Analytical solutions to the free boundary problem of a two-phase model with radial and cylindrical symmetry

In this paper, we study the free boundary problem of an inviscid two-phase model, where we take the pressure function as $P(n,\rho )={\rho }^{\gamma }+n^{\alpha }$ ($\gamma >1$, $\alpha \ge 1$) with $n$ and $\rho$ being the densities of two phases. First, we construct some self-similar analytical solutions for the $N$-dimensional radially symmetric case by using some ansatzs, and investigate the spreading rate of the free boundary by using the method of averaged quantities. Second, we extend the results of the $N$-dimensional radially symmetric case to the three-dimensional cylindrically symmetric case. Third, we present some analytical solutions for the three-dimensional cylindrically symmetric model with a Coriolis force. From the analytical solutions constructed in this paper, we find that the Coriolis force can prevent the free boundary from spreading out infinitely.