2. Physical description

The physical set-up of the problem (see figure 1) consists of a diverging cylindrical piston (Rosenau & Frankenthal Reference Rosenau and Frankenthal1976a ) in a vdW gas under the solid body rotation. The piston may represent an inner expanding surface in an explosion, leading to a disturbance. Under the explosion, a finite amount of energy is released along the axis of the cylindrical piston in the $(r,\theta , z)$ coordinate system that creates an outward-propagating cylindrical shock. The shock represents a simplified model of a rapid pressure pulse in a gaseous medium. The ambient gas is at rest in the radial direction while it moves with variable tangential velocity. Under the effect of viscosity and heat conduction, one-dimensional, unsteady and adiabatic flow of a vdW gas behind the shock is investigated. The normal and tangential viscous stresses are expressed by Newton’s law of viscosity for cylindrical symmetric flow of the gas. The heat conduction flux is taken in terms of Fourier’s law of heat conduction. The vdW gas (vdW equations of state) incorporates pressure and volume corrections to account for intermolecular attraction and the finite size of molecules. These corrections enable realistic modelling of high pressure and moderate temperature flows where deviations from ideal-gas behaviour are significant (Anderson Reference Anderson1989; Emelyanov, Karpenko & Volkov Reference Emelyanov, Karpenko and Volkov2019; Gopal et al. Reference Gopal, Volpiani, Yellapantula and Larsson2021). The system undergoes a solid body rotation with constant angular velocity $\varOmega$ , a common and physically reasonable approximation for preshock equilibrium states where the gas rotates uniformly under centrifugal–pressure balance (Fraenkel Reference Fraenkel1959; Chaturani Reference Chaturani1971; Hopfinger Reference Hopfinger1992; Greitzer, Tan & Graf Reference Greitzer, Tan and Graf2004). It captures the dominant rotational effects without introducing unnecessary complexity. The shock is considered to be isothermal. This assumption is a limiting case of a highly conductive shock and is valid when the relaxation time for temperature is much shorter than the hydrodynamic time scale. Gravitational, magnetic and radiative effects are neglected for simplicity. The proposed physical set-up is justifiable with the following numerical and experimental work on shock wave.

The inclusion of vdW effects and rotation is supported by the molecular dynamic simulations and analytical findings of Liu et al. (Reference Liu, Liu, Fu and Zhou2021), who demonstrated that the pressure correction term enhances compressibility in spinning gases. Moreover, experimental investigations by Duan et al. (Reference Duan, Xiao, Gong, Li, Zeng, Gao and Sun2019) confirm that heat significantly modifies postshock flow properties in high-pressure environments, providing indirect validation for the present dissipative model. Experimental studies on rotating detonations (Kindracki et al. Reference Kindracki, Wolański and Gut2011) and supersonic viscous flows (Kovacs et al. Reference Kovacs, Passaggia, Mazellier and Lago2024) further support the physical feasibility of combining rotation, viscosity and heat conduction in cylindrical shock configurations. Furthermore, Lei Zhao (Reference Zhao, Ma, Chen, Zhang, Bian and Sun2026) experimentally and numerically shows that in nozzle flows, the back pressure significantly affects the shock position, intensity and Mach number. The basic equations and jump conditions over the physical problem are described in the next section.

3. Basic equations and jump conditions

The motivation behind this study is to investigate the effect of viscosity and heat conduction on a diverging cylindrical shock wave in a vdW gas having solid body rotation. The equations of continuity, momentum and energy can be expressed (Fraenkel Reference Fraenkel1959; Chaturani Reference Chaturani1971; Hopfinger Reference Hopfinger1992; Levin & Skopina Reference Levin and Skopina2004; Papanastasiou, Georgiou & Alexandrou Reference Papanastasiou, Georgiou and Alexandrou2021) as

(3.1) \begin{align} & \frac {\partial \rho }{\partial t} +u_{r}\frac {\partial \rho }{\partial r} + \rho \frac {\partial u_{r}}{\partial r} +\frac {\rho u_{r}}{r} =0, \end{align}

(3.2) \begin{align} & \frac {\partial u_r }{\partial t } + u_r\frac {\partial u_r }{\partial r } + \frac {1}{\rho }\frac {\partial p }{\partial r } - \frac {1}{\rho }\frac {\partial \tau _{\textit{rr}}}{\partial r}-\frac {2\mu }{r\rho }\frac {u_{r}}{r}+\frac {2\mu }{\rho r}\frac {\partial u_{r}}{\partial r}-\frac {{u_{\theta }}^2}{r}=0 , \end{align}

(3.3) \begin{align} & \frac {\partial u_{\theta }}{\partial t} +u_{r}\frac {\partial u_{\theta }}{\partial r}-\frac {1}{\rho }\frac {\partial \tau _{r\theta }}{\partial r} -\frac {2\tau _{r\theta }}{\rho r}+\frac {u_{r}u_{\theta }}{r}=0, \end{align}

(3.4) \begin{align} & \frac {\partial e }{\partial t } + u_{r}\frac {\partial e}{\partial r} - \frac {p }{\rho ^2}\biggl (\frac {\partial \rho }{\partial t} + u_{r}\frac {\partial \rho }{\partial r}\biggr ) - \frac {\tau _{\textit{rr}}}{\rho }\Big [\frac {\partial u_{r}}{\partial r}+\frac {u_{r}}{r}\Big ]-\frac {\tau _{r\theta }}{\rho }\Big [-\frac {u_{\theta }}{r}+\frac {\partial u_{\theta }}{\partial r}\Big ] \nonumber\\ & \quad -\frac {2\mu }{\rho }\Big [\Big (\frac {u_{r}}{r}\Big )^2 -\frac {u_{r}}{r}\frac {\partial u_{r}}{\partial r}\Big ] +\frac {1}{\rho }\frac {\partial q }{\partial r}+ \frac {q}{\rho r}=0, \end{align}

where $r$ is space coordinate, $t$ is time, $\rho$ is density, $u_{r}$ is radial component, $u_{\theta }$ is tangential component of velocity, $p$ is pressure, $\tau _{\textit{rr}}$ is normal viscous stress, $\tau _{r\theta }$ is tangential viscous stress and $q$ is heat flux. The governing equations are written in an inertial frame, so no Coriolis or centrifugal terms are required; only the standard geometric term $\rho u^2_{\theta }/r$ accounts for centripetal effects. This approach is consistent with earlier studies on rotating gases (Levin & Skopina Reference Levin and Skopina2004; Nath & Sahu Reference Nath and Sahu2017). For system level rotation having constant angular velocity $\varOmega$ , the tangential component of velocity $u_{\theta }$ in (3.2), (3.3) and (3.4) is given by

(3.5) \begin{equation} u_{\theta }=\varOmega r, \end{equation}

where $r$ is the radial distance from the axis of symmetry. The adopted system level rotation corresponds to an intermediate swirl strength neither so large as to become comparable to the speed of sound, which would render the assumption non-physical, nor so weak as to have a negligible effect on the shock dynamics. Hence, this idealized rotation profile provides a simplified but physically consistent model for analysing the influence of ambient swirl on cylindrical shock propagation. In this case, the vorticity vector $\boldsymbol{\zeta }=({1}/{2}) \boldsymbol{\nabla }\times \boldsymbol{q}$ has the following component:

