<p>This paper investigates the analytical solutions to the relaxed compressible Navier-Stokes equations with spherical symmetry, focusing on the behavior of fluids under the presence of a vacuum free boundary. Under some specific conditions, we show that there exist analytical solutions that meet the continuous density requirement. These solutions are detailed and demonstrate that fluid density and velocity tend to zero everywhere as time approaches infinity. Furthermore, we observe that, unlike the classical compressible Navier-Stokes equations with constant viscosity coefficient, the shear stress tensor can be a non-zero function in the relaxed equations, depending on the choice of initial conditions. This observation underscores the importance of the relaxation effect in fluid dynamics.</p>

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Analytical Solutions to Relaxed Compressible Navier-Stokes Equations with Vacuum Free Boundary

  • Na Wang,
  • Yuxi Hu

摘要

This paper investigates the analytical solutions to the relaxed compressible Navier-Stokes equations with spherical symmetry, focusing on the behavior of fluids under the presence of a vacuum free boundary. Under some specific conditions, we show that there exist analytical solutions that meet the continuous density requirement. These solutions are detailed and demonstrate that fluid density and velocity tend to zero everywhere as time approaches infinity. Furthermore, we observe that, unlike the classical compressible Navier-Stokes equations with constant viscosity coefficient, the shear stress tensor can be a non-zero function in the relaxed equations, depending on the choice of initial conditions. This observation underscores the importance of the relaxation effect in fluid dynamics.