Feasibility analysis of neutral stability for shock waves in a nonideal gas flow about a wedge

Two‐phase states of the generalized van der Waals gas: Conditions of absolute and neutral shock wave stability

“…For state equation (), we can successively get (see Refs. [4, 17]) $$$$\begin{equation} e(V,T)=F(T)-\frac{m+1}{k-1}\frac{A}{(V+C)^{k-1} T^m}, \end{equation}$$$$ $$$$\begin{equation} \begin{split} c^2 &=\frac{V^2{\left(mP-(m+1)RT\big /(V-V_0)\right)}^2}{T{\left(F^{\prime }(T)+\frac{m(m+1)}{k-1}(V+C)(\frac{R}{V-V_0}-\frac{P}{T})\right)}}+\\ &\quad +\frac{RTV^2}{(V-V_0)^2}-\frac{kV^2}{V+C}{\left(\frac{RT}{V-V_0}-P\right)}. \end{split} \end{equation}$$$$Here, $$F(T)$$ is some function.…”

“…In this work, we will deal with generalized van der Waals gas state equation (see Refs. [1, 4, 17]): $$$$\begin{equation} P=\frac{RT}{V-V_0}-\frac{A}{(V+C)^k T^m}, \ V>V_0, \end{equation}$$$$where V 0 , A , k , $$m>0$$, C are some constants, R is a gas constant. Note that for $$k=2$$, $$C=m=0$$, Equation () transitions into a known van der Waals gas state equation, and for $$C=m=0$$, it transitions into the equation from Konukhov et al.…”

“…\end{equation}$$Then (see Refs. [4, 17]), $$$$\begin{equation} \frac{\hat{P}}{\hat{P}_\infty }=1+\tilde{M}^2_\infty \frac{xy}{1+xy},\quad \tilde{M}^2_\infty =M^2_\infty \frac{\hat{c}^2_\infty }{\hat{P}_\infty \hat{V}_\infty }, \end{equation}$$$$ $$$$\begin{equation} \frac{\hat{c}^2_\infty }{\hat{P}_\infty \hat{V}_\infty }=\frac{\Lambda _\infty }{(1-\beta )^2}+ \frac{k}{1+\tilde{\delta }}{\left(1-\frac{\Lambda _\infty }{1-\beta }\right)} \end{equation}$$$$ $$$$\begin{equation} \qquad\qquad\qquad\qquad +\frac{(m-(m+1)\Lambda _\infty /(1-\beta ))^2}{\Lambda _\infty {\left(\frac{1}{\gamma _\i...$$…”

“…We will not present here the numerous calculations results for different gases (they are discussed in detail in Refs. [4, 17]). We will only note that further we will be interested in cases when the matter in the oncoming flow and after the shock wave the following inequalities hold $$\hat{T}_\infty ,\hat{T}<T_{cr}$$.…”

“…the shock wave for the generalized van der Waals gas is absolutely (uniformly) stable, if (see Refs. [4,17])…”