Velocity Potential Function
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Velocity Potential Function
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
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
Properties of Potential Function:
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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2. If the velocity potential (Φ ) satisfies the Laplace equation, it represents the possible steady incompressible irrotational flow.
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