Self-similar solutions of the Euler equations with spherical symmetry

Abstract

We consider self-similar flows arising from the uniform expansion of a spherical piston and preceded by a shock wave front. With appropriate boundary conditions imposed on the piston surface and the spherical shock, the isentropic compressible Euler system is transformed into a nonlinear ODE system. We formulate the problem in a simple form in order to present the analytic proof of the global existence of positive smooth solutions.

Introduction

Spherically symmetric motion is important in the theory of explosion. The simplest way to simulate an explosion is to view it as a spherical piston motion pushing out undisturbed gas ahead of it; as a result, shock waves occur. In this paper, we study the spherical, self-similar flow which arises from the uniformly expanding piston and is preceded by a shock front moving with a constant speed. When the spherical piston expands at a constant speed, with the self-similar assumption, the problem can be reduced to that of solving two coupled nonlinear ODEs with appropriate boundary conditions imposed on the piston surface and the spherical shock. The main purpose here is to provide an analytic proof of the global existence of such solutions for the nonlinear system of ODEs.

We consider Euler’s equations with spherical symmetry:

$$\{\begin{array}{c}\frac{\partial \rho }{\partial t}+\frac{\partial \left(\rho u\right)}{\partial r}=-\frac{2\rho u}{r}\text{,}r>0,t>0\text{,}\hfill \\ \frac{\partial \left(\rho u\right)}{\partial t}+\frac{\partial (\rho u^2+P)}{\partial r}=-\frac{2\rho u^2}{r}\text{,}\hfill \end{array}$$

where $\rho ,u$ and $P$ are the density, velocity and pressure of the gas respectively. Assume that $P\left(\rho \right)={\rho }^{\gamma }$ for isentropic gases and $\gamma \in [1,3]$ is the adiabatic constant. And the speed of sound is $c=\sqrt{P^{\prime }\left(\rho \right)}$. For the motions caused by the expansion of a spherical piston into still gas, the velocity of the flow is radial, and its magnitude, like those of the density, pressure, temperature and entropy, depends on the time $t$ and the distance $r$ from the center of the piston. Here, $t$ is chosen to be zero when $r=0$, and the motion is supposed to be so small that only weak shocks are produced; therefore, the changes in entropy are ignored.

There are many studies of spherical shock waves. Self-similar solutions are well introduced and discussed in the books [1], [2], [3], [4]. The nonlinear ODE problem was first proposed and solved numerically by Taylor [5]. He studied (1.1) for isentropic gases. Due to the self-similar motion, $u$ and $P$ are functions of $x=r/t$ only. He introduced the following variables in order to express (1.1) in nondimensional form: $\xi =u/x,\eta =c^2/x^2,z=lnx$. The nonlinear ODEs are obtained to solve for $\eta \left(\xi \right)$ and $z\left(\xi \right)$ numerically. Taylor’s results show that if the radial velocity of the expanding piston is sufficiently large, the thickness of the layer of disturbed air is close to 6% of the radius of the piston. However, his numerical approach did not obtain suitable solutions for when the nondimensional expansion speed is small. Courant and Friedrich [6] investigated the vector field of the ODE system in a different form, to illustrate the solution curves. Lighthill [7] introduced the velocity potential in order to obtain a nonlinear second-order ODE. By ignoring certain terms in the equation, he obtained an approximate relation between the shock Mach number $M$ and the nondimensional piston velocity $\alpha$. Also, local analytic solutions of the ODE system are pursued by many applied mathematicians, and we refer readers to Sachdev [8] for thorough discussions and [9], [10] for related works. However, rigorous analysis of global existence is absent. In this paper, we introduce a suitable variable in terms of which to rewrite the system of nonlinear ODEs and present the global existence of smooth solutions for the first-order ODEs. Due to the effective form, we notice that the system is singular when the shock Mach number $M=1$. In [6], [7], [5], they show (either by numerical solution or by means of the approximate equation) that $M$ is a monotone function of the prescribed piston velocity $\alpha$, and $M$ approaches 1 when $\alpha$ is close to zero.

