= Polytropic elliptical patch in a shearing sheet
The steady planar velocity $\mathbf u=\alpha y\mathbf e_x-\beta x\mathbf e_y$ supports a bounded polytropic patch when $\alpha=(\beta^2-3\Omega^2)/\beta$ and $\sqrt3\Omega<\beta<2\Omega$. Its <polytropic enthalpy> is $Q=Q_0-(2\Omega-\beta)(\beta x^2+\alpha y^2)/2$ with $Q_0>0$. Its <fluid free boundary> is the material ellipse $\beta x^2+\alpha y^2=2Q_0/(2\Omega-\beta)$. No additional self-gravity potential is part of this model.
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