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Physics Research International / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 782172 | https://doi.org/10.1155/2011/782172

G. Nath, A. K. Sinha, "A Self-Similar Flow behind a Magnetogasdynamic Shock Wave Generated by a Moving Piston in a Gravitating Gas with Variable Density: Isothermal Flow", Physics Research International, vol. 2011, Article ID 782172, 8 pages, 2011. https://doi.org/10.1155/2011/782172

A Self-Similar Flow behind a Magnetogasdynamic Shock Wave Generated by a Moving Piston in a Gravitating Gas with Variable Density: Isothermal Flow

Academic Editor: Neil Sullivan
Received20 Apr 2011
Accepted28 Jul 2011
Published27 Oct 2011

Abstract

The propagation of a cylindrical (or spherical) shock wave in an ideal gas with azimuthal magnetic field and with or without self-gravitational effects is investigated. The shock wave is driven out by a piston moving with time according to power law. The initial density and the initial magnetic field of the ambient medium are assumed to be varying and obeying power laws. Solutions are obtained, when the flow between the shock and the piston is isothermal. The gas is assumed to have infinite electrical conductivity. The shock wave moves with variable velocity, and the total energy of the wave is nonconstant. The effects of variation of the piston velocity exponent (i.e., variation of the initial density exponent), the initial magnetic field exponent, the gravitational parameter, and the Alfven-Mach number on the flow field are obtained. It is investigated that the self-gravitation reduces the effects of the magnetic field. A comparison is also made between gravitating and nongravitating cases.

1. Introduction

The explanation and analysis for the internal motion in stars is one of the basic problems in astrophysics. According to the observational data, the unsteady motion of large mass of the gas was followed by sudden release of energy results flare-ups in novae and supernovae. A qualitative behaviour of the gaseous mass may be discussed with the help of the equations of motion and equilibrium taking gravitational forces into account. Numerical solutions for self-similar adiabatic flows in self-gravitating gas were obtained by Sedov [1] and Carrus et al. [2], independently. Purohit [3] and Singh and Vishwakarma [4] have discussed homothermal flows behind a spherical shock wave in a self-gravitating gas using similarity method. Nath et al. [5] have studied the above problem assuming the flow to be adiabatic and self-similar and obtained the effects of the presence of a magnetic field. Shock wave through a variable density medium have been treated by Sedov [1], Sakurai [6], Rogers [7], Rosenau and Frankenthal [8], Nath et al. [5], Vishwakarma and Yadav [9], Nath [10], and others. Their results are more applicable to the shock formed in the deep interior of stars. Also, Vishwakarma and Singh [11] obtained the similarity solution for the flow behind a shock wave in a gravitating or nongravitating nonuniform gas with heat conduction and radiation heat flux in the case of adiabatic flow.

In aerodynamics, the analogy between the steady hypersonic flow past slender-blunted (planar or axisymmetric) power-law bodies and the one-dimensional unsteady self-similar flow behind a shock driven out by a piston moving with time according to a power-law is well known (see, e.g., [1, 8, 10, 12–19]).

Since at high temperatures that prevail in the problems associated with the shock waves a gas is ionized, electromagnetic effects may also be significant. A complete analysis of such a problem should, therefore, consist of the study of the gas-dynamic flow and the electromagnetic field simultaneously. A detailed study towards gaining a better understanding of the interaction between gasdynamic motion of an electrically conducting medium and magnetic field within the context of hyperbolic system has been carried out by many investigators such as Korobeinikov [20], Shang [21], and Lock and Mestel [22]. A detailed review in the field of magnetogasdynamic flows can be seen in the paper of Shang [21]. Lock and Mestel [22] analyzed the annular self-similar solutions in ideal magnetogasdynamics by casting the ideal magnetogasdynamic equations to a three-dimensional autonomous system in which either the magnetic pressure or the fluid pressure vanishes.

In all of the works, mentioned above, the effect of self-gravitation is not taken into account by any of the authors in the case of isothermal flow with magnetic field.