(3.6) \begin{equation} \zeta _r=0,\quad \zeta _{\theta }=0,\quad \zeta _z = \frac {1}{2r} \frac {\partial (ru_{\theta })}{\partial r}= \varOmega , \end{equation}

which is found to be constant. The normal viscous stress $\tau _{\textit{rr}}$ in (3.2) and (3.4), and tangential viscous stress $\tau _{r\theta }$ in (3.3) and (3.4), play a crucial role in the rate of deformation of the viscous gas flow (Singh & Patel Reference Singh and Patel2019; Papanastasiou et al. Reference Papanastasiou, Georgiou and Alexandrou2021; Brutyan & Ibragimov Reference Brutyan and Ibragimov2022). In the present one-dimensional viscous flow under cylindrical symmetry, the viscous stress components $\tau _{\textit{rr}}$ , $\tau _{r\theta }$ and $\tau _{\theta \theta }$ are taken as

(3.7) \begin{equation} \tau _{\textit{rr}} =\mu \left [\frac {4}{3}\frac {\partial u_r}{\partial r}-\frac {2}{3}\frac {u_{r}}{r}\right ]\!,\quad \tau _{r\theta }= \mu \left [\frac {\partial u_{\theta }}{\partial r}-\frac {u_{\theta }}{r}\right ]\!, \quad \tau _{\theta \theta }= \mu \left [\frac {4}{3}\frac {u_{r}}{r}-\frac {2 }{3}\frac {\partial u_r}{\partial r}\right ]\!, \end{equation}

where $\mu$ is the coefficient of viscosity. The normal stress component $\tau _{\theta \theta }$ in (3.7) will be used to write (3.2) in conservation form in (A19) and (A21). These normal stresses act to compress or expand a fluid element, and are pivotal to the rate of strain analysis in the fluid flow. The viscosity coefficient $\mu$ in (3.7) is assumed to follow a power law in density and temperature (Chapman & Cowling Reference Chapman and Cowling1970; Singh & Patel Reference Singh and Patel2019) as

(3.8) \begin{equation} \mu = \mu _{0} \left (\frac {T}{T_{0}}\right )^{a_{c}} \left (\frac {\rho }{\rho _0}\right )^{b_{c}}, \end{equation}

where $\mu _0, \rho _0, T_0$ are dynamic viscosity, density and temperature at the reference state. The $\mu _0= 1.85\times 10^{-5}\,\text{kg}\, \text{m}^{-1} \text{s}^{-1}$ and $\rho _0= 1.184\, \text{kg}\, \text{m}^{-3}$ at the temperature $298.15$ K for air as a reference medium. The exponents $a_c$ and $b_c$ are taken in its most general form to see dependency of viscosity on temperature and density in the vdW gas. These exponents are selected to ensure consistency with the physical conditions and the nature of the desired similarity solution. The heat flux $q$ in the energy equation (3.4) is governed by the following Fourier’s law (Singh & Patel Reference Singh and Patel2019) of heat conduction:

(3.9) \begin{equation} q=-K \frac {\partial T}{\partial r}, \end{equation}

where $T$ is the absolute temperature and $K$ is the thermal conductivity coefficient that vary in power of temperature and density as (Chapman & Cowling Reference Chapman and Cowling1970; Ghoniem et al. Reference Ghoniem, Kamel, Berger and Oppenheim1982)

(3.10) \begin{equation} K=K_{0}\left (\frac {T}{T_0}\right )^{\beta _c}\left (\frac {\rho }{\rho _0}\right )^{\delta _c}, \end{equation}

where $K_0$ is the thermal conductivity coefficient of reference state. For the reference state as an air, $K_0 =2.62\times 10^{-2}\,\text{kg}\,\text{m}^{-1}\,\text{s}^{-1}\,\text{K}^{-1}$ at a temperature of $298.15 \, \text{K}$ . The exponents $\beta _c$ and $\delta _c$ are chosen in a way that it is compatible with the condition of the problem and required solution. The variation of viscosity and thermal conductivity with temperature and density corresponds to a change in collision frequency and effective mean free path due to the intermolecular force between the molecules of the vdW gas.

For the study of shock propagation in a non-ideal gas, we have assumed that the gas obeys the vdW equation of state of the form (Huete et al. Reference Huete, Velikovich, Martinez-Ruiz and Calvo-Rivera2021; Calvo-Rivera et al. Reference Calvo-Rivera, Huete and Velikovich2022)

(3.11) \begin{equation} p= \frac {\rho R^* T}{(1-b\rho )} -a\rho ^2, \end{equation}

where $R^*$ is the specific gas constant. Here $R^* = 287\,\text{m}^2\,\text{s}^{-2}\,\text{K}^{-1}$ corresponds to a reference state of air at $298.15$ K. The parameter $a$ is the correction due to the intermolecular attraction between the molecules of the gas and $b$ is the correction in the volume of the gas. The vdW constants $a$ and $b$ are evaluated using pseudocritical properties (Monteiro Medeiros et al. Reference Monteiro Medeiros, de Mendonça Silva, do Nascimento Silva and Mendes2020). For the air, the value of the vdW constants are $a = 1.622\times 10^2 \,\text{m}^5\, \text{kg}^{-1}\, \text{s}^{-2}$ and $b= 1.309\times 10^{-3}\,\text{kg}^{-1}\,\text{m}^3$ at critical temperature $T_c = 132.2\,\text{K}$ and pressure $p_c=37.45\times 10^5\, \text{Pa}$ . The rotation modes are completely active since the characteristic rotational temperature for nitrogen is $2.88$ K, for oxygen is $2.07$ K and for carbon dioxide is $91.5$ K (Khapra & Patel Reference Khapra and Patel2022). The vibrational and electronic excitation, dissociation and ionization are crucial in true high temperature flow conditions. But, they become active only when the temperature reaches $800$ K, $2000$ K, $9000$ K, respectively (Anderson Reference Anderson1989; Emelyanov et al. Reference Emelyanov, Karpenko and Volkov2019; Gopal et al. Reference Gopal, Volpiani, Yellapantula and Larsson2021). Therefore, vibration, dissociation and ionization effects are not considered in the present study. For $a=0$ , (3.11) reduces to a simplified vdW equation of state $p= {\rho R^* T}/{(1-b\rho )}$ (Roberts & Wu Reference Roberts and Wu1996; Patel & Singh Reference Patel and Singh2019), and for $a=0, b=0$ , to an ideal gas equation of state $ p=\rho R^* T$ . The specific internal energy $e$ per unit mass of the gas and speed of sound $c$ are expressed as a function of the gas pressure and density as (Huete et al. Reference Huete, Velikovich, Martinez-Ruiz and Calvo-Rivera2021; Calvo-Rivera et al. Reference Calvo-Rivera, Huete and Velikovich2022)

(3.12) \begin{equation} e = \frac { \big(p+a\rho ^2\big)(1-b\rho )}{ \rho (\gamma - 1)}-a\rho ,\quad c^2=\frac {\gamma \big(p+a\rho ^2\big)}{\rho (1-b\rho )}-2a\rho . \end{equation}

For $a=0$ , the specific internal energy $e=p(1-b\rho )/\rho (\gamma -1)$ and the speed of sound $c =\sqrt {{\gamma p}/{\rho (1-b\rho )}}$ correspond to the simplified vdW gas (Patel & Singh Reference Patel and Singh2019). For ideal gas ( $a=b=0$ ), the simple manipulation of the (3.12) yields well-known specific internal energy $e=p/\rho (\gamma -1)$ and the speed of sound $ c=\sqrt {\gamma p/\rho }$ (Patel & Pandey Reference Patel and Pandey2024).

