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Strong Shock Waves in a Dusty-Gas Atmosphere Under Isothermal Conditions: A Power Series Solution

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Abstract

In this paper, a study concerning strong plane and cylindrically symmetric shock waves in a dusty gas atmosphere under isothermal condition has been performed, by taking into account the power series expansion method. The flow medium is assumed to be a mixture of perfect gas and small solid particles. The governing equation and associated boundary conditions are non-dimentionlized using a similarity transformation, and are hence expanded as a power series in \({\mathrm{M}}^{-2}\), where \(\mathrm{M}\) is the upstream mach number. Runga-Kutta method of order four has been used to obtain a numerical solution. The effect of (i) geometry factor and (ii) dust loading parameters on the flow variables behind the shock front has been studied in detail and their behaviour in flow field is illustrated via figures. It has been observed that the in the flow field region the quantity \({\mathrm{J}}_{0}\) increases with increase in \(\mathrm{G}\) (the ratio of density of solid particles to the initial density of perfect gas) while it decreases with increment in \({\upkappa }_{\mathrm{p}}\) (mass concentration of solid particles) and \(\mathrm{j}\) (geometry factor). Hence, \({\upkappa }_{\mathrm{p}}\) has a decaying effect on shock wave, whereas the shock grows with increment in \(\mathrm{G}\) and \(\mathrm{j}\).

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Abbreviations

a a :

Speed of sound ahead of shock wave

\(a_{{isoth.}}^{2}\) :

Isothermal speed of sound

c sp :

Specific heat of solid particles

c v :

Specific heat of the gas at constant volume

c p :

Specific heat of the gas at constant pressure

c pm :

Specific heat of the mixture constant pressure

c vm :

Specific heat of the mixture constant volume

E j :

Energy carried by the shock

e :

Internal energy of the mixture

e a :

Internal energy at the initial stage

f :

Non-dimensional velocity

G :

Ratio of density of solid particles to that of perfect gas

g :

Non-dimensional pressure

h :

Non-dimensional density

J :

Energy integral

j :

Geometry factor

M :

Upstream Mach number

m :

Mass

m sp :

Total mass of solid particles

p :

Pressure in the flow field

p a :

Pressure of mixture at initial stage

R i :

Gas constant

r :

Space coordinate

r n :

Position of shock front

T :

Temperature in the flow field

t :

Time coordinate

u :

Velocity component in radial direction

V :

Volume

V a :

Volume of the mixture at initial stage

V sp :

Volumetric extension of the solid particles

W s :

Velocity of shock propagation

x :

Similarity variable (non-dimensional variable)

y :

Reciprocal of square of Mach number (non-dimensional variable)

Z :

Volume fraction of solid particles

Z a :

Volume fraction of solid particles at initial stage

βsp :

Ratio of specific heat of solid particles

γ:

Ratio of specific heat of gas

Γ:

Ratio of specific heat of mixture

δ :

Abbreviation

Θ:

Non-dimensional temperature

κ p :

Mass concentration of solid particles

λ :

Shock decay parameter

ρ :

Density of the mixture

ρ a :

Density of the mixture at initial stage

ρ sp :

Species density of solid particles

τ :

Isothermal compressibility

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Acknowledgements

The first and third authors acknowledge NIT, Uttarakhand for financial support.

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  1. Department of Mathematics, National Institute of Technology, Srinagar (Garhwal), Uttarakhand, 246174, India

    Danish Amin, D. B. Singh & Vidit Kumar Vats

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  1. Danish Amin
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Amin, D., Singh, D.B. & Vats, V.K. Strong Shock Waves in a Dusty-Gas Atmosphere Under Isothermal Conditions: A Power Series Solution. Int. J. Appl. Comput. Math 7, 174 (2021). https://doi.org/10.1007/s40819-021-01111-5

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