Expanding gas clouds of ellipsoidal shape: new exact solutions

Abstract

We consider a model of a freely expanding gas cloud of tri-axial ellipsoidal shape, Gaussian density profile, proposed by Dyson (1968). The ellipsoids are deformable, but are further constrained here to have principal axes maintaining a fixed orientation in space: the study of the more general rotating flows is deferred to a future work. Our main result is that, when the fluid is a monatomic gas with adiabatic index γ = 5/3, the model is completely integrable by quadratures. Solutions starting from a state of rest are describable by elliptic functions; the generic solution however is a more general transcendent that cannot be reduced to elliptic type.The complete integrability of Dyson's model may be ascribed to the fact that it possesses the Painlevé property (Ince 1956; Ablowitz & Segur 1977), meaning, essentially, that the solutions are meromorphic functions of the independent variable, admitting only pole singularities. However, the correct choice of independent variable here is not just the physical time t: rather, it is the thermasy (van Danzig 1939) u = ∫ Tdt, which is one of the potentials occurring in the Clebsch transformation.Further investigation will be required to test Dyson's full ‘spinning gas cloud’ model for an eventual Painlevé property.