The goal of present problem is to study the propagation of a diverging cylindrical shock wave through a van ver Waals gas undergoing a solid body rotation of constant angular velocity $\varOmega$ . Therefore, the flow variables just ahead of the shock front are taken as (Laumbach & Probstein Reference Laumbach and Probstein1969; Chaturani Reference Chaturani1971; Levin & Skopina Reference Levin and Skopina2004; Nath & Sahu Reference Nath and Sahu2017)

(3.13) \begin{equation} u_{r_1}=0;\quad u_{\theta _1}= \varOmega R;\quad q_1 =0,\quad \tau _{rr_1}=0;\quad \tau _{{r\theta }_1}= 0, \end{equation}

where the subscript ` $1$ ’ indicates the value just ahead of the shock front and $R$ is radius of the shock front. The ambient density $\rho _1$ and shock radius $R$ are not assumed in advance but derived in § 4 through a self-similar solution as intermediate asymptotics.

Following the integral form of conservation laws (A22–A25) given in Appendix A.2, ambient condition (3.13) and (3.8), we have derived the following Rankine–Hugoniot conditions for isothermal shock in a rotating vdW gas under viscous stress and heat conduction (Whitham Reference Whitham1999; Bajargaan & Patel Reference Bajargaan and Patel2018; Patel & Pandey Reference Patel and Pandey2024; Patel & Garg Reference Patel and Garg2024):

(3.14) \begin{align} \rho _1 U &= \rho _2 (U - u_{r_2}), \end{align}

(3.15) \begin{align} p_1 + \rho _{1} U^2 &= p_2 + \rho _{2} (U - u_{r_2})^2 - \tau _{rr_2}, \end{align}

(3.16) \begin{align} u_{\theta _1} &= u_{\theta _2}+\frac {\tau _{{r\theta }_2}}{\rho _1 U}, \end{align}

(3.17) \begin{align} e_1 + \frac {p_1}{\rho _1} + \frac {U^2}{2} -\frac {u_{\theta _1}^2}{2}+ \frac {q_2}{\rho _1 U} &= e_2 + \frac {p_2}{\rho _2} + \frac {(U-u_{r_2})^2}{2}-\frac {u_{\theta _2}^2}{2} - \frac {\tau _{rr_2}}{\rho _2}, \end{align}

(3.18) \begin{align} T_1 = T_2, \end{align}

(3.19) \begin{align} \mu _1 &= \mu _2\Big (\frac {\rho _1}{\rho _2}\Big )^{b_c}, \end{align}

where $U(t)= {\textrm d}R/{\textrm d}t$ is the shock velocity and suffixes $1$ and $2$ denote the value of the flow variables just ahead and behind the shock wave. Following Zel’dovich & Raizer (Reference Zel’dovich and Raizer1966) and Rosenau & Frankenthal (Reference Rosenau and Frankenthal1976b , Reference Rosenau and Frankenthal1978), we have adopted isothermal shock conditions (3.18) as the limiting case where the temperature remains continuous across the shock surface, while a discontinuity appears in the temperature gradient. This arises naturally from the Fourier law of heat conduction (3.9), which excludes a finite temperature jump at the discontinuity. Here the isothermal shock arises in media where thermal conduction or radiative cooling is so efficient that the gas temperature remains essentially constant across the discontinuity. Further, the shock is considered to be isothermal due to the viscous mechanism that converts a portion of kinetic energy of the gas entering into the discontinuity into the heat. Such shocks are particularly relevant in thermally conducting astrophysical flows (e.g. solar wind conduction-driven shocks), laboratory plasmas, moist atmospheric shocks and cooled engineering flows. The introduction of $\tau _{rr_2}$ , $\tau _{{r\theta }_1}$ and $\tau _{{r\theta }_2}$ in jump conditions (3.15)–(3.17) and (3.8) is a novel idea not known in the literature to investigate the effect of viscosity on the shock propagation.

4. Self-similar solution as intermediate asymptotics

Following the concept of intermediate asymptotics introduced by Zel’dovich & Raizer (Reference Zel’dovich and Raizer1966), Rosenau & Frankenthal (Reference Rosenau and Frankenthal1976a ), Barenblatt (Reference Barenblatt1996) and Barenblatt & Zel’dovich (Reference Barenblatt and Zel’dovich2003), we introduce the following similarity transformation to obtain self-similar solutions of (3.1–3.4), (3.7) and (3.9) in terms of the similarity variable $\eta$ :

(4.1) \begin{equation} \eta = \frac {r}{R(t)},\quad \eta _p\leqslant \eta \leqslant 1, \, r_p\leqslant r \leqslant R(t), \end{equation}

where $\eta = 1$ at the shock front and $\eta = \eta _p$ at piston. The shock radius $R(t)$ and ambient density $\rho _1$ are chosen as the basic scales, while the shock velocity $\dot {R}= {{\textrm d}R}/{{\textrm d}t}$ provides the characteristic velocity scale. Accordingly, $\rho _1{\dot {R}}^{2}$ and $\rho _1{\dot {R} }^{3}$ serve as the characteristic scales for pressure (and viscous stresses) and heat flux, respectively. This scaling remains general, as any numerical constants can be taken into the similarity functions. Using these scales, we have introduced the following similarity transformation (Patel & Pandey Reference Patel and Pandey2024) for the solution of (3.1–3.4), (3.7) and (3.9) as

(4.2) \begin{align} u_{r} &=\dot {R}V(\eta ),\quad u_{\theta }=\dot {R}W(\eta ),\quad \rho =\rho _{1}D(\eta ),\quad p=\rho _{1}{\dot {R}}^2P(\eta ), \end{align}

(4.3) \begin{align} \tau _{\textit{rr}} &=\rho _{1}{\dot {R}}^2{\mathcal{T}}_{\textit{rr}}(\eta ),\quad \tau _{r\theta }=\rho _{1}{\dot {R}}^2{\mathcal{T}}_{r\theta }(\eta ),\quad q=\rho _{1}{\dot {R}}^3Q(\eta ), \end{align}

where $V$ , $D$ , $P$ , ${\mathcal{T}}_{\textit{rr}}$ , ${\mathcal{T}}_{r\theta }$ and $Q$ are the functions of non-dimensional similarity variable $\eta$ . Changing the independent variables $r \,\text{and}\, t$ to $\eta$ using

(4.4) \begin{equation} \frac {\partial }{\partial r}= \frac {1}{R}\frac {\partial }{\partial \eta }, \frac {\partial }{\partial t}= \frac {\partial }{\partial t}- \eta \frac {\dot {R}}{R}\frac {\partial }{\partial \eta }, \end{equation}

where $\dot {R}$ denotes the shock velocity $U= {{\textrm d}R}/{{\textrm d}t}$ , (3.1–3.4), (3.7) and (3.9) are transformed into the following:

(4.5) \begin{align} & \frac {\dot {\rho _1}}{\rho _1}D +\frac {\dot {R}}{R}\left [ (V-\eta )\frac {{\textrm d}D}{{\textrm d}\eta }+D\frac {{\textrm d}V}{{\textrm d}\eta }+\frac {VD}{\eta }\right ]=0, \end{align}