The purpose of this study is, therefore, to obtain the self-similar solutions for the propagation of magnetogasdynamic cylindrical (or spherical) shock wave generated by a moving piston in a nonuniform gas with or without self-gravitational effects, in the presence of an azimuthal magnetic field, under isothermal flow condition. The media ahead and behind the shock front are assumed to be inviscid and to behave as thermally perfect gases. The counter pressure (the pressure ahead of the shock) is taken into account. The azimuthal magnetic field and the density in the ambient medium are assumed to vary and obey the power laws. The assumption of isothermal flow is physically realistic, when radiation heat transfer effects are implicitly present. As the shock propagates, the temperature behind it increases and becomes very large so that there is intense transfer of energy by radiation. This causes the temperature gradient to approach zero; that is, the dependent temperature tends to become uniform behind the shock front, and the flow becomes isothermal [10, 17, 20, 23–27]. A detailed mathematical theory of one-dimensional isothermal blast waves in a magnetic field was developed by Lerche [28, 29]. Numerical solutions for the flow field between the shock and the piston are obtained in the case of isothermal flow in Section 3. Effects of viscosity and rotation are not taken into account.

2. Equations of Motion and Boundary Conditions

The fundamental equations governing the unsteady and cylindrically (or spherically) symmetric isothermal flow of an electrically conducting and self-gravitating ideal gas, in the presence of an azimuthal magnetic field, may be written as (c.f. [2, 10, 17, 20, 23–26, 30]) 𝜕𝜌𝜕𝑡+𝑢𝜕𝜌𝜕𝑟+𝜌𝜕𝑢+𝜕𝑟𝑖𝑢𝜌𝑟=0,(1)𝜕𝑢𝜕𝑡+𝑢𝜕𝑢+1𝜕𝑟𝜌𝜕𝑝𝜕𝑟+ğœ‡â„Žğœ•â„Ž+ğœ•ğ‘Ÿğœ‡â„Ž2𝑟+𝐺𝑚𝑟𝑖=0,(2)ğœ•â„Žğœ•ğ‘¡+ğ‘¢ğœ•â„Žğœ•ğ‘Ÿ+â„Žğœ•ğ‘¢ğœ•ğ‘Ÿ+(𝑖−1)â„Žğ‘¢ğ‘Ÿ=0,(3)𝜕𝑚𝜕𝑟=2𝜋𝑖𝜌𝑟𝑖,(4)𝜕𝑇𝜕𝑟=0,(5) where 𝑟 and 𝑡 are independent space and time coordinates, 𝑢 is the fluid velocity, 𝜌 the density, 𝑝 the pressure, ℎ the azimuthal magnetic field, 𝑇 the temperature, 𝜇 the magnetic permeability, 𝑚 the mass contained in a unit cylinder of radius 𝑟 or in a sphere of radius 𝑟, and the dimension of 𝑚 is taken as [𝑚]=𝑀𝐿(𝑖−2), and 𝑖 takes the values 2 and 1 for the respective cases of spherical and cylindrical symmetries, and 𝐺 the gravitational constant. In nongravitating case, (4) and the term 𝐺𝑚/𝑟𝑖 in (2) do not occur.

The electrical conductivity of the gas is assumed to be infinite. Therefore, the diffusion term from the magnetic field equation is omitted, and the electrical resistivity is ignored. Also, the effect of viscosity on the flow of the gas is assumed to be negligible.

The above system of equations should be supplemented with an equation of state. An ideal gas behaviour of the medium is assumed, so that 𝑝𝑝=Γ𝜌𝑇,𝑒=,(𝛾−1)𝜌(6) where Γ is the gas constant, and 𝛾 is the ratio of specific heats.

A strong cylindrical (or spherical) shock wave is supposed to be propagating in the undisturbed ideal gas with variable density 𝜌=𝐴𝑟𝑤 in the presence of an azimuthal magnetic field ℎ=𝐵𝑟−𝛼, where 𝐴,𝑤,𝐵, and 𝛼 are constant.

The flow variables immediately ahead of the shock front are 𝑢1=0,𝜌=𝜌1=𝐴𝑟𝑤𝑠,𝑚=𝑚1=2𝜋𝑖𝐴(𝑟𝑖+𝑤+1)𝑠𝑖+𝑤+1,ℎ=ℎ1=𝐵𝑟𝑠−𝛼,𝑝=𝑝1=𝜇𝐵2(1−𝛼)𝑟2𝛼𝑠−2𝛼−𝜋𝐴2𝐺𝑖𝑟(𝑤+𝑖+1)(𝑤+1)𝑠2𝑤+2+constant,(7) where −(𝑖+1)<𝑤<−1, 𝛼=−(𝑤+1), 𝑟𝑠 is the shock radius, and the subscript “1” refers to the conditions immediately ahead of the shock.

