streamfunction

The streamfunction is a concept used in fluid dynamics to describe the flow of an incompressible fluid. It is a scalar function, often denoted by the Greek letter psi (ψ), whose level curves represent the streamlines of the flow. A streamline is a curve that is everywhere tangent to the velocity vector of the fluid. In two-dimensional, incompressible flow, the streamfunction is defined such that the velocity components in the x and y directions, u and v, are related to the partial derivatives of ψ. Specifically, u = ∂ψ/∂y and v = -∂ψ/∂x. This definition ensures that the continuity equation, which states that mass is conserved, is automatically satisfied.

The streamfunction is particularly useful for visualizing fluid flow patterns and for solving certain types of fluid dynamics problems. For example, if the streamfunction is known, the velocity at any point can be readily calculated. Furthermore, the difference in streamfunction values between two streamlines is equal to the volume flow rate per unit depth between those streamlines. This means that streamlines cannot cross each other, as the flow rate between them would be constant. The streamfunction is also related to the vorticity of the flow, with vorticity being proportional to the Laplacian of the streamfunction. Its application extends to various fields, including aerodynamics, hydrodynamics, and meteorology.