Analytical Solutions to the Cylindrically Symmetric Compressible Navier–Stokes Equations with Density-Dependent Viscosity and Vacuum Free Boundary

Abstract

In this paper, we investigate the analytical solutions to the cylindrically symmetric compressible Navier–Stokes equations with density-dependent viscosity and vacuum free boundary. The shear and bulk viscosity coefficients are assumed to be a power function of the density and a positive constant, respectively, and the free boundary is assumed to move in the radial direction with the radial velocity, which will affect the angular velocity but does not affect the axial velocity. We obtain a global analytical solution by using some ansatzs and reducing the original partial differential equations into a nonlinear ordinary differential equation about the free boundary. The free boundary is shown to grow at least sub-linearly in time and not more than linearly in time for the analytical solution by using a new averaged quantity.

Appendix

In this appendix, we will give the derivation of system (1.4). In view of

$$\begin{aligned}{} & {} \frac{\partial }{\partial x_{1}}\left( u\frac{x_{1}}{r}-v\frac{x_{2}}{r}\right) +\frac{\partial }{\partial x_{2}}\left( u\frac{x_{2}}{r}+v\frac{x_{1}}{r}\right) +\frac{\partial w}{\partial x_{3}} \nonumber \\{} & {} =u_{r}\frac{x_{1}^{2}}{r^{2}}+u\frac{1}{r}-u\frac{x_{1}^{2}}{r^{3}}-v_{r}\frac{x_{1}x_{2}}{r^{2}}+v\frac{x_{1}x_{2}}{r^{3}}\ \ \ \ \ \ \nonumber \\{} & {} +u_{r}\frac{x_{2}^{2}}{r^{2}}+u\frac{1}{r}-u\frac{x_{2}^{2}}{r^{3}}+v_{r}\frac{x_{1}x_{2}}{r^{2}}-v\frac{x_{1}x_{2}}{r^{3}}\ \nonumber \\{} & {} =u_{r}+\frac{1}{r}u,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{aligned}$$

(A.1)

we get

$$\begin{aligned}{} & {} \textrm{div}(\rho {\textbf{U}})=\nabla \rho \cdot {\textbf{U}}+\rho \textrm{div}{\textbf{U}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\{} & {} =\left( \frac{\partial \rho }{\partial x_{1}},\frac{\partial \rho }{\partial x_{2}},\frac{\partial \rho }{\partial x_{3}}\right) \cdot \left( u\frac{x_{1}}{r}-v\frac{x_{2}}{r},u\frac{x_{2}}{r}+v\frac{x_{1}}{r},w\right) \nonumber \\{} & {} \ \ \ \ \ \ \ \ +\rho \left[ \frac{\partial }{\partial x_{1}}\left( u\frac{x_{1}}{r}-v\frac{x_{2}}{r}\right) +\frac{\partial }{\partial x_{2}}\left( u\frac{x_{2}}{r}+v\frac{x_{1}}{r}\right) +\frac{\partial w}{\partial x_{3}}\right] \nonumber \\{} & {} =(\rho u)_{r}+\frac{1}{r}\rho u.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{aligned}$$

(A.2)

We substitute (A.2) into (1.1) to obtain (1.4)$_{1}$.

By (1.1), we can rewrite (1.2) as

$$\begin{aligned} \rho ({\textbf{U}}_{t}+{\textbf{U}}\cdot \nabla {\textbf{U}})+\nabla P(\rho )-\textrm{div}(\mu (\nabla {\textbf{U}}+\nabla {\textbf{U}}^{T})) -\nabla (\lambda \textrm{div}{\textbf{U}})={\textbf{0}}. \end{aligned}$$

(A.3)

We calculate ${\textbf{U}}\cdot \nabla {\textbf{U}}$, $\nabla P(\rho )$, $\textrm{div}(\mu (\nabla {\textbf{U}}+\nabla {\textbf{U}}^{T}))$ and $\nabla (\lambda \textrm{div}{\textbf{U}})$ as follows.