The jump conditions at the magnetogasdynamic shock wave are given by the principle of conservation of mass, momentum, and energy across the shock; namely, 𝜌1𝑉=𝜌2𝑉−𝑢2,ℎ1𝑉=ℎ2𝑉−𝑢2,𝑝1+12ğœ‡â„Ž21+𝜌1𝑉2=𝑝2+12ğœ‡â„Ž22+𝜌2𝑉−𝑢22,𝑒1+𝑝1𝜌1+12𝑉2+ğœ‡â„Ž21𝜌1âˆ’ğ‘ž1𝜌1𝑉=𝑒2+𝑝2𝜌2+12𝑉−𝑢22+ğœ‡â„Ž22𝜌2âˆ’ğ‘ž2𝜌1𝑉,𝑚1=𝑚2,(8) where the subscript “2” denotes the conditions immediately behind the shock front, 𝑉(=𝑑𝑟𝑠/𝑑𝑡) denotes the velocity of the shock front, and â€œğ‘žâ€ is the radiation heat flux.

From (8), we obtain 𝑢2𝜌=(1−𝛽)𝑉,2=𝜌1𝛽,𝑝2=1(1−𝛽)+𝛾𝑀2+𝑀𝐴−2211−𝛽2𝜌1𝑉2,𝑚2=𝑚1,ℎ2=ℎ1𝛽,(9) where 𝑀=(𝜌1𝑉2/𝛾𝑝1)1/2 is the shock-Mach number referred to the frozen speed of sound (𝛾𝑝1/𝜌1)1/2, and 𝑀𝐴=(𝜌1𝑉2/ğœ‡â„Ž21)1/2 is the Alfven-Mach number. The quantity 𝛽(0<𝛽<1) is obtained by the relation 𝛽22(𝛾+1)−𝛽𝑀2+𝛾1+𝑀𝐴−2−1+(𝛾−2)𝑀𝐴−2+𝛽(1−𝛽)+1/𝛾𝑀2+𝑀𝐴−2/21−1/𝛽2î€·ğ‘žî€¸î€»(𝛾−1)2âˆ’ğ‘ž1(1−𝛽)𝑝2𝑉=0.(10) As the shock is strong, we assume (ğ‘ž2âˆ’ğ‘ž1) to be negligible in comparison with the product of 𝑝2 and 𝑉 [10, 17, 18, 23, 27, 31, 32]. Therefore, (10) reduces to 𝛽22(𝛾+1)−𝛽𝑀2+𝛾1+𝑀𝐴−2−1+(𝛾−2)𝑀𝐴−2=0.(11) Equation (5) together with (6) gives 𝑝𝑝2=𝜌𝜌2.(12)

