Group theoretic method for analyzing interaction of a discontinuity wave with a strong shock in an ideal gas

Pandey, Manoj

doi:10.1007/s00033-009-0030-2

Cited by 10 publications

(2 citation statements)

References 19 publications

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“…Let us suppose that the

C^{1}

discontinuity is propagating along the fastest characteristic

d x false/ d t = u + C

. Then the transport equation for the weak discontinuity is given by (see related studies)

l^{false(1 false)} ({}, \frac{d normalΠ}{d t} + false(V_{x} + normalΠ false) false(\nabla λ_{1} false) normalΠ) + {false(false(\nabla l^{false(1 false)} false) normalΠ false)}^{t r} \frac{d V}{d t} + false(l^{false(1 false)} normalΠ false) false(false(\nabla λ_{1} false) V_{x} + λ_{x}^{false(1 false)} false) - false(\nabla false(l^{false(1 false)} f false) false) normalΠ = 0,

where

V = {false(ρ, u false)}^{t r}

f = {false(0, β false)}^{t r}

, and

normalΠ = ω false(t false) r^{false(1 false)}

denote the jump in

V_{x}

across the

C^{1}

discontinuity and are collinear to the right eigenvector

r^{false(1 false)}

, with

ω false(t false)

as the amplitude of the discontinuity wave. Using Equations , and in , we obtain the following transport equation for the wave amplitude

ω false(t false)

\frac{d ω}{d t}

…”

Section: Evolution Of C1 Discontinuity Wavementioning

confidence: 99%

Symmetry analysis and optimal systems of generalized Chaplygin gas equations with a source term

Pradhan

Pandey

2020

Math Methods in App Sciences

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The system of generalized Chaplygin gas equations with a coulomblike friction term has been investigated by using the famous Lie symmetry method. A direct and systematic algorithm based on the adjoint transformation and invariants of the admitted Lie algebras is then used to construct one-and two-dimensional optimal system of the Chaplygin gas equations. Inequivalent classes of group invariant solutions are then obtained using the one-dimensional optimal system. Further, the evolutionary behaviour of the weak discontinuity wave within the state characterized by one of the group invariant solutions is investigated in detail, and certain observations are noted in respect to their contrasting behaviour. KEYWORDSLie symmetries, first -order hyperbolic systems, Chaplygin gas equations, weak discontinuities MSC CLASSIFICATION 70G65; 35L40

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“…Let us suppose that the

C^{1}

discontinuity is propagating along the fastest characteristic

d x false/ d t = u + C

. Then the transport equation for the weak discontinuity is given by (see related studies)

l^{false(1 false)} ({}, \frac{d normalΠ}{d t} + false(V_{x} + normalΠ false) false(\nabla λ_{1} false) normalΠ) + {false(false(\nabla l^{false(1 false)} false) normalΠ false)}^{t r} \frac{d V}{d t} + false(l^{false(1 false)} normalΠ false) false(false(\nabla λ_{1} false) V_{x} + λ_{x}^{false(1 false)} false) - false(\nabla false(l^{false(1 false)} f false) false) normalΠ = 0,

where

V = {false(ρ, u false)}^{t r}

f = {false(0, β false)}^{t r}

, and

normalΠ = ω false(t false) r^{false(1 false)}

denote the jump in

V_{x}

across the

C^{1}

discontinuity and are collinear to the right eigenvector

r^{false(1 false)}

, with

ω false(t false)

as the amplitude of the discontinuity wave. Using Equations , and in , we obtain the following transport equation for the wave amplitude

ω false(t false)

\frac{d ω}{d t}

…”

Section: Evolution Of C1 Discontinuity Wavementioning

confidence: 99%

Symmetry analysis and optimal systems of generalized Chaplygin gas equations with a source term

Pradhan

Pandey

2020

Math Methods in App Sciences

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“…Besides similarity methods, another use of Lie symmetries admitted by given PDEs consists in introducing some invertible point transformations that map the original system to an equivalent one, admitting special solutions [4]. Using this procedure, Donato and Oliveri [5] obtained exact solutions to axisymmetric MHD equations, Pandey et al [6] and Pandey [7] obtained exact solutions of magnetogasdynamic equations and perfect gas involving shocks, Raja Sekhar and Sharma obtained similarity solutions of the modified shallow water equations and discussed Riemann problem in the case of ideal magnetogasdynamics (see, [8] and [9]). In [10] various classes of exact solutions to ideal magnetogasdynamic equation of a perfect gases are determined by introducing some transformations, referred to as substitution principle, that map the given equations to an equivalent form.…”

Section: Introductionmentioning

confidence: 99%

Evolution of Weak Discontinuities in Non-ideal Magnetogasdynamic Equations

Pandey

2015

Int. J. Appl. Comput. Math

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In this paper classical Lie group of transformations are used to obtain an exact particular solution to the equations of non-ideal magnetogasdynamics, which exhibits space-time dependence. Appropriate canonical variables are characterized that transform the governing system to an equivalent autonomous form, the autonomous system is solved explicitly to obtain exact particular solution of the original system. The particular solution to the governing system, which exhibits space-time dependence, is used to study the evolutionary behaviour of the weak discontinuities.

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Evolution of $$C^{1}$$-wave and its collision with the blast wave in one-dimensional non-ideal gas dynamics

et al. 2020

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Group theoretic method for analyzing interaction of a discontinuity wave with a strong shock in an ideal gas

Cited by 10 publications

References 19 publications

Symmetry analysis and optimal systems of generalized Chaplygin gas equations with a source term

Symmetry analysis and optimal systems of generalized Chaplygin gas equations with a source term

Evolution of Weak Discontinuities in Non-ideal Magnetogasdynamic Equations

Evolution of $$C^{1}$$-wave and its collision with the blast wave in one-dimensional non-ideal gas dynamics

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