(4.6) \begin{align} & \frac {R\ddot {R}}{\dot {R}^2}V+\frac {{\textrm d}V}{{\textrm d}\eta }(V-\eta )+\frac {1}{D}\frac {{\textrm d}P}{{\textrm d}\eta }-\frac {1}{D}\frac {{\textrm d}{\mathcal{T}}_{\textit{rr}}}{{\textrm d}\eta }-\frac {W^2}{\eta }\nonumber \\ &\quad +\frac {2\mu _1}{\rho _1R\dot {R}D} \frac {(\gamma M^2)^{a_c}\left (P+ \frac {\bar {a} D^2}{\gamma M^2}\right )^{a_c}(1-\bar {b}D)^{a_c} D^{b_c -a_c}}{(1+\bar {a})^{a_c}(1-\bar {b})^{a_c}}\left [\frac {V}{\eta ^2}-\frac {1}{\eta }\frac {{\textrm d}V}{{\textrm d}\eta }\right ]=0, \end{align}

(4.7) \begin{align} & \frac {R\ddot {R}}{\dot {R}^2}W +\frac {{\textrm d} W}{{\textrm d} \eta }(V-\eta )+\frac {WV}{\eta }-\frac {1}{D}\left [\frac {{\textrm d}{\mathcal{T}}_{r\theta }}{{\textrm d}\eta }+\frac {2{\mathcal{T}}_{r\theta }}{\eta }\right ]=0, \end{align}

(4.8) \begin{align} & (1-\overline {b}D)\frac {{\textrm d} P}{{\textrm d}\eta }-\frac {P}{D}\frac {{\textrm d}D}{{\textrm d}\eta }\left [\gamma -\frac {\overline {a}(1-2\overline {b}D)D^2}{\gamma M^2P}+\frac {\overline {a}(\gamma -1)D^2}{\gamma M^2 P}\right ]\nonumber \\& +\frac {\gamma -1}{V-\eta }\Big [-\frac {\dot {\rho _1} R}{\rho _1 \dot {R}}P-{\mathcal{T}}_{\textit{rr}}\left (\frac {{\textrm d}V}{{\textrm d}\eta }+\frac {V}{\eta }\right )-{\mathcal{T}}_{r\theta }\left (-\frac {W}{\eta }+\frac {{\textrm d}W}{{\textrm d}\eta }\right )\nonumber \\& -\frac {2\mu _1}{\rho _1 R\dot {R}}\frac {(\gamma M^2)^{a_c}\left (P+ \frac {\bar {a} D^2}{\gamma M^2}\right )^{a_c}(1-\bar {b}D)^{a_c} D^{b_c -a_c}}{(1+\bar {a})^{a_c}(1-\bar {b})^{a_c}}\left (\frac {V^2}{\eta ^2}-\frac {V}{\eta }\frac {{\textrm d}V}{{\textrm d}\eta }\right )\frac {{\textrm d}Q}{{\textrm d}\eta }+\frac {Q}{\eta }\Big ]\nonumber \\& +\frac {R\ddot {R}}{\dot {R}^2}\frac {2}{(V-\eta )}\left [P(1-\overline {b}D)+\frac {\overline {a}D^2(1-\overline {b}D)}{\gamma M^2}-\overline {a}(\gamma -1)D^2\right ]=0, \end{align}

(4.9) \begin{align} & {\mathcal{T}}_{\textit{rr}}= \frac {\mu _1}{\rho _1 R \dot {R} }\frac {(\gamma M^2)^{a_c}\left (P+ \frac {\bar {a}D^2}{\gamma M^2}\right )^{a_c}(1-\bar {b}D)^{a_c} D^{b_c -a_c}}{(1+\bar {a})^{a_c}(1-\bar {b})^{a_c}}\left [\frac {4}{3}\frac {{\textrm d}V}{{\textrm d}\eta }-\frac {2}{3}\frac {V}{\eta }\right ], \end{align}

(4.10) \begin{align} & {\mathcal{T}}_{r\theta }= \frac {\mu _1}{\rho _1 R \dot {R}}\frac {(\gamma M^2)^{a_c}\left (P+ \frac {\bar {a}D^2}{\gamma M^2}\right )^{a_c}(1-\bar {b}D)^{a_c} D^{b_c -a_c}}{(1+\bar {a})^{a_c}(1-\bar {b})^{a_c}}\left [-\frac {W}{\eta }+\frac {{\textrm d} W}{{\textrm d}\eta }\right ], \end{align}

(4.11) \begin{align} & Q(\eta )= -\frac {K_0 \rho ^{\delta _c -1}_1 \dot {R}^{2\beta _c-1}}{{R^*}^{\beta _c+1}T^{\beta _c}_0\rho ^{\delta _c}_0 R} D^{\delta _c-\beta _c}(1-\overline {b}D)^{\delta _c}\left (P+\frac {\overline {a}D^2}{\gamma M^2}\right )^{\beta _c} \nonumber\\ & \quad \left [\frac {(1-\overline {b}D)}{D}\frac {{\textrm d}P}{{\textrm d}\eta }+\frac {{\textrm d}D}{{\textrm d}\eta }\left (-\frac {P}{D^2}+\frac {\overline {a}(1-2\overline {b}D)}{\gamma M^2}\right )\right ], \end{align}

where $M=\sqrt {{\rho _1{\dot {R}}^2}/{\gamma p_1}}$ is shock Mach number, non-idealness parameters $\bar {a}= {a\rho ^2_1}/{p_1}$ corresponds to pressure correction $a$ , and $\bar {b}= \rho _1b$ corresponds to volume correction $b$ in the vdW gas, ${\mu _1}/{\rho _1R \dot {R}}$ in equations (4.6), (4.8), (4.9) and (4.10) is the inverse of shock Reynolds number $\textit{Re}_s$ to describe the effect of viscosity, and $({K_0 \rho ^{\delta _c -1}_1 \dot {R}^{2\beta _c-1}})/({{R^*}^{\beta _c+1}T^{\beta _c}_0\rho ^{\delta _c}_0 R})$ in (4.11) is the heat conductive parameter $\varGamma _c$ to see the effect of heat conduction. Here $\mu _1$ , $\rho _1$ , $R$ and $\dot {R}$ corresponds to the coefficient of viscosity, density of the gas, characteristic length and characteristic velocity of the medium (Zel’dovich & Raizer Reference Zel’dovich and Raizer1966). For the existence of a self-similar solution, the governing system of (4.5)–(4.11) must be independent of time $t$ . This will lead the self-similar solution as a limiting case of an intermediate asymptotic solution which does not significantly depend on initial/boundary conditions of the problem but evolves as the interaction of convection dissipation and conduction processes in the intermediate range of the flow field. This leads the coefficients in the equations (4.6)–(4.11) to be constant. Therefore,

(4.12) \begin{equation} \frac {\ddot {R}R}{\dot {R}^2} = \text{constant}, \end{equation}

which yields the shock radius $R(t)$ in the following general form:

(4.13) \begin{equation} R = \begin{cases} A t^{\alpha }, & \text{for constant} \neq 1, \\[6pt] B^*e^{\sigma t}, & \text{for constant} = 1, \end{cases} \end{equation}

where $A$ , $B^*$ and $\sigma$ are the dimensional constants, and $\alpha$ is a real number. Depending on the constant $\neq 1$ , $\alpha$ can be positive, negative or zero, but for constant $= 1$ , $\sigma$ is either positive or negative.