3. Self-Similarity Transformations

The inner boundary of the flow field behind the shock is assumed to be an expanding surface (piston). In the frame work of self-similarity [1], the velocity 𝑢𝑝=𝑑𝑟𝑝/𝑑𝑡 of the piston is assumed to follow a power law which results in [10, 16–19] 𝑢𝑝=𝑑𝑟𝑝𝑑𝑡=𝑈0𝑡𝑡0𝑛,(13) where 𝑟𝑝 is the radius of the piston, and 𝑡0 denotes the time at a reference state, 𝑈0 is the piston velocity at 𝑡=𝑡0, and 𝑛 is a constant. The consideration of the ambient pressure 𝑝1 and the ambient magnetic field ℎ1 imposes a restriction on “𝑛”. Also, cylindrical and spherical geometries (i.e., for 𝑖=1,2) do not permit 𝑛≤−1 for physical reasons. Thus, using (17), we obtain −(1/(𝑖+1))<𝑛<0. For −(1/(𝑖+1))<𝑛<0, the piston velocity suddenly rises at 𝑡=0 from zero to infinite velocity leading to the formation of a strong shock in the initial phase. The piston is then decelerated. Concerning the shock boundary conditions, self-similarity requires that the velocity of the shock 𝑉=𝑑𝑟𝑠/𝑑𝑡 is proportional to the velocity of the piston; that is, 𝑉=𝑑𝑟𝑠𝑑𝑡=𝐶𝑈0𝑡𝑡0𝑛,(14) where “𝐶” is a dimensionless constant. Using (14), the time and space coordinate can be changed into a dimensionless self-similarity variable 𝜂 as 𝑟𝜂=𝑟𝑠=(𝑛+1)𝑡0𝑛𝐶𝑈0𝑟𝑡𝑛+1.(15) Evidently, 𝜂=𝜂𝑝=𝑟𝑝/𝑟𝑠 at the piston, and 𝜂=1 at the shock. To obtain the similarity solutions, we write the unknown variables in the following form [10, 16–18]: 𝑟𝑢=𝑡𝑈(𝜂),𝜌=𝜌1𝐷(𝜂),𝑚=𝑚1𝑁𝑟(𝜂),𝑝=2𝑡2𝜌1√𝑃(𝜂),ğœ‡â„Ž=𝜌11/2𝑟𝑡𝐻(𝜂),(16) where 𝑈, 𝐷, 𝑁, 𝑃, and 𝐻 are functions of 𝜂 only.

For the existence of similarity solutions, “𝑀” and 𝑀𝐴 should be constants; therefore, 𝑛𝛼=−(𝑤+1)=−𝑛+1.(17) Thus, 𝑀2=2𝑛{𝑛(𝑖+1)+𝑖}𝜇𝐵2−(2𝑛+1)𝜇𝐵2{𝑛(𝑖+1)+𝑖}−2𝜋𝐺𝐴2𝑖(𝑛+1)2𝛾𝑀2𝐴,(18) where −(1/(𝑖+1))<𝑛<0. Equation (18) shows that the solutions of the present problem can be reduced to the case in which the gas is nongravitating (i.e., the case in which 𝐺=0; the solution obtained by Nath [10] in the case of nonrotating medium for 𝑖=1).

Also, the total energy of the disturbance is given by 𝐸=2𝜋𝑖𝑟𝑠𝑟𝑝𝜌12𝑢2+𝑝+𝜌(𝛾−1)ğœ‡â„Ž2−2𝜌𝐺𝑚𝑟𝑖−1𝑟𝑖𝑑𝑟.(19) Applying the similarity transformations (16) in (19), we obtain 𝐸=2𝜋𝑖𝐴𝑟𝑠[𝑖+3𝑛/(𝑛+1)]𝑈0𝐶(𝑛+1)𝑡𝑛02/(𝑛+1)𝐽,(20) where ∫𝐽=1𝜂𝑝[𝑃/(𝛾−1)+(1/2)𝐷𝑈2+𝐻2/2−(𝑖(𝑛+1)2𝐺0/𝛾𝑀2(𝑖+𝑤+1)𝜂𝑖+1)𝑁𝐷]𝜂𝑖+2𝑑𝜂, and 𝐺0=2𝜋𝐴2𝐺/[𝜇𝐵2(1−𝛼)/2𝛼−𝜋𝑖𝐺𝐴2/(𝑤+1)(𝑤+1+𝑖)] is the gravitational parameter.

Hence, the total energy of the shock wave is nonconstant and varies as 𝑟𝑠[𝑖+3𝑛/(𝑛+1)].

Equation (12) with the aid of (16) and (9) yields a relation between 𝑃 and 𝐷 in the form 𝑃(𝜂)=𝐿𝐷(𝜂)𝜂2,(21) where 𝐿=[(1−𝛽)+1/𝛾𝑀2+(𝑀𝐴−2/2)(1−1/𝛽2)](𝑛+1)2𝛽.

