Short Answer:

The stream function–potential function relationship explains the connection between two mathematical functions used to describe two-dimensional, incompressible, and irrotational fluid flow. The stream function (ψ) represents lines of constant flow, while the potential function (φ) represents lines of constant velocity potential. Both functions are related through partial derivatives of velocity components, and their curves intersect at right angles in the flow field.

This relationship helps to determine velocity, direction, and flow characteristics without directly measuring the fluid motion. It simplifies the analysis of complex fluid flow problems and is very useful in theoretical fluid mechanics.

Detailed Explanation :

Stream Function–Potential Function Relationship

In two-dimensional steady and incompressible flow, the motion of a fluid can be described using two mathematical functions — the stream function (ψ) and the velocity potential function (φ). Both these functions are tools used to simplify fluid flow analysis and visualize flow patterns without tracing individual fluid particles. The relationship between them provides important information about the nature of the flow, especially when it is irrotational.

1. Stream Function (ψ)

The stream function is a scalar function defined for two-dimensional, incompressible flow. It helps describe the motion of the fluid such that the flow velocity components can be derived from it.

If  and  are the velocity components in the x and y directions respectively, then the stream function is defined as:

In this equation:

•  is constant along a streamline.
• The difference in stream function values between two streamlines gives the flow rate per unit width between them.

For incompressible flow, this definition ensures that the continuity equation is automatically satisfied:

Hence, the stream function provides a convenient way to visualize flow and calculate flow rates.

2. Velocity Potential Function (φ)

The velocity potential function is another scalar function used to describe the flow field. It is applicable when the flow is irrotational, meaning there is no rotation of fluid particles about their axes.

It is defined such that the velocity components are the negative partial derivatives of φ:

For irrotational flow, the condition of zero rotation (vorticity = 0) must be satisfied:

Thus, φ exists only when the flow is irrotational. The surfaces of constant φ are called equipotential lines.

3. Relationship Between Stream Function and Potential Function

In two-dimensional, steady, incompressible, and irrotational flow, there exists a direct relationship between the stream function (ψ) and potential function (φ). The velocity components are common to both definitions, so:

From the stream function:

From the potential function:

By comparing these two sets of equations, we get:

These relations are known as the Cauchy–Riemann equations in fluid mechanics.

4. Orthogonality of Streamlines and Equipotential Lines

From the Cauchy–Riemann relations, it can be shown that:

and

The product of these two slopes equals -1, which indicates that streamlines and equipotential lines are orthogonal (perpendicular) to each other.
Therefore, at every point in the flow field:

This orthogonality is a key characteristic of potential flow fields and helps in graphical flow representation.

5. Laplace’s Equation for φ and ψ

Both φ and ψ satisfy the Laplace equation, which confirms that they are harmonic functions.

For φ:

For ψ:

Since both satisfy Laplace’s equation, they are related through the Cauchy–Riemann conditions and can be expressed as conjugate harmonic functions. This means if one function is known, the other can be determined mathematically.

6. Physical Significance of the Relationship

• The potential function φ gives information about the magnitude and direction of the velocity field.
• The stream function ψ gives information about the flow pattern and streamlines.
• The relationship between φ and ψ helps in completely defining a two-dimensional incompressible, irrotational flow.
• Their orthogonality ensures that no fluid crosses an equipotential line; hence, flow always occurs tangentially to streamlines.

This relationship is used in aerodynamics, hydrodynamics, and fluid simulation to solve for velocity and pressure distribution without directly using complex fluid dynamic equations.

Conclusion

The stream function–potential function relationship forms the mathematical foundation for analyzing ideal fluid flow. The two functions are related through the Cauchy–Riemann equations, and their curves — streamlines and equipotential lines — always intersect at right angles. Both functions satisfy Laplace’s equation and are harmonic in nature, making them extremely valuable in solving two-dimensional flow problems. This relationship provides a clear and simple method to represent incompressible, irrotational flow fields graphically and analytically.