Well-posed Euler model of shock-induced two-phase flow in bubbly liquid

Abstract

A well-posed mathematical model of non-isothermal two-phase two-velocity flow of bubbly liquid is proposed. The model is based on the two-phase Euler equations with the introduction of an additional pressure at the gas bubble surface, which ensures the well-posedness of the Cauchy problem for a system of governing equations with homogeneous initial conditions, and the Rayleigh–Plesset equation for radial pulsations of gas bubbles. The applicability conditions of the model are formulated. The model is validated by comparing one-dimensional calculations of shock wave propagation in liquids with gas bubbles with a gas volume fraction of 0.005–0.3 with experimental data. The model is shown to provide satisfactory results for the shock propagation velocity, pressure profiles, and the shock-induced motion of the bubbly liquid column.

Appendix

$$\begin{aligned} T= & {} \left( \begin{array}{ccccccc} \rho ^0_2 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \phi ^0_2 &{}\quad 0 \\ -1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \rho ^0_2 \phi ^0_2 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \rho _1 \phi ^0_1 &{}\quad 0 &{}\quad 0\\ \displaystyle {\frac{p^0+(\gamma - 1) p^0_{\mathrm {i}}}{\phi ^0_2 }} &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0&{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0&{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0&{}\quad 1\\ \end{array} \right) \end{aligned}$$

(38)

$$\begin{aligned} A= & {} \left( \begin{array}{ccccccc} \rho ^0_2 u^0_2 &{}\quad 0 &{}\quad 0 &{}\quad \rho ^0_2 \phi ^0_2 &{}\quad 0 &{}\quad u^0_2 \phi ^0_2 &{}\quad 0 \\ -u^0_1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \phi ^0_1 &{}\quad 0 &{}\quad 0\\ p^0 - p^0_{\mathrm {i}} &{}\quad \phi ^0_2 &{}\quad 0 &{}\quad \rho ^0_2 u^0_2 \phi ^0_2 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ p^0_{\mathrm {i}} - p^0 &{}\quad 0 &{}\quad \phi ^0_1 &{}\quad 0 &{}\quad \rho _1 u_1 \phi ^0_1 &{}\quad 0 &{}\quad 0\\ \displaystyle {\frac{(p^0+(\gamma - 1) p^0_{\mathrm {i}}) u^0_2}{\phi ^0_2 }} &{}\quad u^0_2 &{}\quad 0 &{}\quad \gamma p^0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ \displaystyle {u^0_2} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad u^0_2\\ \end{array} \right) \nonumber \\ \end{aligned}$$

(39)

where $p^0 = p^0_2 = p^0_1$, $p^0_{\mathrm {i}} = p^0 + C_\mathrm {s}(\phi ^0_2, \mathrm {Re}_{12}^0) \rho _1 (u^0_1 - u^0_2)^2$, $\displaystyle {R^0 = \left( \frac{3 \phi ^0_2}{4 \pi N}\right) ^{\frac{1}{3}}}$. For matrix Q there are no zero elements only for $\displaystyle {Q_{7,2} = - \frac{1}{\rho _1 R^0}}, \displaystyle {Q_{7,3} = \frac{1}{\rho _1 R^0}}, Q_{6, 7} = - 4\pi N \left( R^0\right) ^2$.

The polynomials are written in the following form using the relations $c^2_2 \rho _2 = p_2 \gamma$ and (2):

$$\begin{aligned} P_1(\lambda )= & {} \rho ^0_2 \big (\varkappa \rho _1(\lambda - u^0_1)^2\phi ^0_2 + \varkappa \rho _1 C_s(\phi ^0_2, \mathrm {Re}_{12}^0) (\Delta ^0)^2 \nonumber \\&\quad \phi ^0_2 \phi ^0_1\left( \lambda - u^0_2 \right) ^2 \big ) \left( (\lambda - u^0_2)^2 - (c^0_2)^2 \right) \end{aligned}$$

(40)

$$\begin{aligned} P_2(\lambda )= & {} \varkappa \rho ^0_2 (\lambda - u^0_2)^2 \gamma p^0_{\mathrm {i}} \phi ^0_1 \end{aligned}$$

(41)

where $\displaystyle {\varkappa = \frac{3 \phi ^0_2}{(R^0\beta )^2\rho _1}}$

(42)