$$\begin{aligned}{} & {} {\textbf{U}}\cdot \nabla {\textbf{U}}=(\left( u\frac{x_{1}}{r}-v\frac{x_{2}}{r}\right) \left( u_{r}\frac{x_{1}^{2}}{r^{2}} +u\frac{1}{r}-u\frac{x_{1}^{2}}{r^{3}} -v_{r}\frac{x_{1}x_{2}}{r^{2}}+v\frac{x_{1}x_{2}}{r^{3}}\right) \nonumber \\{} & {} \qquad +\left( u\frac{x_{2}}{r}+v\frac{x_{1}}{r}\right) \left( u_{r}\frac{x_{1}x_{2}}{r^{2}}-u\frac{x_{1}x_{2}}{r^{3}}-v_{r}\frac{x_{2}^{2}}{r^{2}} -v\frac{1}{r}+v\frac{x_{2}^{2}}{r^{3}}\right) , \nonumber \\{} & {} \qquad \left( u\frac{x_{1}}{r}-v\frac{x_{2}}{r}\right) \left( u_{r}\frac{x_{1}x_{2}}{r^{2}}-u\frac{x_{1}x_{2}}{r^{3}}+v_{r}\frac{x_{1}^{2}}{r^{2}} +v\frac{1}{r}-v\frac{x_{1}^{2}}{r^{3}}\right) \nonumber \\{} & {} \qquad +\left( u\frac{x_{2}}{r}+v\frac{x_{1}}{r}\right) \left( u_{r}\frac{x_{2}^{2}}{r^{2}}+u\frac{1}{r}-u\frac{x_{2}^{2}}{r^{3}} +v_{r}\frac{x_{1}x_{2}}{r^{2}}-v\frac{x_{1}x_{2}}{r^{3}}\right) , \nonumber \\{} & {} \qquad \left( u\frac{x_{1}}{r}-v\frac{x_{2}}{r}\right) w_{r}\frac{x_{1}}{r}+\left( u\frac{x_{2}}{r}+v\frac{x_{1}}{r}\right) w_{r}\frac{x_{2}}{r})\ \ \nonumber \\{} & {} \quad =\left( uu_{r}\frac{x_{1}}{r}-uv_{r}\frac{x_{2}}{r}-uv\frac{x_{2}}{r^{2}}-v^{2}\frac{x_{1}}{r^{2}},\ \right. \nonumber \\{} & {} \qquad \left. uu_{r}\frac{x_{2}}{r}+uv_{r}\frac{x_{1}}{r}+uv\frac{x_{1}}{r^{2}}-v^{2}\frac{x_{2}}{r^{2}},uw_{r}\right) ,\ \ \ \end{aligned}$$

(A.4)

$$\begin{aligned}{} & {} \nabla P(\rho )=[P(\rho )]_{r}\left( \frac{x_{1}}{r},\frac{x_{2}}{r},0\right) , \end{aligned}$$

(A.5)

the first component of $\textrm{div}(\mu (\nabla {\textbf{U}}+\nabla {\textbf{U}}^{T}))$ is

