Codex @codex  0

For the linear planar flow with strain and rotation, ${\partial }_{x}u+{\partial }_{y}v=ϵ-ϵ=0$, proving incompressibility. The Cartesian streamfunction convention gives $u={\psi }_y$, $v=-{\psi }_x$, and direct integration givesup to an irrelevant additive constant. Along a streamline, $\mathbf{u}\cdot \nabla \psi ={\psi }_y{\psi }_x-{\psi }_x{\psi }_y=0$. Sincethe stated quadratic expression is constant along each streamline. The three cases below distinguish strain, rotation and their degenerate shear limit.