Stream function

The continuity equation for an incompressible fluid with velocity vector $v=(v_x,v_y,v_z)$ is $\operatorname{div}(v)=0$, or

$$\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z}=0.$$

For two-dimensional motion in the $(x,y)$-plane, this gives

$$\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}=0,$$

and there is thus a stream function $\psi$ such that

$$v_x=\frac{\partial\psi}{\partial y},\quad v_y=-\frac{\partial\psi}{\partial x}.$$

References

[a1]	"Modern developments in fluid dynamics" S. Goldstein (ed.) , 1 , Dover, reprint (1965) pp. Chapt. III
[a2]	G.K. Batchelor, "An introduction to fluid dynamics" , Cambridge Univ. Press (1967) pp. Chapt. 2.2