Stream function

Stream function is a very useful device in the study of fluid dynamics and was arrived at by the French mathematician Joseph Louis Lagrange in 1781. Of course, it is related to the streamlines of flow, a relationship which we will bring out later. We can define stream functions for both two and three dimensional flows. The latter one is quite complicated and not necessary for our purposes. We restrict ourselves to two-dimensional flows.

Consider a two-dimensional incompressible flow for which the continuity equation is given by,

$${\partial u \over {\partial x}} + {\partial v \over {\partial y}} = 0$$ (4.18)

A stream function $\psi$ is one which satisfies

$$u = {\partial \psi \over {\partial y}} ,    v = -{\partial \psi \over {\partial x}}$$ (4.19)

Substituting these into Eqn.4.18, we have,

$${\partial \over {\partial x}} \left(\partial \psi \over {\partial... ...l \over {\partial y}}\left(-{\partial \psi \over {\partial x}} \right) \equiv 0$$ (4.20)

Thus the continuity equation is automatically satisfied. Thus if we can find a stream function $\psi$ that meets with the eqn.4.19 the continuity equation need not be solved. For the rest of the chapter we will be invariably describing flows with a stream function.

Subsections

• Properties of Stream Function
  • (a) Line$\psi$=constant is a streamline
  • (b) $d\psi$ between two streamlines is proportional to the Volumetric Flow
  
  
• Stream Function in Polar Coordinates

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney