# Similarity Solution for Magnetogasdynamic Shock Waves in a Weakly Conducting Perfect Gas by Using the Lie Group Invariance Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Similarity Analysis by Using Invariance Group

## 4. Similarity Solution

**Case I**: Let $a\ne 0$, $c=0$; then, from Equation (21), we have the infinitesimal generators as

## 5. Results and Discussion

#### 5.1. The Implications of a Rise in the Exponent of Ambient Density ${\lambda}_{1}$

#### 5.2. The Implications of a Rise in the Ratio of Specific Heats $\gamma $

#### 5.3. The Implications of a Rise in the Magnetic Reynolds Number ${R}_{m}$

#### 5.4. The Implications of a Rise in the Parameters ${M}_{{a}_{z}}^{-2}$ or ${M}_{{a}_{\theta}}^{-2}$

## 6. Conclusions

- The adiabatic exponent $\gamma $ of the gas has decaying effects on the shock strength. The parameters $\gamma $, ${R}_{m}$, ${M}_{{a}_{z}}^{-2}$, or ${M}_{{a}_{\theta}}^{-2}$ have similar effects on the shock strength.
- With an increase in magnetic Reynolds number ${R}_{m}$, pressure $p/{p}_{1}$, the density $\rho /{\rho}_{1}$ and fluid velocity $u/{u}_{1}$ increase, but the axial magnetic induction ${B}_{z}/{B}_{{z}_{1}}$ decreases. The parameters ${M}_{{a}_{z}}^{-2}$, ${M}_{{a}_{\theta}}^{-2}$, and magnetic Reynolds number ${R}_{m}$ have similar effects on $\rho /{\rho}_{1}$, $p/{p}_{1}$, $u/{u}_{1}$, and ${B}_{z}/{B}_{{z}_{1}}$.
- The ambient density exponent ${\lambda}_{1}$ has a decaying impact on shock strength for ${\lambda}_{1}<0$, whereas shock strength increases for ${\lambda}_{1}>0$.
- The change in ambient density exponent from negative to positive, the shock strength, density $\rho /{\rho}_{1}$, and axial magnetic induction ${B}_{z}/{B}_{{z}_{1}}$ decrease, but the pressure $p/{p}_{1}$ and fluid velocity $u/{u}_{1}$ increase, and the azimuthal magnetic induction ${B}_{\theta}/{B}_{{\theta}_{1}}$ remains unchanged.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Distribution of the flow variables in the region behind the shock front for different values of $\gamma $ and ${\lambda}_{1}$ when ${R}_{m}=0.1$, ${M}_{{a}_{z}}^{-2}=0.01$, and ${M}_{{a}_{\theta}}^{-2}=0.01$ for power law shock path case: (

**a**) density $\frac{\rho}{{\rho}_{1}}$, (

**b**) pressure $\frac{p}{{p}_{1}}$, (

**c**) axial magnetic induction $\frac{{B}_{z}}{{B}_{{z}_{1}}}$, (

**d**) azimuthal magnetic induction $\frac{{B}_{\theta}}{{B}_{{\theta}_{1}}}$, (

**e**) radial velocity $\frac{u}{{u}_{1}}$, 1. $\gamma =4/3$, ${\lambda}_{1}=-1.5$; 2. $\gamma =4/3$, ${\lambda}_{1}=-1$; 3. $\gamma =4/3$, ${\lambda}_{1}=0$; 4. $\gamma =4/3$, ${\lambda}_{1}=1$; 5. $\gamma =4/3$, ${\lambda}_{1}=1.5$; 6. $\gamma =5/3$, ${\lambda}_{1}=-1.5$; 7. $\gamma =5/3$, ${\lambda}_{1}=-1$; 8. $\gamma =5/3$, ${\lambda}_{1}=0$; 9. $\gamma =5/3$, ${\lambda}_{1}=1$; 10. $\gamma =5/3$, ${\lambda}_{1}=1.5$.

