A velocity potential function is a mathematical function that describes the velocity field of a fluid flow. It is a scalar function that satisfies the condition of irrotational flow. In other words, it describes a flow field where the fluid particles move without any rotation.


Irrotational flow refers to a flow field where the fluid particles move along smooth paths without any rotation or angular momentum. In such a flow, the vorticity (a measure of rotation) is zero everywhere. This means that the fluid particles move purely in a translational motion without any swirling or rotational motion.


For a flow to be irrotational, the following conditions must be satisfied:
- The flow must be steady, which means that the velocity field does not change with time.
- The fluid must be inviscid, which means that there is no internal friction or viscosity in the fluid.
- The flow must be incompressible, which means that the density of the fluid remains constant.
- The flow must be free from any external moments or torques.


In an irrotational flow, a velocity potential function can be defined. This function, denoted by φ, is a scalar function of the position coordinates (x, y, z) that satisfies the condition of irrotational flow. It can be mathematically expressed as:

∇²φ = 0

Where ∇² is the Laplacian operator, and φ is the velocity potential function.

The significance of the velocity potential function lies in the fact that it allows us to express the velocity components of the flow in terms of a single scalar function. This simplifies the analysis of the flow field and makes it easier to determine various flow parameters.


In summary, a velocity potential function exists only when the flow is irrotational. Irrotational flow refers to a flow field without any rotation or swirling motion. The velocity potential function is a scalar function that satisfies the condition of irrotational flow and allows us to describe the flow field in terms of a single function.