$$\begin{aligned}{} & {} \frac{\partial }{\partial x_{1}}\left[ \mu \left( 2u_{r}\frac{x_{1}^{2}}{r^{2}}+2u\frac{1}{r}-2u\frac{x_{1}^{2}}{r^{3}}-2v_{r}\frac{x_{1}x_{2}}{r^{2}}+2v\frac{x_{1}x_{2}}{r^{3}}\right) \right] \ \nonumber \\{} & {} \qquad +\frac{\partial }{\partial x_{2}}\left[ \mu \left( 2u_{r}\frac{x_{1}x_{2}}{r^{2}}-2u\frac{x_{1}x_{2}}{r^{3}}+v_{r}\frac{x_{1}^{2}-x_{2}^{2}}{r^{2}}+v\frac{x_{2}^{2}-x_{1}^{2}}{r^{3}}\right) \right] +\frac{\partial }{\partial x_{3}}\left( \mu w_{r}\frac{x_{1}}{r}\right) \nonumber \\{} & {} \quad =\mu _{r}\frac{x_{1}}{r}\left( 2u_{r}\frac{x_{1}^{2}}{r^{2}}+2u\frac{1}{r}-2u\frac{x_{1}^{2}}{r^{3}}-2v_{r}\frac{x_{1}x_{2}}{r^{2}}+2v\frac{x_{1}x_{2}}{r^{3}}\right) \nonumber \\{} & {} \qquad +\mu (2u_{rr}\frac{x_{1}^{3}}{r^{3}}+4u_{r}\frac{x_{1}}{r^{2}}-4u_{r}\frac{x_{1}^{3}}{r^{4}}+2u_{r}\frac{x_{1}}{r^{2}}-2u\frac{x_{1}}{r^{3}}-2u_{r}\frac{x_{1}^{3}}{r^{4}}-4u\frac{x_{1}}{r^{3}}\ \nonumber \\{} & {} \qquad +6u\frac{x_{1}^{3}}{r^{5}}-2v_{rr}\frac{x_{1}^{2}x_{2}}{r^{3}}-2v_{r}\frac{x_{2}}{r^{2}}+4v_{r}\frac{x_{1}^{2}x_{2}}{r^{4}}+2v_{r}\frac{x_{1}^{2}x_{2}}{r^{4}}+2v\frac{x_{2}}{r^{3}}-6v\frac{x_{1}^{2}x_{2}}{r^{5}})\nonumber \\{} & {} \qquad +\mu _{r}\frac{x_{2}}{r}\left( 2u_{r}\frac{x_{1}x_{2}}{r^{2}}-2u\frac{x_{1}x_{2}}{r^{3}}+v_{r}\frac{x_{1}^{2}-x_{2}^{2}}{r^{2}}+v\frac{x_{2}^{2}-x_{1}^{2}}{r^{3}}\right) \ \nonumber \\{} & {} \qquad +\mu (2u_{rr}\frac{x_{1}x_{2}^{2}}{r^{3}}+2u_{r}\frac{x_{1}}{r^{2}}-4u_{r}\frac{x_{1}x_{2}^{2}}{r^{4}}-2u_{r} \frac{x_{1}x_{2}^{2}}{r^{4}}\ \nonumber \\{} & {} \qquad -2u\frac{x_{1}}{r^{3}}+6u\frac{x_{1}x_{2}^{2}}{r^{5}}+v_{rr}\frac{(x_{1}^{2}-x_{2}^{2})x_{2}}{r^{3}}-2v_{r}\frac{x_{2}}{r^{2}}\ \nonumber \\{} & {} \qquad -2v_{r}\frac{(x_{1}^{2}-x_{2}^{2})x_{2}}{r^{4}}+v_{r}\frac{(x_{2}^{2}-x_{1}^{2})x_{2}}{r^{4}}+2v\frac{x_{2}}{r^{3}} +3v\frac{(x_{1}^{2}-x_{2}^{2})x_{2}}{r^{5}})\nonumber \\{} & {} \quad =\mu _{r}(2u_{r}\frac{x_{1}}{r}-v_{r}\frac{x_{2}}{r}+v\frac{x_{2}}{r^{2}}) +\mu (2u_{rr}\frac{x_{1}}{r}+2u_{r}\frac{x_{1}}{r^{2}}-2u\frac{x_{1}}{r^{3}}-v_{rr}\frac{x_{2}}{r}-v_{r}\frac{x_{2}}{r^{2}}+v\frac{x_{2}}{r^{3}}),\nonumber \\ \end{aligned}$$

(A.6)

the second component of $\textrm{div}(\mu (\nabla {\textbf{U}}+\nabla {\textbf{U}}^{T}))$ is