Although the majority of the problems have a power law character, the present study is focused on the analysis of the exponential shock due to its application in gas dynamic, physical and biological systems. An explosion leading to a strong disturbance of radius varying with time in a gas, ignition delay in combustion and propulsion systems, and arterial pulse in blood flow are some of the examples in fluid flow with exponential shock. Therefore, in the present problem, we consider diverging exponential shock wave of radius $R(t)$ given by

(4.14) \begin{equation} R= B^*\exp (\sigma t), \sigma \gt 0, \end{equation}

where the constant $B^*$ is a dimensional characteristic of dimension of $R$ at $t=0$ related to various physical sources where the sudden expansion occurs such as the outer layer of the star (stellar corona) or a condensed explosive of the charged particle or a diaphragm containing a highly pressurized gas or a pressure pulse wave in an artery.

One of the essential requirements for solving the present gas flow problem is to provide a crucial boundary condition on the gas flow. The motion of the entire system with exponential shocks in (4.14) is assumed to be self-similar, therefore, the propagation of the piston or inner expanding surface generated by the explosion must follow an exponential law as

(4.15) \begin{equation} r_p = B\exp (\sigma t), \quad \sigma \gt 0, \end{equation}

where $r_p$ is radius of the piston, $B$ is a dimensional constant that depends on $B^*$ and non-dimensional radius of the piston (see (4.16) below). The constants $ B$ and $B^*$ in (4.13) and (4.14) are dependent on each other and the following relation can be obtained using (4.1):

(4.16) \begin{equation} \eta _p = \frac {B}{B^*}. \end{equation}

Similar boundary-driven similarity formulations for axisymmetric shock propagation have been developed in magnetohydrodynamics by Rosenau & Frankenthal (Reference Rosenau and Frankenthal1976a ) and Zel’dovich & Raizer (Reference Zel’dovich and Raizer1966), within the Barenblatt–Zel’dovich framework of intermediate asymptotics, where the shock evolution is determined by invariance of the governing equations (see (4.12) and (4.17)) and kinematic compatibility (see (4.50)) rather than by conservation of a finite explosion energy (Ling & Balachandar Reference Ling and Balachandar2018). In the present study, the non-idealness, viscosity and heat conduction effects enter parametrically without introducing additional macroscopic length scales.

For investigation of a self-similar solution as a limit of intermediate asymptotics, the requirement of $ {\dot {\rho _1}}/{\rho _1}\propto {\dot {R}}/{R}$ in (4.5) leads to the ambient density

(4.17) \begin{equation} \rho _1 = \rho _0 \exp {(\delta t)}, \end{equation}

where $\rho _0$ and $\delta$ are the dimensional constants. The value of density exponent $\delta$ depends on the pressure and volume correction in the vdW gas (see (4.24)). Here the effect of variable ambient density is crucial, as it significantly influences shock propagation, wave interactions and distribution of flow variable behind the shock wave in practical gas dynamics and blast wave (Kumar, Singh & Kumar Reference Kumar, Singh and Kumar2016; Lei Zhao Reference Zhao, Ma, Chen, Zhang, Bian and Sun2026). The velocity of a shock whose radius varies exponentially with time as in (4.14) is written as

(4.18) \begin{equation} \dot {R}= U = \sigma R(t). \end{equation}

By using the ambient density $\rho _1$ , ambient conditions in (3.13) into (3.2), we obtain the ambient pressure gradient as

(4.19) \begin{equation} \frac {{\textrm d}p_1}{{\textrm d}R} =\rho _1 \varOmega ^2 R. \end{equation}

Equation (4.19) represents the classical equilibrium state of a rotating medium, where radial pressure gradient is exactly balanced by centrifugal force (Fraenkel Reference Fraenkel1959; Chaturani Reference Chaturani1971). Substituting the expressions for the shock radius $R(t)$ and ambient density $\rho _1$ into (4.19), we obtain ambient pressure as

(4.20) \begin{equation} p_1= \frac {\rho _0\varOmega ^2\sigma R^{\frac {\delta +2\sigma }{\sigma }}}{(2\sigma +\delta ){B^*}^{\frac {\delta }{\sigma }}}, \quad 2\sigma +\delta \neq 0. \end{equation}

Now, we investigate the conditions under which shock Mach number $M$ , heat conductive parameters $\varGamma _c$ , non-idealness parameters $\bar {a}$ and $\bar {b}$ , and shock Reynolds number $\textit{Re}_s$ in the system of (4.6)–(4.11) must be constant for the similarity solution.

Using (4.17)–(4.18) and (4.20), the shock-Mach number $M$ can be written as

(4.21) \begin{equation} M^2=\frac {\rho _1 \dot {R}^2}{\gamma p_1}=\frac {\sigma (\delta +2\sigma )}{\gamma \varOmega ^2}, \end{equation}

which is dimensionless constant. The existence of a constant Mach number in the present work is opposite to the variation of Mach number as a function of shock radius in the Chester–Chisnell–Whitham theory. The shock Mach number $M$ depends on the ratio $\gamma$ of specific heats, exponent of ambient density $\delta$ , exponent of shock radius $\sigma$ and constant angular velocity $\varOmega$ of the solid body rotation. From (4.10) and (4.18), the heat conductive parameter $\varGamma _c$ is a function of shock radius $R$ and shock velocity $U$ as

(4.22) \begin{equation} \varGamma _c = \frac {K_0\sigma \rho ^{\delta _c -1}_1 U^{2\beta _c-2}}{{R^*}^{\beta _c+1}T^{\beta _c}_0\rho ^{\delta _c}_0}, \end{equation}

which is independent of time $t$ if $\delta _c = 1$ and $ \beta _c = 1$ . Thus we have

(4.23) \begin{equation} \varGamma _c = \frac {K_0 \sigma }{T_0\rho _0 R^{{*}^{2}}}, \end{equation}

where $\varGamma _c$ is the non-dimensional heat conductive parameter that depends on the thermal conductivity $K_0$ and shock radius exponent $\sigma$ .

The pressure correction $\bar {a}$ and volume correction $\bar {b}$ depend on shock radius $R$ as

(4.24) \begin{equation} \bar {a}= \frac {a\rho ^2_1}{p_1} = \frac {a(2\sigma +\delta )\rho _0 R^{\frac {(\delta -2\sigma )}{\sigma }}}{{B^*}^{\delta /\sigma } \varOmega ^2 \sigma } \quad \text{and} \quad \bar {b}= b\rho _1 = b\rho _0\exp (\delta t)= \frac {\rho _0 R^{\frac {\delta }{\sigma }}}{{B^*}^{\frac {\delta }{\sigma }}}b. \end{equation}

Now, $\bar {a}$ is constant (dimensionless) when $\delta =2\sigma$ ( $\sigma \gt 0$ ) and $\bar {b}$ is constant (dimensionless) when $\delta =0$ . Therefore, from (4.17), the solution of the present problem in vdW gas exists under pressure and volume corrections separately. Under the above conditions, the dimensionless pressure correction $\bar {a}$ and volume correction $\bar {b}$ become

(4.25) \begin{equation} \bar {a}= \frac {4\rho _0 a}{(B^*\varOmega )^2}, \, \bar {b} = \rho _0 b. \end{equation}

