Short Answer:

A velocity potential function is a scalar function used to describe the flow of a fluid, where the velocity components at any point can be obtained by taking the partial derivatives of this function. It is applicable only for irrotational flow, where there is no rotation of fluid particles.

In simple words, the velocity potential function represents a mathematical way to define how fluid velocity changes from one point to another. It is denoted by the Greek letter φ (phi), and its derivatives give the velocity components of the flow in different directions.

Detailed Explanation:

Velocity Potential Function

The velocity potential function is an important concept in fluid mechanics, particularly in the study of irrotational and incompressible fluid flow. It helps describe the velocity field of a fluid mathematically using a single scalar function instead of multiple velocity components.

This function greatly simplifies the analysis of fluid motion because, once it is known, the velocity at any point in the flow can be easily determined by differentiation. The function is applicable to flows where the rotation of fluid particles is absent, that is, in irrotational flow conditions.

1. Definition of Velocity Potential Function

The velocity potential function (φ) is defined as a scalar function of space and time such that the velocity components of the fluid are the negative partial derivatives of φ with respect to the spatial coordinates.

For a three-dimensional flow:

where:

•  = velocity components in the  directions respectively,
•  = velocity potential function.

The negative sign indicates that the velocity is in the direction of decreasing φ. This means that fluid flows from a region of higher velocity potential to a region of lower velocity potential.

2. Physical Meaning of Velocity Potential Function

The physical meaning of the velocity potential function is that it represents the potential energy per unit mass associated with the motion of the fluid. In simpler terms, it gives a way to describe the velocity field without directly dealing with each velocity component.

In a flow field:

• A constant value of φ represents a surface known as an equipotential surface.
• The velocity vector is always perpendicular to the equipotential surface.
• The difference in φ between two points represents the potential change or work done per unit mass in moving from one point to another.

Thus, in regions of high φ, the velocity is lower, and in regions of low φ, the velocity is higher — showing that fluids move from high potential to low potential areas.

3. Mathematical Expression and Derivation

For a two-dimensional flow (x–y plane):

Substituting these in the continuity equation for incompressible flow,

we get,

This equation is known as Laplace’s equation, and it governs the behavior of the velocity potential function in an incompressible, irrotational flow. Any function that satisfies this equation is a valid velocity potential function.

4. Conditions for Existence of Velocity Potential Function

The velocity potential function exists only under specific conditions:

1. Irrotational Flow:
   The rotation or vorticity of the fluid must be zero. For a flow to be irrotational,

where   is the velocity vector.

1. Continuity Equation Must Hold:
   The flow must satisfy the continuity equation for incompressible fluids.
2. Single-Valued Function:
   The velocity potential must have a single value at each point in the flow field, ensuring that velocity is continuous and defined everywhere.

These conditions ensure that the flow can be represented using φ and that the flow field is physically possible.

5. Relationship Between Stream Function and Velocity Potential Function

In two-dimensional irrotational flow, both stream function (ψ) and velocity potential function (φ) exist and are related through the Cauchy–Riemann equations:

This relationship implies that:

• Streamlines (ψ = constant) and equipotential lines (φ = constant) are always perpendicular to each other.
• ψ represents the flow direction, while φ represents the energy potential in the same flow field.

Thus, the combination of both φ and ψ gives a complete description of two-dimensional irrotational flow.

6. Example of Velocity Potential Function

Consider a velocity potential function given by:

Then the velocity components are:

The continuity equation for incompressible flow is:

Hence, this particular φ does not represent an incompressible flow because it does not satisfy the continuity equation.

A valid potential function must always satisfy Laplace’s equation, ensuring that:

7. Importance of Velocity Potential Function

1. Simplifies Flow Analysis:
   It allows the entire velocity field to be described using a single function, reducing the complexity of calculations.
2. Helps Visualize Equipotential Surfaces:
   The equipotential lines provide a clear visualization of how fluid velocity changes across the field.
3. Automatic Continuity Satisfaction:
   If φ satisfies Laplace’s equation, the continuity condition is automatically fulfilled.
4. Useful in Irrotational Flow Problems:
   Used in solving problems of ideal flow, potential flow past bodies, and aerodynamics.
5. Foundation for Complex Potential:
   In two-dimensional analysis, combining φ and ψ forms a complex potential function that helps solve real flow problems using complex mathematics.

Conclusion

In conclusion, the velocity potential function (φ) is a scalar function that defines the velocity field of an irrotational and incompressible fluid flow. The velocity components can be obtained from its partial derivatives, and it always satisfies Laplace’s equation. Equipotential lines represent regions of equal potential, while the flow direction is always perpendicular to them. The velocity potential function is a powerful mathematical tool that simplifies fluid flow analysis, especially in aerodynamics and potential flow studies, providing a complete and elegant representation of irrotational motion.