Interaction of a characteristic shock with a weak discontinuity in a relaxing gas

Jena, J.; Sharma, V. D.

doi:10.1007/s10665-007-9182-2

Cited by 26 publications

(4 citation statements)

References 23 publications

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“…Typically weak discontinuities are solutions that admit a jump in the first derivative of the solution across the characteristic curve which propagates at the local characteristic speed. Let us assume that the

C^{1}

discontinuity is moving along the fastest characteristic

\frac{d x}{d t} = u + c .

Then, the transport equation for the

C^{1}

discontinuity is given by

b^{false(3 false)} ({}, \frac{d normalΩ}{d t} + ((), {double-struckQ}_{x} + normalΩ) ((), \nabla μ^{false(3 false)}) normalΩ) + {((), ((), \nabla b^{false(3 false)}) normalΩ)}^{t r} \frac{d double-struckQ}{d t} + ((), b^{false(3 false)} normalΩ) ((), ((), \nabla μ^{false(3 false)}) {double-struckQ}_{x} + μ_{x}^{false(3 false)}) - ((), \nabla ((), b^{false(3 false)} N false(double-struckQ false))) normalΩ = 0,

where

normalΩ = ϕ false(t false) d^{false(3 false)}

represents the jump in

{double-struckQ}_{x}

across the

C^{1}

discontinuity that is collinear to the right eigenvector

d^{false(3 false)}

and

ϕ false(t false)

is the amplitude of the discontinuity wave. Making use of invariant solution obtained from the optimal class

W_{4}

(refer to Table ) in Equation , we obtain the following first order Bernoulli type equation for

ϕ false...

…”

Section: Evolution Of Weak Discontinuitymentioning

confidence: 99%

“…Boillat and Ruggeri discussed the influence of an incident wave on a strong shock when it is free to propagate. Interaction between blast wave and weak wave in the context of planar and radially symmetric flows has been discussed by Radha et al In previous studies, authors studied the collision of weak discontinuity with characteristic shock for various physical problems. Minhajul and Raja Sekhar discussed collision between weak discontinuity and elementary waves of Riemann problem for the multiphase flows.…”

Section: Introductionmentioning

confidence: 99%

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

Sahoo

Sekhar

2020

Math Methods in App Sciences

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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

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C^{1}

discontinuity is moving along the fastest characteristic

\frac{d x}{d t} = u + c .

Then, the transport equation for the

C^{1}

discontinuity is given by

b^{false(3 false)} ({}, \frac{d normalΩ}{d t} + ((), {double-struckQ}_{x} + normalΩ) ((), \nabla μ^{false(3 false)}) normalΩ) + {((), ((), \nabla b^{false(3 false)}) normalΩ)}^{t r} \frac{d double-struckQ}{d t} + ((), b^{false(3 false)} normalΩ) ((), ((), \nabla μ^{false(3 false)}) {double-struckQ}_{x} + μ_{x}^{false(3 false)}) - ((), \nabla ((), b^{false(3 false)} N false(double-struckQ false))) normalΩ = 0,

where

normalΩ = ϕ false(t false) d^{false(3 false)}

represents the jump in

{double-struckQ}_{x}

across the

C^{1}

discontinuity that is collinear to the right eigenvector

d^{false(3 false)}

and

ϕ false(t false)

is the amplitude of the discontinuity wave. Making use of invariant solution obtained from the optimal class

W_{4}

(refer to Table ) in Equation , we obtain the following first order Bernoulli type equation for

ϕ false...

…”

Section: Evolution Of Weak Discontinuitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Sahoo

Sekhar

2020

Math Methods in App Sciences

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“…Radha et al 2 and Jena 3 used the general theory, which was developed by Brun 4 and Jeffrey 5 to study the problems of wave interactions. One‐dimensional impact of elementary waves for unsteady flow of different gases and two‐phase flows, which yield more rich results, we refer to works studied in previous studies 6‐9 and the references cited therein. Propagation and interaction of different type of non‐linear waves in the reactive mixture of ideal gases have been analyzed by Singh and Jena 10‐13 and in context of non‐ideal relaxing gas previous studies 14,15 .…”

Section: Introductionmentioning

confidence: 99%

Lie symmetries for analyzing interaction of a characteristic shock with a singular surface in a non‐ideal reacting gas with dust particles

Shah

Singh

2020

Math Methods in App Sciences

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Using the invariance group properties of the system describing one‐dimensional unsteady flow of a non‐ideal dusty reacting gas, the evolutions of characteristic shock and singular surface are investigated. The effects of non‐idealness, reaction mechanism, and dusty gas parameters on the characteristic shock and singular surface are analyzed. Further, the amplitude of reflected wave and (or) transmitted wave along with bounce in the acceleration of shock, which generated from the interaction of a characteristic shock with a singular surface, are obtained.

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“…The relaxation mechanism in gas is basically the rate of process of attaining equilibrium through nonequilibrium which occurs due to some external forces and the time taken to get the equilibrium state back is known as the relaxation time [29][30][31]. In general, the energy in a gas molecule is distributed among translational modes, vibrational modes, and rotational modes.…”

Section: Introductionmentioning

confidence: 99%

Interaction of a Singular Surface with a Characteristic Shock in a Relaxing Gas with Dust Particles

Mittal

Jena

2019

Zeitschrift Für Naturforschung A

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A system of hyperbolic differential equations outlining one-dimensional planar, cylindrical symmetric and spherical symmetric flow of a relaxing gas with dust particles is considered. Singular surface theory used to study different aspects of wave propagation and its culmination to the steepened form. The evolutionary behavior of the characteristic shock is studied. A particular solution of the governing system of equations is used to discuss the steepened wave form, characteristic shock and their interaction. The results of the interaction between the steepened wave front and the characteristic shock using the general theory of wave interaction are discussed. Also, the influence of relaxation and dust parameters on the steepened wave front, the formation of a characteristic shock, reflected and transmitted waves after interaction and a jump in shock acceleration are investigated.

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Interaction of a characteristic shock with a weak discontinuity in a relaxing gas

Cited by 26 publications

References 23 publications

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

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

Lie symmetries for analyzing interaction of a characteristic shock with a singular surface in a non‐ideal reacting gas with dust particles

Interaction of a Singular Surface with a Characteristic Shock in a Relaxing Gas with Dust Particles

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