# Converging Cylindrical Symmetric Shock Waves in a Real Medium with a Magnetic Field

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations and Shock Conditions

## 3. Lie Group Analysis

## 4. Similarity Solutions

## 5. Imploding Shocks

## 6. Numerical Results and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Flow patterns of non-dimensional (

**a**) velocity (U); (

**b**) density $(\Lambda )$; (

**c**) pressure (P); and (

**d**) magnetic pressure (E); for cylindrically symmetric flow ($m=1$) with $\gamma =1.4$ and ${C}_{0}=0.02$.

**Figure 2.**Flow patterns of non-dimensional (

**a**) velocity (U); (

**b**) density $(\Lambda )$; (

**c**) pressure (P); and (

**d**) magnetic pressure (E) for cylindrically symmetric flow ($m=1$) with $\gamma =1.4$ and $b=0.02$.

**Figure 3.**Flow patterns of non-dimensional (

**a**) velocity (U); (

**b**) density $(\Lambda )$; (

**c**) pressure (P); and (

**d**) magnetic pressure E for cylindrically symmetric flow ($m=1$) with $b=0.01$ and ${C}_{0}=0.02$ .

**Table 1.**Computed values of similarity exponent $\delta $ for different values of $\gamma $, b and ${C}_{0}$ in non-ideal magnetogasdynamics for cylindrically symmetric flow.

$\mathit{\gamma}$ | b | ${\mathit{C}}_{0}$ | Computed $\mathit{\delta}$ | Guderley $\mathit{\delta}$ | Arora & Sharma [27] |
---|---|---|---|---|---|

1.4 | 0.25 | 0 | 0.71517095 | 0.71159620 | 0.7157703 |

1.4 | 0.2 | 0 | 0.73163770 | 0.73010800 | 0.72834400 |

1.4 | 0.1 | 0 | 0.77410944 | 0.77567800 | 0.77411040 |

1.4 | 0.02 | 0 | 0.8225961 | 0.82001151 | - |

1.4 | 0.01 | 0 | 0.82865997 | 0.82801210 | - |

1.4 | 0.001 | 0 | 0.8349957 | 0.83401995 | - |

1.4 | 0.2 | 0.02 | 0.72995791 | 0.72559179 | - |

1.4 | 0.1 | 0.02 | 0.76995941 | 0.76011995 | - |

1.4 | 0.02 | 0.02 | 0.81499587 | 0.81401121 | - |

1.4 | 0.01 | 0.02 | 0.81859961 | 0.81015911 | - |

1.4 | 0.001 | 0.02 | 0.82089959 | 0.82159921 | - |

1.4 | 0.2 | 0.05 | 0.72854121 | 0.72550110 | - |

1.4 | 0.1 | 0.05 | 0.75592815 | 0.75125994 | - |

1.4 | 0.02 | 0.05 | 0.8055967 | 0.80154497 | - |

1.4 | 0.01 | 0.05 | 0.80599987 | 0.80202170 | - |

1.4 | 0.001 | 0.05 | 0.80600151 | 0.80559210 | - |

1.66 | 0.2 | 0 | 0.75195451 | 0.75179000 | 0.75170810 |

1.66 | 0.1 | 0 | 0.77969441 | 0.77873000 | 0.77939910 |

1.66 | 0.02 | 0 | 0.80755197 | 0.80959170 | - |

1.66 | 0.01 | 0 | 0.81159549 | 0.81125900 | - |

1.66 | 0.001 | 0 | 0.81559594 | 0.81599959 | - |

1.66 | 0.2 | 0.02 | 0.74559197 | 0.74059917 | - |

1.66 | 0.1 | 0.02 | 0.77459125 | 0.77091887 | - |

1.66 | 0.02 | 0.02 | 0.80615917 | 0.80011759 | - |

1.66 | 0.01 | 0.02 | 0.80910059 | 0.80159520 | - |

1.66 | 0.001 | 0.02 | 0.81195419 | 0.81999595 | - |

1.66 | 0.2 | 0.05 | 0.73559188 | 0.73139146 | - |

1.66 | 0.1 | 0.05 | 0.76017755 | 0.76001151 | - |

1.66 | 0.02 | 0.05 | 0.80415990 | 0.80019880 | - |

1.66 | 0.01 | 0.05 | 0.80554279 | 0.80111595 | - |

1.66 | 0.001 | 0.05 | 0.80750951 | 0.80799150 | - |

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**MDPI and ACS Style**

Devi, M.; Arora, R.; Rahman, M.M.; Siddiqui, M.J.
Converging Cylindrical Symmetric Shock Waves in a Real Medium with a Magnetic Field. *Symmetry* **2019**, *11*, 1177.
https://doi.org/10.3390/sym11091177

**AMA Style**

Devi M, Arora R, Rahman MM, Siddiqui MJ.
Converging Cylindrical Symmetric Shock Waves in a Real Medium with a Magnetic Field. *Symmetry*. 2019; 11(9):1177.
https://doi.org/10.3390/sym11091177

**Chicago/Turabian Style**

Devi, Munesh, Rajan Arora, Mustafa M. Rahman, and Mohd Junaid Siddiqui.
2019. "Converging Cylindrical Symmetric Shock Waves in a Real Medium with a Magnetic Field" *Symmetry* 11, no. 9: 1177.
https://doi.org/10.3390/sym11091177