The pressure correction $\bar {a}$ not only depends on reference ambient density $\rho _0$ and vdW pressure correction $a$ but also on the initial ambient tangential velocity $B^* \varOmega$ of the reference state. The vdW constants $a=162.1 \text{m}^5\, \text{kg}^{-1}\text{s}^{-2}$ and $b=1.309\times 10^{-3}\text{kg}\,\text{m}^{-3}$ corresponds to the air ( $\rho _0 =1.184\,\text{kg}\,\text{m}^{-3}$ ) at their critical temperature and pressure (Monteiro Medeiros et al. Reference Monteiro Medeiros, de Mendonça Silva, do Nascimento Silva and Mendes2020) for the reference state. The $\bar {a}=0.0036$ and $\bar {a}=0.013$ correspond to the ambient tangential velocity of $465\, \text{m}\text{s}^{-1}$ of air at the Earth’s surface and $240\text{m}\,\text{s}^{-1}$ at the surface of Mars. The ambient tangential velocity of $280\, \text{m}\text{s}^{-1}$ and $350\, \text{m}\text{s}^{-1}$ of the reference state corresponds to the pressure correction of $\bar {a}=0.01$ and $\bar {a}=0.0063$ , respectively. The reference state density is $\rho _0 = 1.184\,\text{kg}\,\text{m}^{-3}$ , which corresponds to the volume correction $\bar {b}=0.0015$ .

Using (3.11), (4.17), (4.18) and (4.20) shock Reynolds number $\textit{Re}_s$ can be written as

(4.26) \begin{equation} \textit{Re}_s = \frac {\rho _1\dot {R}R}{\mu _1}= \frac {{(R^* T_0(2\sigma +\delta ))}^{a_c}\sigma {\rho _0}R^{\frac {-\delta (b_c -1)+ 2\sigma (1-a_c)}{\sigma }}}{\mu _0(\varOmega ^2\sigma )^{a_c}(1+\bar {a})^{a_c}(1-\bar {b})^{a_c}{B^*}^{\frac {-\delta (b_c -1)}{\sigma }}}. \end{equation}

Equations (4.24) and (4.26) show that similar to the Chester–Chisnell–Whitham theory, the pressure correction $\bar {a}$ , volume correction $\bar {b}$ and shock Reynolds number $\textit{Re}_s$ depend on the shock curvature in the non-self-similar solution. For self-similar solution, $\textit{Re}_s$ must be constant. Therefore, the shock-Reynolds number $\textit{Re}_s$ becomes constant under the condition $-(b_c-1)\delta +(1- a_c)2\sigma =0$ and is given by

(4.27) \begin{equation} \textit{Re}_s = \frac {{(R^* T_0(2\sigma +\delta ))}^{a_c}\sigma {\rho _0}}{\mu _0(\varOmega ^2\sigma )^{a_c}(1+\bar {a})^{a_c}(1-\bar {b})^{a_c}{B^*}^{\frac {-\delta (b_c -1)}{\sigma }}}. \end{equation}

Equation (4.27) indicates that the dimensionless shock Reynolds number $\textit{Re}_s$ depends on the reference density, temperature and viscosity, as well as on the non-idealness parameters $\bar {a},\, \bar {b}$ , the viscosity exponents $a_c, \, b_c$ , the exponents associated with the ambient density and shock radius, the rotational velocity and the initial shock radius $B^*$ . Therefore, the shock Reynolds number is a characteristic parameter of viscous shock dynamics for a rotating vdW gas.

Using $\tau _{r\theta }$ from (3.7) and shock Reynolds number $\textit{Re}_s= \rho _1 R \dot {R}/\mu _1$ , solving (3.16) for $u_{\theta _2}$ , we obtain

(4.28) \begin{equation} \frac {u_{\theta _2}}{R} = \frac {u_{\theta _1}}{R} +\frac {C}{R^{\textit{Re}_s \beta ^{b_c}}}, \end{equation}

where $C$ is the constant of integration. Under the solid-body rotation, this (4.28) leads to

(4.29) \begin{equation} u_{\theta _2} = u_{\theta _1}. \end{equation}

Using the equation of state (3.11), the shock Reynolds number $\textit{Re}_{s}$ , shock Mach number $M$ and (3.19), the flow variables just behind the shock front are obtained by solving (3.14)–(3.18) as

(4.30) \begin{align} & \rho _{2}=\frac {\rho _{1}}{\beta }, \, u_{2}=(1-\beta )U,\, p_{2}= \rho _{1}U^2\left [1-\beta + \frac {1}{\gamma M^2}+\frac {2(1-\beta )}{3 R_{es} \beta ^{b_c}}\right ], \, u_{\theta _2}= u_{\theta _1}, \end{align}

(4.31) \begin{align} & \tau _{rr_2}= \frac {2(1-\beta )\rho _{1}U^2}{3\textit{Re}_{s}\beta ^{b_c}},\, \tau _{r\theta _2}= 0, \end{align}

(4.32) \begin{align} & q_{2} =\rho _{1}U^3 \left [\frac {\overline {b}(1-\beta )+2\overline {a}\beta -\overline {a}\overline {b}(1+\beta )}{\gamma M^2(\beta -\overline {b})}-\frac {2\overline {a}}{\gamma M^2 \beta }+\frac {\beta ^2-1}{2}-\frac {2\beta (1-\beta )}{3\textit{Re}_{s}\beta ^{b_c}}\right ]. \end{align}

The $\beta$ in the above equations is given by

(4.33) \begin{align} & \beta ^4 +\beta ^{4-b_c}\left (\frac {2}{3 \textit{Re}_{s}}\right ) -\beta ^3\left (1+\frac {1}{\gamma M^2}+ \bar {b}\right )- \beta ^{3- b_c}\frac {2}{3 \textit{Re}_{s}} (1+\bar {b}) \nonumber \\ &\quad +\, \beta ^2\left (\bar {b}\left (1+\frac {1}{\gamma M^2} \right ) + \frac {(1+\bar {a})(1-\bar {b})}{\gamma M^2}\right )+ \beta ^{2-b_c}\frac {2}{3\textit{Re}_{s}}\bar {b}-\frac {\beta \bar {a}}{\gamma M^2}+\frac {\bar {a}\bar {b}}{\gamma M^2}=0. \end{align}

The density $\beta = \rho _2/\rho _1$ depends on non-idealness parameters $\bar {a}$ and $\bar {b}$ , shock Mach number $M$ , shock Reynolds number $\textit{Re}_s$ , ratio of specifics heats $\gamma$ and exponent $b_c$ . For different values of the parameter $\bar {a}$ , $\bar {b}$ , $M$ , $\textit{Re}_s$ , $\gamma$ and $b_c$ within the physical limit, (4.33) gives four different values of $\beta$ out of which we take only one which lies within the required range $0\lt \beta \lt 1$ satisfying the physical limit of the present study. The governing equations (4.30)–(4.33) differ from those obtained by Patel & Pandey (Reference Patel and Pandey2024) only through the multiplicative coefficient of the shock Reynolds number $\textit{Re}_s$ for non-rotating ideal gas ( $\overline {a},\, \overline {b}=0, \, b_c =0$ ). This difference arises due to the geometric effects in the divergence of the velocity and the associated viscous stresses. In non-viscous flow ( $\textit{Re}_s \rightarrow \infty$ ), these system equations ((4.30)–(4.33)) coincide with Nath & Sahu (Reference Nath and Sahu2017) in absence of radiation for $\bar {a}=0$ . These comparisons justify the generalization of the jump conditions (4.30)–(4.33) for the viscous flow of a rotating non-ideal gas behind exponential cylindrical shock wave.