In this work, we assume that the spherical piston moves outward at a constant speed $c_0\alpha$ and the gas flow is headed by a weak shock moving at a constant speed $c_{0}M$. Here, $M$ is the shock Mach number, $M>1,\alpha$ is the nondimensional piston velocity and $\alpha >0$ is small. The constant $c_0(=\sqrt{P^{\prime }\left({\rho }_0\right)})$ denotes the speed of sound for the undisturbed gas. We introduce a nondimensional variable $\xi$:

$$\xi =\frac{r}{R\left(t\right)}\text{,}$$

where $R\left(t\right)(=c_{0}Mt)$ is the shock radius at time $t$. Notice that $\xi =1$ represents the shock locus. Fig. 1 shows the symbols that we use in this paper. Our goal is to seek the solutions of system (1.1) in the following self-similar form:

$$\{\begin{array}{c}u(r,t)=c_{0}Mf\left(\xi \right)\text{,}\hfill \\ \rho (r,t)={\rho }_{0}h\left(\xi \right)\text{.}\hfill \end{array}$$

By the above assumption of self-similarity, system (1.1) of partial differential equations is transformed into a system of nonlinear ordinary differential equations:

$$\{\begin{array}{c}{f}^{\prime }\left(\xi \right)=\frac{2f{h}^{\gamma -1}}{{\left[M(\xi -f)\right]}^2\xi -h^{\gamma -1}\xi }\text{,}\hfill \\ h^{\prime }\left(\xi \right)=\frac{2M^{2}fh(\xi -f)}{{\left[M(\xi -f)\right]}^2\xi -h^{\gamma -1}\xi }\text{.}\hfill \end{array}$$

For system (1.1), there are two families of elementary waves. Due to the Rankine–Hugoniot condition and the entropy condition [6], we can obtain the flow velocity and density immediately behind the two-shock wave as follows. We need to solve the following equation:

$$\frac{1}{\gamma M}{y}^{\gamma +1}-(\frac{1}{\gamma M}+M)y+M=0\text{.}$$

There is only one root greater than 1. We denote this root by $y_1$ and set the initial condition of system (1.3) as follows:

$$\{\begin{array}{c}f\left(1\right)=1-\frac{1}{y_1}\text{,}\hfill \\ h\left(1\right)=y_1\text{.}\hfill \end{array}$$

Furthermore, the kinematic condition at the piston requires that the flow velocity on the piston surface is the same as the piston velocity, which gives us

$$f\left({\xi }_2\right)={\xi }_2\text{.}$$

We note that ${\xi }_1=1$ and

$${\xi }_2=\frac{\text{ the radius of the piston}}{\text{ the radius of the shock}}=\frac{\alpha }{M}\text{.}$$

Now the problem can be treated as an initial value problem. For a given $\gamma \in [1,3]$ and given $M>1$, we first solve system (1.3) with the initial condition (1.5). Then system (1.3) is integrated from $\xi =1$ backward until the value $\xi ={\xi }_2$ for which (1.6) holds is reached. We obtain the following main result.

Theorem 1.1

Consider the ODE system (1.3) satisfying the conditions (1.5)–(1.6). For $\gamma \ge 1$ with any given $M>1$, there exists a unique positive smooth solution $(f\left(\xi \right),h\left(\xi \right))$ for $\xi \in [{\xi }_2,1]$ . Moreover, $f\left(\xi \right)$ and $h\left(\xi \right)$ are decreasing functions for $\xi \in [{\xi }_2,1]$.

To prove the main theorem, we carefully investigate the common denominator of system (1.3) together with $\xi -f\left(\xi \right)$ to show that the solutions are actually decreasing. The existence and uniqueness of the global solution for $\gamma =1$ is shown in Section 2, and the proof for $\gamma >1$ is in Section 3. The conclusions are finally given in Section 4. We also present the numerical solutions for $f\left(\xi \right)$ and $h\left(\xi \right)$ of system (1.3) in Sections 2 Existence for, 3 Existence for to give a better understanding of the structure of the solution.