scholarly journals SIMILARITY SOLUTIONS FOR STRONG SHOCKS IN A NON-IDEAL GAS

2012 ◽  
Vol 17 (3) ◽  
pp. 351-365 ◽  
Author(s):  
Rajan Arora ◽  
Amit Tomar ◽  
Ved Pal Singh
Keyword(s):  
Power Law ◽  
Initial Density ◽  
Similarity Solutions ◽  
Ideal Gas ◽  
Spherically Symmetric ◽  
State Dependent ◽  
Group Theoretic Method ◽  
Special Cases ◽  
Dependent Form ◽  
Flow Variables

A group theoretic method is used to obtain an entire class of similarity solutions to the problem of shocks propagating through a non-ideal gas and to characterize analytically the state dependent form of the medium ahead for which the problem is invariant and admits similarity solutions. Different cases of possible solutions, known in the literature, with a power law, exponential or logarithmic shock paths are recovered as special cases depending on the arbitrary constants occurring in the expression for the generators of the transformation. Particular case of collapse of imploding cylindrically and spherically symmetric shock in a medium in which initial density obeys power law is worked out in detail. Numerical calculations have been performed to obtain the similarity exponents and the profiles of the flow variables behind the shock, and comparison is made with the known results.

scholarly journals Similarity solutions of the steady state cosmic-ray equation of transport

1976 ◽  
Vol 19 (4) ◽  
pp. 432-451 ◽  
Author(s):  
G. M. Webb
Keyword(s):  
Diffusion Coefficient ◽  
Steady State ◽  
Radial Distance ◽  
Cosmic Ray ◽  
Analytic Solutions ◽  
State Equation ◽  
Similarity Solutions ◽  
Symmetric Model ◽  
Spherically Symmetric ◽  
Special Cases

AbstractSimilarity solutions of the steady-state equation of transport for the distribution function F0 of cosmic rays in the interplanetary region are obtained by theuse of transformation groups. The solutions are derived in detail for a spherically-symmetric model of the interplanetary region with an effective radial diffusion coefficient κ = κ0(p)rb with r the heliocentric radial distance. p the particle momentum, κ0(p) an arbitary function of p, and the solar wind velocity is radial and of constant speed V. Solutions for which the similarity variable η is a function of r only are also derived; these are of particular impoartance when the F0 is specified on a boundary of given radius. Non spherically-symmetric solutions can also be obtained by group methods and examples of such solutions are listed, without derivation, for the equation of transport incorporating the effects of anisotropic diffusion (diffusion coefficient κ1 in the radial direction and κ2 normal to it). The solutions are the most extensive steady-state analytic solutions yet obtained, and contain previous analytic solutions as special cases.

Polytropic solutions to the problem of spherically symmetric flow of an ideal gas

1980 ◽  
Vol 58 (8) ◽  
pp. 1085-1092 ◽  
Author(s):  
D. Summers
Keyword(s):  
Differential Equations ◽  
Fluid Dynamic ◽  
Mass Density ◽  
Ideal Gas ◽  
Radial Coordinate ◽  
Polytropic Index ◽  
Spherically Symmetric ◽  
Polytropic Model ◽  
Special Cases ◽  
Complete Set

An ideal, compressible gas is considered in steady, spherically symmetric, purely radial motion in the presence of a gravitating point mass situated at the origin of coordinates. The gas pressure p and mass density ρ are assumed to satisfy a simple polytrope law, [Formula: see text], where β is the polytropic index which is assumed to take on any real value; the self-gravitation of the gas is neglected. The model equations, which are expressed in a form appropriate to both the expansion and accretion cases, reduce to two non-linear ordinary differential equations, for which Bernoulli integrals are readily found. Singular points of the differential equations are analyzed, and the complete set of asymptotic solutions for the velocity, temperature and Mach number are given for λ → 0 and λ → ∞, where λ [Formula: see text] (radial coordinate)−1, as well as a special class of solutions valid for λ → constant (≠ 0). Families of velocity profiles are sketched which are representative of the complete range of β. The polytropic model, special cases of which have been used successfully in astrophysics in stellar wind and accretion problems, is here cast in general fluid-dynamic terms so that the complete set of solutions obtained may be applicable to a wide class of gas expansion and accretion problems.

scholarly journals Similarity Solutions of Strong Shock Waves for Isothermal Flow in an Ideal Gas

2019 ◽  
Vol 4 (5) ◽  
pp. 1094-1107
Author(s):  
Astha Chauhan ◽  
Rajan Arora
Keyword(s):  
Strong Shock ◽  
Unsteady Motion ◽  
Similarity Solutions ◽  
Ideal Gas ◽  
Isothermal Flow ◽  
Linear Partial Differential Equations ◽  
The One ◽  
Similar Solutions ◽  
Flow Variables ◽  
Strong Shock Waves

Self-similar solutions of the system of non-linear partial differential equations are obtained using the Lie group of invariance technique. The system of equations governs the one dimensional and unsteady motion for the isothermal flow of an ideal gas. The medium has been taken the uniform. From the expressions of infinitesimal generators involving arbitrary constants, different cases arise as per the choice of the arbitrary constants. In this paper, the case of a collapse of an implosion of a cylindrical shock wave is shown in detail along with the comparison between the similarity exponent obtained by Guderley's method and by Crammer's rule. Also, the effects of the adiabatic index and the ambient density exponent on the flow variables are illustrated through the figures. The flow variables are computed behind the leading shock and are shown graphically.

