Potential Flow Theory
Let us imagine a pathline of a fluid particle shown in Fig. 20.1.
Rate of spin of the particle is ωz . The flow in which this spin is zero throughout is known as irrotational flow .
For irrotational flows,
For irrotational flows,
Fig 20.1 Pathline of a Fluid Particle
Velocity Potential and Stream Function
Since for irrotational flows .
.
the velocity for an irrotational flow, can be expressed as the gradient of a scalar function called the velocity potential, denoted by Φ
 | (20.2) |
Combination of Eqs (20.1) and (20.2) yields
 | (20.3) |
For irrotational flows
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For two-dimensional case (as shown in Fig 20.1)
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which is again Laplace's equation.
- From Eq. (20.3) we see that an inviscid, incompressible, irrotational flow is governed by Laplace's equation.
- Laplace's equation is linear, hence any number of particular solutions of Eq.(20.3) added together will yield another solution .
- A complicated flow pattern for an inviscid, incompressible, irrotational flow can be synthesized by adding together a number of elementary flows ( provided they are also inviscid, incompressible and irrotational)----- The Superposition Principle
The analysis of Laplace's Eq. (20.3) and finding out the potential functions are known as Potential Flow Theory and the inviscid, incompressible, irrotational flow is often called as Potential Flow .
There are some elementary flows which constitute several complex potential-flow problems.