$$\begin{aligned}{} & {} \frac{\partial }{\partial x_{1}}\left[ \mu \left( 2u_{r}\frac{x_{1}x_{2}}{r^{2}}-2u\frac{x_{1}x_{2}}{r^{3}}+v_{r}\frac{x_{1}^{2}-x_{2}^{2}}{r^{2}}+v\frac{x_{2}^{2}-x_{1}^{2}}{r^{3}}\right) \right] \nonumber \\{} & {} \qquad +\frac{\partial }{\partial x_{2}}\left[ \mu \left( 2u_{r}\frac{x_{2}^{2}}{r^{2}}+2u\frac{1}{r}-2u\frac{x_{2}^{2}}{r^{3}}+2v_{r}\frac{x_{1}x_{2}}{r^{2}}-2v\frac{x_{1}x_{2}}{r^{3}}\right) \right] \ \nonumber \\{} & {} \quad =\mu _{r}\frac{x_{1}}{r}\left( 2u_{r}\frac{x_{1}x_{2}}{r^{2}}-2u\frac{x_{1}x_{2}}{r^{3}}+v_{r}\frac{x_{1}^{2}-x_{2}^{2}}{r^{2}}+v\frac{x_{2}^{2}-x_{1}^{2}}{r^{3}}\right) \ \nonumber \\{} & {} \qquad +\mu (2u_{rr}\frac{x_{1}^{2}x_{2}}{r^{3}}+2u_{r}\frac{x_{2}}{r^{2}}-4u_{r}\frac{x_{1}^{2}x_{2}}{r^{4}}-2u_{r}\frac{x_{1}^{2}x_{2}}{r^{4}}-2u\frac{x_{2}}{r^{3}}+6u\frac{x_{1}^{2}x_{2}}{r^{5}} +v_{rr}\frac{(x_{1}^{2}-x_{2}^{2})x_{1}}{r^{3}} \nonumber \\{} & {} \qquad +2v_{r}\frac{x_{1}}{r^{2}}-2v_{r}\frac{(x_{1}^{2}-x_{2}^{2})x_{1}}{r^{4}}+v_{r}\frac{(x_{2}^{2}-x_{1}^{2})x_{1}}{r^{4}}-2v\frac{x_{1}}{r^{3}} +3v\frac{(x_{1}^{2}-x_{2}^{2})x_{1}}{r^{5}}) \nonumber \\{} & {} \qquad +\mu _{r}\frac{x_{2}}{r}\left( 2u_{r}\frac{x_{2}^{2}}{r^{2}}+2u\frac{1}{r}-2u\frac{x_{2}^{2}}{r^{3}}+2v_{r}\frac{x_{1}x_{2}}{r^{2}}-2v\frac{x_{1}x_{2}}{r^{3}}\right) \ \nonumber \\{} & {} \qquad +\mu (2u_{rr}\frac{x_{2}^{3}}{r^{3}}+4u_{r}\frac{x_{2}}{r^{2}}-4u_{r}\frac{x_{2}^{3}}{r^{4}}+2u_{r}\frac{x_{2}}{r^{2}}-2u\frac{x_{2}}{r^{3}}-2u_{r}\frac{x_{2}^{3}}{r^{4}}-4u\frac{x_{2}}{r^{3}}\ \nonumber \\{} & {} \qquad +6u\frac{x_{2}^{3}}{r^{5}}+2v_{rr}\frac{x_{1}x_{2}^{2}}{r^{3}}+2v_{r}\frac{x_{1}}{r^{2}}-4v_{r}\frac{x_{1}x_{2}^{2}}{r^{4}}-2v_{r}\frac{x_{1}x_{2}^{2}}{r^{4}}-2v\frac{x_{1}}{r^{3}}+6v\frac{x_{1}x_{2}^{2}}{r^{5}})\ \nonumber \\{} & {} \quad =\mu _{r}(2u_{r}\frac{x_{2}}{r}+v_{r}\frac{x_{1}}{r}-v\frac{x_{1}}{r^{2}}) +\mu (2u_{rr}\frac{x_{2}}{r}+2u_{r}\frac{x_{2}}{r^{2}}-2u\frac{x_{2}}{r^{3}}+v_{rr}\frac{x_{1}}{r}+v_{r}\frac{x_{1}}{r^{2}}-v\frac{x_{1}}{r^{3}}),\nonumber \\ \end{aligned}$$

(A.7)

the third component of $\textrm{div}(\mu (\nabla {\textbf{U}}+\nabla {\textbf{U}}^{T}))$ is

$$\begin{aligned}{} & {} \frac{\partial }{\partial x_{1}}(\mu w_{r}\frac{x_{1}}{r})+\frac{\partial }{\partial x_{2}}(\mu w_{r}\frac{x_{2}}{r})=(\mu w_{r})_{r}\frac{x_{1}^{2}}{r^{2}}\nonumber \\{} & {} \qquad +\mu w_{r}\left( \frac{1}{r}-\frac{x_{1}^{2}}{r^{3}}\right) +(\mu w_{r})_{r}\frac{x_{2}^{2}}{r^{2}}+\mu w_{r}\left( \frac{1}{r}-\frac{x_{2}^{2}}{r^{3}}\right) \nonumber \\{} & {} \quad =(\mu w_{r})_{r}+\mu w_{r}\frac{1}{r}, \end{aligned}$$

(A.8)

$$\begin{aligned}{} & {} \nabla (\lambda \textrm{div}{\textbf{U}})=\lambda \left( (u_{r}+\frac{u}{r})_{r}\frac{x_{1}}{r},(u_{r}+\frac{u}{r})_{r}\frac{x_{2}}{r},0\right) . \end{aligned}$$

(A.9)