The propagation of shock causes a sudden change in thermodynamic flow variables. Two important characteristics associated with shock propagation are shock compression and shock strength. The relative change in density is known as shock compression and is defined as the ratio of the density increase across the shock front to the ambient density. From (4.30), the shock compression $C_s$ is derived as

(4.34) \begin{equation} C_{s} = \frac {\rho _2 -\rho _1}{\rho _1} = \frac {1-\beta }{\beta }, \end{equation}

where $\beta = \rho _1/\rho _2$ is the density ratio of the gas across the shock front. It plays an important role in various physical processes, including the formation of stars from interstellar gas, achieving the high densities required to initiate nuclear fusion in inertial confinement fusion fuel pellets. Shock strength, on the other hand, quantifies the relative increase of the pressure across the shock front to the ambient pressure. On using (4.30) and (3.19), the strength $Z_s$ can be derived as

(4.35) \begin{equation} Z_s = \frac {p_2 - p_1}{p_1} = \gamma M^2 (1-\beta )\left [1-\frac {2}{3 \textit{Re}_s \beta ^{b_c}}\right ]. \end{equation}

The strength $Z_s$ depends on density ratio $\beta$ , shock Mach number $M$ and shock Reynolds number $\textit{Re}_s$ , and reduces to $ \gamma M^2 (1-\beta )$ for non-viscous flow ( $\textit{Re}_s \to \infty$ ) (Whitham Reference Whitham1999; Anderson Reference Anderson2021). The shock strength characterizes blast pressure from explosions, thermal loads and aerodynamic drag on high-speed vehicles, structural stresses during re-entry, energy release and transfer in detonation waves, and in shock wave lithotripsy, etc.

The total energy $E$ of the flow field behind the shock is described by the following equation:

(4.36) \begin{equation} E = E_0\exp (\lambda t) ; \quad \lambda \gt 0, \end{equation}

where $E_0$ and $\lambda$ are constants. The positive values of $\lambda$ correspond to a class of solutions where the total energy increases exponentially as time progresses.

The total energy of the disturbance between the cylindrical piston and the corresponding cylindrical shock wave of radius $R(t)$ is given by

(4.37) \begin{equation} E = 2\pi \int _{r_p}^{R}\rho \left [ e +\frac {u_r^2}{2}+\frac {u_{\theta }^2}{2}\right ] r{\textrm d}r, \end{equation}

where $r_p$ is the radius of the piston or inner expanding surface. Using (3.12), the self-similarity transformations (4.2)–(4.3), and the relation (4.18) in (4.37), we get

(4.38) \begin{equation} E = 2\pi \rho _0\sigma ^2 B{^*}^{3}N \exp (\delta +4\sigma )t = 2\pi \rho _0 \sigma ^2 B{^*}^{\delta /\sigma }N( R(t))^{\delta /\sigma +4} , \end{equation}

where

(4.39) \begin{equation} N = 2\pi \int _{\eta _p}^{1} \left [\frac {(P+\bar {a}/\gamma M^2 D^2)(1-\overline {b}D)}{(\gamma -1)}-\frac {\bar {a}}{\gamma M^2} D^2+ \frac {V^2 D}{2}+\frac {W^2 D}{2}\right ]\eta {\textrm d}\eta , \end{equation}

where $\eta _p\leqslant \eta \leqslant 1$ , $\eta _p = {r_p}/{R(t)}$ at the piston. From (4.36) and (4.38), we obtain $ \lambda =\delta +4\sigma$ , $\delta \geqslant 0$ , $\sigma \gt 0$ , demonstrating that the total energy is not constant. Equation (4.38) indicates that the total energy of the gas between piston and shock is proportional to the $(\delta /\sigma +4)^{th}$ power of the shock radius $R(t)$ . For the constant ambient density $(\delta =0)$ in volume-corrected vdW gas $(\bar {a}=0)$ , this condition coincides with Nath & Sahu (Reference Nath and Sahu2017) and Bajargaan & Patel (Reference Bajargaan and Patel2018). This comparison shows the generalization of the total energy of the disturbance in vdW gas. The effect of $\bar {a}$ and $\bar {b}$ in (4.38) and (4.39) are not known in the earlier literature.

The present similarity is not governed by energy conservation as in the Taylor–Sedov blast wave and the second-type explosion similarity discussed by Waxman & Shvarts (Reference Waxman and Shvarts1993), or by the far and near field energy-based scaling employed for finite sources by Ling & Balachandar (Reference Ling and Balachandar2018). Instead, the similarity here is determined by kinematic compatibility and invariance of the governing equations, consistent with the Zel’dovich & Raizer (Reference Zel’dovich and Raizer1966) and Barenblatt (Reference Barenblatt1996) concept of intermediate asymptotics for boundary-driven flows and the piston-driven similarity formulations of Rosenau & Frankenthal (Reference Rosenau and Frankenthal1976a ).

The increase in total energy may happen due to the pressure extended on the gas flow by the piston. This situation may appear during the formation of cylindrical spark channels in the exploding wires. The time-dependent energy input in the usual cases of spark breakdown is a more realistic assumption than the instantaneous increase of energy input (Freeman & Craggs Reference Freeman and Craggs1969; Nath & Sahu Reference Nath and Sahu2017). The energy addition due to pressure is now modified by the pressure correction term $\bar {a}$ .

Under the above discussion of a self-similar solution as a limit of the intermediate asymptotic, the system of (4.5)–(4.11) reduces to

(4.40) \begin{align} & (V-\eta )\frac {{\textrm d}D}{{\textrm d}\eta } + D\left [\frac {\delta }{\sigma } + \frac {{\textrm d}V}{{\textrm d}\eta }\right ] + \frac {VD}{\eta } = 0, \end{align}

(4.41) \begin{align} & V + \frac {{\textrm d}V}{{\textrm d}\eta }(V-\eta ) + \frac {1}{D}\frac {{\textrm d}P}{{\textrm d}\eta } - \frac {1}{D}\frac {{\textrm d}{\mathcal{T}}_{\textit{rr}}}{{\textrm d}\eta } - \frac {W^2}{\eta } + \nonumber \\&\quad \frac {2{(\gamma M^2)}^{a_c}}{R_{es}(1+\bar {a})^{a_c}(1-\bar {b})^{a_c}}\times \left (\left (P+\frac {\bar {a} D^2}{\gamma M^2}\right )(1-\bar {b}D)\right )^{a_c} D^{b_c -a_c-1} \left [\frac {V}{\eta ^2}-\frac {1}{\eta }\frac {{\textrm d}V}{{\textrm d}\eta }\right ] = 0, \end{align}

(4.42) \begin{align} & W + \frac {{\textrm d}W}{{\textrm d}\eta }(-\eta +V) + \frac {WV}{\eta } - \frac {1}{D}\left [\frac {{\textrm d}{\mathcal{T}}_{r\theta }}{{\textrm d}\eta }+\frac {2{\mathcal{T}}_{r\theta }}{\eta }\right ] = 0, \end{align}