Similarity solutions for cylindrical shock wave in rotating non-ideal gas using Lie group theoretic method

2021 ◽  
Author(s):  
Sumeeta Singh
Keyword(s):  
Shock Wave ◽  
Power Law ◽  
Lie Group ◽  
Numerical Solutions ◽  
Fluid Velocity ◽  
Ideal Gas ◽  
Velocity Increase ◽  
Cylindrical Shock Wave ◽  
Exponential Law ◽  
Group Theoretic Method

In this paper, the propagation of cylindrical shock wave in rotating non-ideal gas under adiabatic flow condition using Lie group of transformation method is investigated. The density is assumed to be constant and azimuthal fluid velocity is assumed to be varying in the undisturbed medium. The arbitrary constants appearing in the expressions for the infinitesimals of the Local Lie group of transformations bring about two different cases of solutions i.e. with a power-law and exponential-law shock paths. Numerical solutions are obtained for both the cases. Distribution of gasdynamical quantities is illustrated through figures. It is obtained that the reduced flow variables pressure and azimuthal fluid velocity decrease in general, whereas density and radial fluid velocity increase in case of power-law shock path. The reduced azimuthal fluid velocity decreases, whereas reduced density, pressure and radial fluid velocity increase in case of exponential-law shock path. Also, it is obtained that shock strength decreases with increase in value of adiabatic exponent or gas non-idealness parameter, whereas it increases due to increase in ambient azimuthal fluid velocity exponent.

scholarly journals The Propagation of Hydromagnetic Cylindrical Shock Waves in Weak Magnetic Field, With A Self-Gravitating Gas

2021 ◽  
pp. 311-323
Author(s):  
Dr. Sarvesh Chandra Yadav
Keyword(s):  
Magnetic Field ◽  
Shock Waves ◽  
Shock Front ◽  
Initial Density ◽  
Analytical Formula ◽  
Weak Shock ◽  
Propagation Distance ◽  
Ideal Gas ◽  
Flow Variables ◽  
Cylindrical Shock Waves

<p>The effects of overtaking disturbances behind the flow on the propagation of diverging cylindrical shock Waves through an ideal gas in presence of a magnetic field having =constant=  and an Initial density distribution  where  is a constant,  is the density at the plane / exes of symmetry: The analytical formula for flow variables representing both the position form viz; weak and strong cases at shock waves have been obtained. Their numerical estimates at permissible shock front locations have been obtained.</p> <p>There numerical estimates at permissible shock front location's have been Calculated and compared with earlier result describing in Free Propagation through figures. After inclusion of E.O.D. noted that there is no change at flow variable with parameters and . However, the trends of variation with propagation distance r, for shock strength, shock velocity and particle velocity are not change in case of weak shock with work Magnetic field<strong>(wswmf).</strong></p>

State-dependent signalling in queueing networks

1994 ◽  
Vol 26 (02) ◽  
pp. 436-455 ◽  
Author(s):  
W. Henderson ◽  
B. S. Northcote ◽  
P. G. Taylor
Keyword(s):  
Queueing Networks ◽  
State Dependence ◽  
Product Form ◽  
Batch Size ◽  
Single Server ◽  
Negative Customers ◽  
Affect Control ◽  
State Dependent ◽  
Special Cases ◽  
Equilibrium Distributions

It has recently been shown that networks of queues with state-dependent movement of negative customers, and with state-independent triggering of customer movement have product-form equilibrium distributions. Triggers and negative customers are entities which, when arriving to a queue, force a single customer to be routed through the network or leave the network respectively. They are ‘signals' which affect/control network behaviour. The provision of state-dependent intensities introduces queues other than single-server queues into the network. This paper considers networks with state-dependent intensities in which signals can be either a trigger or a batch of negative customers (the batch size being determined by an arbitrary probability distribution). It is shown that such networks still have a product-form equilibrium distribution. Natural methods for state space truncation and for the inclusion of multiple customer types in the network can be viewed as special cases of this state dependence. A further generalisation allows for the possibility of signals building up at nodes.

Analytical solution for unsteady flow behind ionizing shock wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field

2021 ◽  
Vol 76 (3) ◽  
pp. 265-283
Author(s):  
G. Nath
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Analytical Solution ◽  
Axial Magnetic Field ◽  
Order Approximation ◽  
Ideal Gas ◽  
Field Region ◽  
First Order ◽  
First Order Approximation ◽  
Flow Variables

Abstract The approximate analytical solution for the propagation of gas ionizing cylindrical blast (shock) wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field is investigated. The axial and azimuthal components of fluid velocity are taken into consideration and these flow variables, magnetic field in the ambient medium are assumed to be varying according to the power laws with distance from the axis of symmetry. The shock is supposed to be strong one for the ratio C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ to be a negligible small quantity, where C 0 is the sound velocity in undisturbed fluid and V S is the shock velocity. In the undisturbed medium the density is assumed to be constant to obtain the similarity solution. The flow variables in power series of C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ are expanded to obtain the approximate analytical solutions. The first order and second order approximations to the solutions are discussed with the help of power series expansion. For the first order approximation the analytical solutions are derived. In the flow-field region behind the blast wave the distribution of the flow variables in the case of first order approximation is shown in graphs. It is observed that in the flow field region the quantity J 0 increases with an increase in the value of gas non-idealness parameter or Alfven-Mach number or rotational parameter. Hence, the non-idealness of the gas and the presence of rotation or magnetic field have decaying effect on shock wave.

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