A two-dimensional incompressible velocity field can be written either as , or with both signs reversed. The continuity equation then holds identically.
For axisymmetric incompressible flow without swirl, and . These identities enforce incompressibility, and measures axial volume flux from the symmetry axis when its additive constant is chosen there.
For , the stream function gives hyperbolic streamlines and a saddle when , while gives elliptical streamlines and a centre.
The incompressible linear flow has streamfunction . Its velocity matrix squares to . Thus strain-dominated flow has saddle trajectories, rotation-dominated flow has closed elliptical trajectories, and equal nonzero strain and rotation gives simple shear flow along parallel lines with a stationary line. Pure rotation is counterclockwise for positive .
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A stream function is a mathematical tool used in fluid mechanics to describe the flow of incompressible fluids. It is a scalar function whose contours represent the flow lines of the fluid. When the flow is two-dimensional, the stream function can help visualize the flow, as the flow velocity components can be obtained from this function. ### Key Characteristics of Stream Functions: 1. **Incompressible Flow**: Stream functions are primarily used for incompressible flow scenarios.