(4.43) \begin{align} & (1-\overline {b}D)\frac {{\textrm d} P}{{\textrm d}\eta }-\frac {P}{D}\frac {{\textrm d}D}{{\textrm d}\eta }\left [\gamma -\frac {\overline {a}(1-2\overline {b}D)D^2}{\gamma M^2P}+\frac {\overline {a}(\gamma -1)D^2}{\gamma M^2 P}\right ]+\frac {\gamma -1}{V-\eta } \Big [\frac {-\delta }{\sigma }P- \nonumber \\& {\mathcal{T}}_{\textit{rr}} \Big (\frac {{\textrm d}V}{{\textrm d}\eta }+\frac {V}{\eta }\Big ) -{\mathcal{T}}_{r\theta }\Big (-\frac {W}{\eta }+\frac {{\textrm d}W}{{\textrm d}\eta }\Big )-\frac {2(\gamma M^2)^{a_c}\Big (\big (P+\frac {\bar {a} D^2}{\gamma M^2}\big )(1-\bar {b}D)\Big )^{a_c} D^{b_c -a_c}}{\textit{Re}_{s}(1+\bar {a})^{a_c}(1-\bar {b})^{a_c}} \nonumber \\& \left (\!\frac {V^2}{\eta ^2}-\frac {V}{\eta }\frac {{\textrm d}V}{{\textrm d}\eta }\! \right ) +\frac {{\textrm d}Q}{{\textrm d}\eta } \!+\! \frac {Q}{\eta }\Big ] \!+\! \frac {2}{(V-\eta )}\! \left [P(1-\overline {b}D) \!+\! \frac {\overline {a}D^2(1-\overline {b}D)}{\gamma M^2} \!-\! \overline {a}(\gamma -1)D^2\! \right ]=0, \end{align}

(4.44) \begin{align} & {\mathcal{T}}_{\textit{rr}} = \frac {(\gamma M^2)^{a_c}\left [\left (P+\frac {\bar {a} D^2}{\gamma M^2}\right )(1-\bar {b}D)\right ]^{a_c} D^{b_c -a_c}}{\textit{Re}_{s}(1+\bar {a})^{a_c}(1-\bar {b})^{a_c}}\left [\frac {4}{3}\frac {{\textrm d}V}{{\textrm d}\eta }-\frac {2}{3}\frac {V}{\eta }\right ], \end{align}

(4.45) \begin{align} & {\mathcal{T}}_{r\theta } = \frac {(\gamma M^2)^{a_c}\left [\left (P+\frac {\bar {a} D^2}{\gamma M^2}\right )(1-\bar {b}D)\right ]^{a_c} D^{b_c -a_c}}{\textit{Re}_{s}(1+\bar {a})^{a_c}(1-\bar {b})^{a_c}}\left [-\frac {W}{\eta }+\frac {{\textrm d} W}{{\textrm d}\eta }\right ], \end{align}

(4.46) \begin{align} & Q(\eta ) = -\varGamma _c (1-\overline {b}D)\left (P+\frac {\overline {a}D^2}{\gamma M^2}\right )\left [\frac {(1-\overline {b}D)}{D}\frac {{\textrm d}P}{{\textrm d}\eta }+\frac {{\textrm d}D}{{\textrm d}\eta }\left (-\frac {P}{D^2}+\frac {\overline {a}(1-2\overline {b}D)}{\gamma M^2}\right )\right ]. \end{align}

The (4.40)–(4.46) reduces into the system of equations (Nath & Sahu Reference Nath and Sahu2017) for the inviscid flow $(\textit{Re}_s \rightarrow \infty )$ of a simplified vdW gas ( $\bar {a}=0\, \text{and}\, \bar {b}\neq 0$ ) under constant ambient density in absence of radiation heat fluxes. These comparisons justify the generalization of system of equations in rotating vdW gas of viscous flow influenced by cylindrical shock wave under the ambient variable density.

Using the similarity transformations (4.2)–(4.3) and the (4.21), the shock jump conditions (4.30)–(4.32) can be transformed into the conditions at the shock front $(\eta = 1)$ as

(4.47) \begin{align} D(1) &=\frac {1}{\beta }, \quad V(1)=(1-\beta ),\quad P(1)= \left [1-\beta + \frac {1}{\gamma M^2}+\frac {2(1-\beta )}{3 R_{es}\beta ^{b_c}}\right ], \end{align}

(4.48) \begin{align} {\mathcal{T}}_{\textit{rr}}(1) &= \frac {2(1-\beta )}{3R_{es}\beta ^{b_c}},\quad {\mathcal{T}}_{r\theta }(1)=0,\quad W(1)= \frac {\varOmega }{\sigma } = \sqrt {\frac {\delta /\sigma +2}{\gamma M^2}}, \end{align}

(4.49) \begin{align} Q(1) &= \left [\frac {\overline {b}(1-\beta )+2\overline {a}\beta -\overline {a}\overline {b}(1+\beta )}{\gamma M^2(\beta -\overline {b})}-\frac {2\overline {a}}{\gamma M^2 \beta }+\frac {\beta ^2-1}{2}-\frac {2\beta (1-\beta )}{3R_{es}\beta ^{b_c}}\right ]. \end{align}

In non-viscous flow ( $\textit{Re}_s \rightarrow \infty$ ), these systems coincide with Nath & Sahu (Reference Nath and Sahu2017) with volume correction only and with constant ambient density in the absence of radiation. These comparisons justify the generalization of the boundary conditions in viscous rotating vdW gas with pressure- and volume-corrected vdW gas under the variable density.

At the piston or inner expanding surface, it is necessary for the fluid velocity to be equal to the velocity of the inner expanding surface itself. This kinematic condition, derived from (4.2), can be expressed as follows:

(4.50) \begin{equation} V(\eta _p) = \eta _p. \end{equation}

The adiabatic compressibility $C_{\textit{adi}}$ of the non-ideal gas behind the shock wave can be calculated Moelwyn-Hughes (Reference Moelwyn-Hughes1961), Bajargaan et al. (Reference Bajargaan, Patel and Singh2021) as

(4.51) \begin{equation} C_{\textit{adi}} = \frac {1}{\rho }\left (\frac {\partial \rho }{\partial p}\right )_S =\frac {1}{\rho c^2}= \frac {1}{\frac {\gamma (p+a \rho ^2)}{1-b\rho }-2a\rho ^2}, \end{equation}

where $S$ is the entropy per unit mass of the non-ideal gas.

Normalizing the flow variables $\rho$ , $u$ , $p$ , $q$ , $\tau$ and $C_{\textit{adi}}$ with their respective value at the shock, we obtain

(4.52) \begin{align} \frac {\rho }{\rho _{2}} &= \frac {D(\eta )}{D(1)}, \quad \frac {u}{u_2} = \frac {V(\eta )}{V(1)},\quad \frac {p}{p_2} = \dfrac {P(\eta )}{P(1)},\quad \dfrac{u_{\theta}}{u_{\theta_2}} = \dfrac {W(\eta)}{W(1)}, \end{align}

(4.53) \begin{align} \frac {q}{q_2} &= \frac {Q(\eta )}{Q(1)},\quad \frac {\tau _{\textit{rr}}}{\tau _{rr_2}} = \frac {{\mathcal{T}}_{\textit{rr}}(\eta )}{{\mathcal{T}}_{\textit{rr}}(1)},\quad \tau _{r\theta } ={\mathcal{T}}_{r\theta }(\eta ),\quad \frac {C_{\textit{adi}}}{C_{adi,2}} = \frac {C_{\textit{adi}}(\eta )}{C_{adi,2}(1)}. \end{align}

Numerical integration of the differential (4.40)–(4.46) with the boundary conditions (4.47)–(4.49) gives the solution of the differential equations of the present problem.