Analytical solutions to the cylindrically symmetric compressible Navier-Stokes equations with free boundary

summary:The compressible Navier-Stokes equations with Coriolis force are usually used to describe the large-scale flow motions in a thin layer of viscous fluids under the influence of the Coriolis rotational force, such as the motions of some geophysical flows and oceanic flows. We consider the vacuum free boundary problem of this system with cylindrical symmetry. We take the viscosity coefficients as $\mu (\rho )=\rho ^{\theta }$, $\lambda (\rho )=(\theta -1)\rho ^{\theta }$, where $\rho $ denotes the density of the fluid and $\theta $ is a constant. We construct some self-similar analytical solutions when $\gamma =\theta >1$ or $\gamma >1$, $\theta =\frac {1}{2}$, where $\gamma $ is the adiabatic exponent. Compared with the analytical solution to the system without the Coriolis force, the free boundary of solution constructed in this paper does not spread out infinitely in time. This indicates that the Coriolis rotational force plays a crucial role in preventing the free boundary from spreading out. Moreover, when $\theta =1$ and $\gamma =2$, under the stress-free boundary condition, we construct an analytical solution for the problem without the Coriolis force.