**Figure 2.**Distribution of the flow variables in the region behind the shock front for different values of ${R}_{m}$, ${M}_{{a}_{z}}^{-2}$, and ${M}_{{a}_{\theta}}^{-2}$ when $\gamma =5/3$ and ${\lambda}_{1}=-0.25$ for power law shock path case. (

**a**) Density $\frac{\rho}{{\rho}_{1}}$, (

**b**) pressure $\frac{p}{{p}_{1}}$, (

**c**) axial magnetic induction $\frac{{B}_{z}}{{B}_{{z}_{1}}}$, (

**d**) azimuthal magnetic induction $\frac{{B}_{\theta}}{{B}_{{\theta}_{1}}}$, (

**e**) radial velocity $\frac{u}{{u}_{1}}$, 1. ${R}_{m}=0.01$, ${M}_{{a}_{z}}^{-2}=0.5$, ${M}_{{a}_{\theta}}^{-2}=0.5$; 2. ${R}_{m}=0.1$, ${M}_{{a}_{z}}^{-2}=0.01$, ${M}_{{a}_{\theta}}^{-2}=0.01$; 3. ${R}_{m}=0.1$, ${M}_{{a}_{z}}^{-2}=0.01$, ${M}_{{a}_{\theta}}^{-2}=0.5$; 4. ${R}_{m}=0.1$, ${M}_{{a}_{z}}^{-2}=0.5$, ${M}_{{a}_{\theta}}^{-2}=0.01$; 5. ${R}_{m}=0.1$, ${M}_{{a}_{z}}^{-2}=0.5$, ${M}_{{a}_{\theta}}^{-2}=0.5$.

**Table 1.**Position of inner expanding surface ${\eta}_{p}$ at different values of $\gamma $ and ${\lambda}_{1}$ when ${M}_{{a}_{z}}^{-2}=0.01$, ${M}_{{a}_{\theta}}^{-2}=0.01$, and ${R}_{m}=0.1$ for power law shock path case.

$\mathit{\gamma}$ | ${\mathit{\lambda}}_{1}$ | ${\mathit{\eta}}_{\mathit{p}}$ Inner Expanding Surface (IES) |
---|---|---|

$4/3$ | −1.5 | 0.928305 |

$4/3$ | −1.0 | 0.895738 |

$4/3$ | 0.0 | 0.000012 |

$4/3$ | 1.0 | 0.843549 |

$4/3$ | 1.5 | 0.889101 |

$5/3$ | −1.5 | 0.923135 |

$5/3$ | −1.0 | 0.893536 |

$5/3$ | 0.0 | 0.00001 |

$5/3$ | 1.0 | 0.754601 |

$5/3$ | 1.5 | 0.813806 |

**Table 2.**Position of inner expanding surface ${\eta}_{p}$ at different values of ${R}_{m}$, ${M}_{{a}_{z}}^{-2}$, and ${M}_{{a}_{\theta}}^{-2}$ when $\gamma =5/3$ and ${\lambda}_{1}=-0.25$ for power law shock path case.

${\mathit{R}}_{\mathit{m}}$ | ${\mathit{M}}_{{\mathit{a}}_{\mathit{z}}}^{-2}$ | ${\mathit{M}}_{{\mathit{a}}_{\mathit{\theta}}}^{-2}$ | ${\mathit{\eta}}_{\mathit{p}}$ Inner Expanding Surface (IES) |
---|---|---|---|

0.01 | 0.01 | 0.01 | 0.586147 |

0.01 | 0.01 | 0.50 | 0.584657 |

0.01 | 0.50 | 0.01 | 0.584657 |

0.01 | 0.50 | 0.50 | 0.583242 |

0.1 | 0.01 | 0.01 | 0.585625 |

0.1 | 0.01 | 0.50 | 0.571565 |

0.1 | 0.50 | 0.01 | 0.571565 |

0.1 | 0.50 | 0.50 | 0.558567 |

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**MDPI and ACS Style**

Nath, G.; S, K.V.
Similarity Solution for Magnetogasdynamic Shock Waves in a Weakly Conducting Perfect Gas by Using the Lie Group Invariance Method. *Symmetry* **2023**, *15*, 1640.
https://doi.org/10.3390/sym15091640

**AMA Style**

Nath G, S KV.
Similarity Solution for Magnetogasdynamic Shock Waves in a Weakly Conducting Perfect Gas by Using the Lie Group Invariance Method. *Symmetry*. 2023; 15(9):1640.
https://doi.org/10.3390/sym15091640

**Chicago/Turabian Style**

Nath, Gorakh, and Kadam V S.
2023. "Similarity Solution for Magnetogasdynamic Shock Waves in a Weakly Conducting Perfect Gas by Using the Lie Group Invariance Method" *Symmetry* 15, no. 9: 1640.
https://doi.org/10.3390/sym15091640