Combining (A.4)–(A.9), we obtain the three components of (1.2) are

$$\begin{aligned}{} & {} \rho \left( u_{t}\frac{x_{1}}{r}-v_{t}\frac{x_{2}}{r}+uu_{r}\frac{x_{1}}{r}-uv_{r}\frac{x_{2}}{r} -uv\frac{x_{2}}{r^{2}}-v^{2}\frac{x_{1}}{r^{2}}\right) +[P(\rho )]_{r}\frac{x_{1}}{r}\nonumber \\{} & {} \qquad -\mu _{r}\left( 2u_{r}\frac{x_{1}}{r} -v_{r}\frac{x_{2}}{r}+v\frac{x_{2}}{r^{2}}\right) \nonumber \\{} & {} \qquad -\mu \left( 2u_{rr}\frac{x_{1}}{r}+2u_{r}\frac{x_{1}}{r^{2}}-2u\frac{x_{1}}{r^{3}}-v_{rr}\frac{x_{2}}{r}-v_{r}\frac{x_{2}}{r^{2}}+v\frac{x_{2}}{r^{3}}\right) \nonumber \\{} & {} \qquad -\lambda \left( u_{r}+\frac{u}{r}\right) _{r}\frac{x_{1}}{r}=0,\ \end{aligned}$$

(A.10)

$$\begin{aligned}{} & {} \rho \left( u_{t}\frac{x_{2}}{r}+v_{t}\frac{x_{1}}{r}+uu_{r}\frac{x_{2}}{r}+uv_{r}\frac{x_{1}}{r}+uv\frac{x_{1}}{r^{2}}-v^{2}\frac{x_{2}}{r^{2}}\right) \nonumber \\{} & {} \qquad +[P(\rho )]_{r}\frac{x_{2}}{r}-\mu _{r}\left( 2u_{r}\frac{x_{2}}{r} +v_{r}\frac{x_{1}}{r}-v\frac{x_{1}}{r^{2}}\right) \nonumber \\{} & {} \qquad -\mu (2u_{rr}\frac{x_{2}}{r}+2u_{r}\frac{x_{2}}{r^{2}}-2u\frac{x_{2}}{r^{3}}+v_{rr}\frac{x_{1}}{r}+v_{r}\frac{x_{1}}{r^{2}}-v\frac{x_{1}}{r^{3}})-\lambda \left( u_{r}+\frac{u}{r}\right) _{r}\frac{x_{2}}{r}=0, \end{aligned}$$

(A.11)

$$\begin{aligned}{} & {} \rho (w_{t}+uw_{r})-(\mu w_{r})_{r}-\mu w_{r}\frac{1}{r}=0, \end{aligned}$$

(A.12)

respectively. By (A.10) and (A.11), we know that

$$\begin{aligned} \left\{ \begin{array}{ll} \left[ \rho \left( u_{t}+uu_{r}-\frac{v^{2}}{r}\right) +P_{r}-\left( 2\mu _{r}u_{r}+2\mu u_{rr}+2\mu u_{r}\frac{1}{r}-2\mu u\frac{1}{r^{2}}\right) -\lambda \left( u_{r}+\frac{u}{r}\right) _{r}\right] \frac{x_{1}}{r}\\ -\left[ \rho \left( v_{t}+uv_{r}+\frac{uv}{r}\right) -\left( \mu _{r}v_{r}-\mu _{r}v\frac{1}{r}+\mu v_{rr}+\mu v_{r}\frac{1}{r}-\mu \frac{v}{r^{2}}\right) \right] \frac{x_{2}}{r}=0,\\ \left[ \rho \left( u_{t}+uu_{r}-\frac{v^{2}}{r}\right) +P_{r}-\left( 2\mu _{r}u_{r}+2\mu u_{rr}+2\mu u_{r}\frac{1}{r}-2\mu u\frac{1}{r^{2}}\right) -\lambda \left( u_{r}+\frac{u}{r}\right) _{r}\right] \frac{x_{2}}{r}\\ +\left[ \rho \left( v_{t}+uv_{r}+\frac{uv}{r}\right) -\left( \mu _{r}v_{r}-\mu _{r}v\frac{1}{r}+\mu v_{rr}+\mu v_{r}\frac{1}{r}-\mu \frac{v}{r^{2}}\right) \right] \frac{x_{1}}{r}=0,\end{array}\right. \nonumber \\ \end{aligned}$$

(A.13)

which leads to (1.4)$_{2}$ and (1.4)$_{3}$ by the Crammer rule.