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Potential Flow Theory

Incompressible flow is a constant density flow.

Let us visualize a fluid element of defined mass, moving along a streamline in an incompressible flow.

Due to constant density, we can write

Irrotational Flow

If the fluid element does not rotate as it moves along the streamline, or to be precise, if its motion is translational (and deformation with no rotation) only, the flow is termed as irrotational.

The rate of rotation of the fluid element can be measured as the average rate of rotation of two perpendicular line segments.

The average rate of rotation ω_z about z-axis is expressed in terms of the gradients of velocity components as

Similarly, the other two components of rotation are

ω_x, ω_y and ω_z are components of

In a two-dimensional flow, ω_z is the only non-trivial component of the rate of rotation called in-plane component of vorticity and computed as

Thus for irrotational flow, vorticity is zero i.e.

Potential Flow Theory

Let us imagine a path line of a fluid particle shown in Fig. 1.

Rate of spin of the particle is ω_z. The flow in which this spin is zero throughout is known as irrotational flow .

For irrotational flows,

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


Combination of Eqs (1) and (2) yields

For irrotational flows



For two-dimensional case (as shown in Fig 1)

which is again Laplace's equation.

From Eq. (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. (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. (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.