Application of Lie groups to compressible model of two-phase flows

We present Lie symmetry analysis for investigating the shock‐wave structure of hyperbolic differential equations of polyatomic gases. With the application of symmetry analysis, we derive particular exact group invariant solutions for the governing system of partial differential equations (PDEs). In the next step, the evolutionary behavior of weak shock along with the characteristic shock and their interaction is investigated. Finally, the amplitudes of reflected wave, transmitted wave, and the jump in shock acceleration influenced by the incident wave after interaction are evaluated for the considered system of equations.

“…Next, by using Lie group analysis and the generalized symmetry method, we introduce the following symmetric group of transformations:

ϕ_{1} = (a_{3} + a_{4}) t + a_{5}, ϕ_{2} = a_{2} t + a_{3} x + a_{6}, ψ_{1} = a_{1} ρ, ψ_{2} = a_{2} - a_{4} v, ψ_{3} = - 2 a_{4} T,

where a 2 , a 3 , a 4 , a 5 , and a 6 are the arbitrary constants and the characteristic equations associated with the infinitesimals transformations is given by the following:

\frac{d t}{ϕ_{1}} = \frac{d x}{ϕ_{2}} = \frac{d ρ}{ψ_{1}} = \frac{d v}{ψ_{2}} = \frac{d T}{ψ_{3}} .

…”

Section: Symmetry Analysis and Exact Solutionsmentioning

confidence: 99%

Section: Symmetry Analysis and Exact Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Weak shock waves and its interaction with characteristic shocks in polyatomic gas

Zeidan

Bira

2019

“…Lie group analysis is one of the systematic techniques to handle such problems. For the theory of Lie group analysis and its applications to differential equations, we refer to other works . There are only few papers that are devoted to group analysis of FDEs and group properties of FDEs are much less understood.…”

Section: Introductionmentioning

confidence: 99%

Exact solutions for some time‐fractional evolution equations using Lie group theory

Bira

Sekhar

Zeidan

2018

In this paper, some classes of nonlinear partial fractional differential equations arising in some important physical phenomena are considered. Lie group method is applied to investigate the symmetry group of transformations under which the governing time-fractional partial differential equation remains invariant. The symmetry generators are used for constructing similarity variables, which leads to a reduced ordinary differential equation of Erdélyi-Kober fractional derivatives. Furthermore, a particular exact solution for each governing equation(s) is constructed. Moreover, the physical significance of the solution is investigated graphically based on numerical simulations in order to highlight the importance of the study.

Optimal classification, exact solutions, and wave interactions of Euler system with large friction

Sahoo

Sekhar

2020

Self Cite

We derive exact solutions of one-dimensional Euler system that accounts for gravity together with large friction. Certain optimal classes of subalgebra using Lie symmetry analysis are obtained for this system. We apply the reduction procedure to reduce the Euler system to a system of ordinary differential equations in terms of new similarity variable for each class of subalgebras leading to invariant solutions. The evolution of characteristic shock and its interaction with the weak discontinuity by using one of the invariant solutions is studied. Further, the properties of reflected and transmitted waves and jump in acceleration influenced by the incident wave have been characterized. KEYWORDS euler system with large friction, invariant solution, lie symmetry analysis, optimal classification, wave interactions, weak discontinuity MSC CLASSIFICATION 35A30; 35C05; 35L60; 35Q35; 76D33 Math Meth Appl Sci. 2020;43:5744-5757. wileyonlinelibrary.com/journal/mma