<p>In this article, we explore the optimal classification for cylindrical shock wave in a self-gravitating rotating ideal gas with variable density considering the influence of either azimuthal or axial magnetic field for obtaining all possible similarity solution with the use of Lie group invariance method. The flow is considered as unsteady isothermal flow. Optimal classification of sub-algebras has been done for obtaining the inequivalence optimal classes for five-dimensional Lie algebra. We have analyzed optimal classes of sub-algebras where similarity solutions exist and identified three cases: power law shock path, exponential law shock path and a specific case of exponential law shock path. Using the Lie group method, we transformed PDEs into ODEs and solved them numerically with the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$4^{th}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>4</mn> <mrow> <mi mathvariant="italic">th</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>-order Runge-Kutta method in Mathematica. Also, we obtain the particular solution in analytical form in power law shock path case with axial magnetic field. This study examine the effects of gravitation parameter, shock Cowling number, adiabatic index, rotational parameter and ambient density index on the associated flow variables and shock strength. An increase in the shock Cowling number, adiabatic index and rotational parameter weakens the shock, while a higher gravitational parameter and ambient density variation index strengthen it. Our findings demonstrate that shock wave exhibit greater strength in the presence of axial magnetic field but weaker strength in the presence of azimuthal magnetic field. The method developed in this manuscript can be applied in analysis of data from exploding wire experiments, axially symmetric hypersonic flow problems and pinch effect. It is also applicable in theoretical physics, particularly for modelling astrophysical systems such as star-forming clouds, accretion disks, and neutron stars to obtain all possible similarity solutions. The solutions obtained here can be used to analyze the measurements made by spacecraft in the solar wind and in the vicinity of Earth’s magnetosphere.</p>

Optimal Classification of Lie Sub-Algebras and Self-Similar Solution Using Group Invariance Method for Shock Wave in Self-Gravitating Ideal Gas with Magnetic Field and Variable Density in Rotating Medium: Isothermal Flow

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发表日期：

2026年2月5日

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G. Nath,
Harshita

摘要

In this article, we explore the optimal classification for cylindrical shock wave in a self-gravitating rotating ideal gas with variable density considering the influence of either azimuthal or axial magnetic field for obtaining all possible similarity solution with the use of Lie group invariance method. The flow is considered as unsteady isothermal flow. Optimal classification of sub-algebras has been done for obtaining the inequivalence optimal classes for five-dimensional Lie algebra. We have analyzed optimal classes of sub-algebras where similarity solutions exist and identified three cases: power law shock path, exponential law shock path and a specific case of exponential law shock path. Using the Lie group method, we transformed PDEs into ODEs and solved them numerically with the $4^{th}$ $4^{th}$ -order Runge-Kutta method in Mathematica. Also, we obtain the particular solution in analytical form in power law shock path case with axial magnetic field. This study examine the effects of gravitation parameter, shock Cowling number, adiabatic index, rotational parameter and ambient density index on the associated flow variables and shock strength. An increase in the shock Cowling number, adiabatic index and rotational parameter weakens the shock, while a higher gravitational parameter and ambient density variation index strengthen it. Our findings demonstrate that shock wave exhibit greater strength in the presence of axial magnetic field but weaker strength in the presence of azimuthal magnetic field. The method developed in this manuscript can be applied in analysis of data from exploding wire experiments, axially symmetric hypersonic flow problems and pinch effect. It is also applicable in theoretical physics, particularly for modelling astrophysical systems such as star-forming clouds, accretion disks, and neutron stars to obtain all possible similarity solutions. The solutions obtained here can be used to analyze the measurements made by spacecraft in the solar wind and in the vicinity of Earth’s magnetosphere.