7B (source code)
= 7B
{scope}
{title2=Fluid Dynamics}
For the <linear planar flow with strain and rotation>, $\partial_xu+\partial_yv=\epsilon-\epsilon=0$, proving <incompressibility>. The <Cartesian streamfunction> convention gives $u=\psi_y$, $v=-\psi_x$, and direct integration gives
$$
\boxed{\psi=\epsilon xy-\frac\gamma2(x^2+y^2)}
$$
up to an irrelevant additive constant. Along a <streamline>, $\mathbf u\cdot\nabla\psi=\psi_y\psi_x-\psi_x\psi_y=0$. Since
$$
4\psi=(\epsilon-\gamma)(x+y)^2-(\epsilon+\gamma)(x-y)^2,
$$
the stated quadratic expression is constant along each streamline. The three cases below distinguish strain, rotation and their degenerate shear limit.
Back to article page
