In two-dimensional incompressible fluid flow, the continuity equation is essential for ensuring mass conservation, meaning that any change in fluid entering or exiting a region is balanced by a corresponding change elsewhere. For incompressible flow, where density remains constant, this requirement simplifies to the condition that the divergence of the velocity field must be zero. Mathematically, this is expressed as,

where u and v represent the horizontal and vertical velocity components, respectively.

To inherently satisfy this equation, we introduce the stream function ψ, which allows the velocity components to be defined in terms of ψ as:

These definitions automatically satisfy the continuity equation, as the mixed partial derivatives of ψ cancel out.

The stream function ψ is constant along streamlines, which are the paths that fluid particles follow within the flow. Streamlines can therefore be visualized as contour lines of ψ, providing a clear representation of fluid motion. Each streamline is tangent to the velocity vector at any point along its path, illustrating the direction of flow without the need for calculating velocities at each point individually.

Moreover, the difference in ψ\psiψ values between two streamlines represents the volumetric flow rate per unit depth between them, enabling direct calculation of flow rates. This property significantly simplifies the analysis of two-dimensional incompressible flows by removing the need to solve separate equations for u and v. As a result, the stream function ψ is a powerful tool for modeling and interpreting fluid behavior in applications involving incompressible, steady flow.