flow.

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