In fluid dynamics, a stream function is a mathematical function used to describe the flow of an incompressible fluid. It is especially useful in two-dimensional, steady, and incompressible flows.
Definition of Stream Function:
For a two-dimensional flow in the x-y plane, the velocity components (u and v) in the x and y directions can be expressed in terms of the stream function, ψ (psi), as follows:
– u = ∂ψ/∂y
– v = -∂ψ/∂x
where:
– u is the velocity in the x-direction,
– v is the velocity in the y-direction,
– ψ is the stream function.
Properties of the Stream Function:
1. Incompressibility: The continuity equation for incompressible flow (∇·v = 0) is automatically satisfied in terms of the stream function. This means the stream function inherently ensures mass conservation in the fluid flow.
2. Streamlines: The streamlines, which represent the paths followed by fluid particles, are given by the contour lines of the stream function. That is, lines where ψ = constant represent streamlines.
3. Irrotational Flow: If the flow is irrotational, the stream function can also be related to the potential function (φ) via ψ = ∂φ/∂y for irrotational, incompressible flow.
Usage:
– The stream function is especially useful in solving problems involving fluid flow with complex boundary conditions. It simplifies the analysis of the flow, as it directly gives the streamlines, which are crucial for understanding the overall flow behavior.
– The stream function is also used in conjunction with other methods like the velocity potential in solving potential flow problems, especially when analyzing the flow around objects or in specific regions.
Example:
If a stream function is given as ψ(x, y) = x² – y², then:
– u = ∂ψ/∂y = -2y
– v = -∂ψ/∂x = -2x
Thus, in this flow, the velocity components are u = -2y and v = -2x, describing a specific type of flow (possibly a rotation or a type of potential flow). The streamlines would be the curves where x² – y² = constant, representing hyperbolas.
This is just a basic outline of how a stream function is used to describe flow. It plays a crucial role in simplifying the analysis of incompressible flow systems.