By use of (16) and (21), (1) to (4) can be transformed and simplified to[]𝑈−(𝑛+1)𝑑𝐷𝑑𝜂+𝐷𝑑𝑈+𝑑𝜂(𝑛+1)𝑤𝐷𝜂+(𝑖+1)𝐷𝑈𝜂[]=0,(22)𝑈−(𝑛+1)𝑑𝑈+𝐻𝑑𝜂𝐷𝑑𝐻+𝐿𝑑𝜂𝐷𝜂2𝑑𝐷+𝑑𝜂𝑈(𝑈−1)𝜂+2𝐻2+𝑖𝐷𝜂(𝑛+1)2𝐺0𝛾𝑀2(𝑖+𝑤+1)𝜂𝑖+1=0,(23)𝑑𝑁𝑑𝜂−𝜂𝑖[]𝐷(𝑖+1+𝑤)=0,(24)𝑈−(𝑛+1)𝑑𝐻𝑑𝜂+𝐻𝑑𝑈+𝐻𝑑𝜂𝜂𝑤2𝑈−1+(𝑛+1)2+𝐻𝑈(𝑖−1)𝜂=0.(25)

From (22)–(25), we obtain 𝑑𝐷𝐷𝑑𝜂=−[]𝑈−(𝑛+1)𝑑𝑈+𝑑𝜂(𝑖+1)𝑈𝜂+(𝑛+1)𝑤𝜂,(26)𝑑𝐻𝐻𝑑𝜂=−[]𝑈−(𝑛+1)𝑑𝑈+𝑑𝜂2(𝑖+1)𝑈−2+(𝑛+1)𝑤2𝜂,(27)𝑑𝑁𝑑𝜂=𝜂𝑖𝐷(𝑖+1+𝑤),(28)𝑑𝑈=𝐻𝑑𝜂2𝜂2𝑤(𝑖+1)𝑈−1+(𝑛+1)2[(]+𝐿𝐷𝑖+1)𝑈+(𝑛+1)𝑤−𝜂2[]×𝑈−(𝑛+1)𝑈(𝑈−1)𝐷+2𝐻2+𝑖(𝑛+1)2𝐺0𝐷𝛾𝑀2(𝑖+𝑤+1)𝜂𝑖𝜂{𝑈−(𝑛+1)}2𝐷𝜂2−𝐻2𝜂2.−𝐿𝐷(29) Using the self-similarity transformations (16), the shock conditions (9) transform into 1𝑈(1)=(1−𝛽)(𝑛+1),𝐷(1)=𝛽,1𝑃(1)=(1−𝛽)+𝛾𝑀2+𝑀𝐴−2211−𝛽2(𝑛+1)2,𝐻(1)=(𝑛+1)𝛽𝑀𝐴,𝑁(1)=1.(30) The piston path coincides at 𝜂𝑝=𝑟𝑝/𝑟𝑠 with a particle path. Using (13) and (16), the relation 𝑈𝜂𝑝=(𝑛+1)(31) can be derived. In addition to shock conditions (30), the kinematic condition (31) at the piston surface must be satisfied.

Normalizing the variables 𝑢, 𝑝, 𝜌, 𝑚, and ℎ with their respective values at the shock, we obtain 𝑢𝑢2=𝑈(𝜂)𝑝𝑈(1)𝜂,𝑝2=𝜌𝜌2𝑚=𝛽𝐷(𝜂),𝑚2=𝑁(𝜂),â„Žğ‘(1)ℎ2=𝐻(𝜂)𝐻(1)𝜂.(32)

4. Results and Discussion

Distributions of the flow variables in the flow field behind the shock front are obtained by the numerical integration of (26)–(29) in the self-gravitating case and from (26), (27), and (29) in the nongravitating case with the boundary conditions (30) by the Runge-Kutta method of the fourth order. The expression for the shock-Mach number “𝑀” in the self-gravitating case with magnetic field (𝑀𝐴−2≠0) is 𝑀2=𝜇𝐵2𝜇𝐵2((1−𝛼)/2𝛼)−𝜋𝐺𝐴2𝛾𝑀𝑖/(𝑖+𝑤+1)(𝑤+1)2𝐴,(33) and in the nongravitating case with magnetic field (𝑀𝐴−2≠0), 𝐺=0,𝑀2=2𝛼𝑀(1−𝛼)𝛾2𝐴,(34) where “𝛼” is the exponent in the law of the variation of initial magnetic field. Also, the expression for the shock-Mach number “𝑀” in the nonmagnetic self-gravitating case (i.e., 𝑀𝐴−2=0) is 𝑀2=𝑛[]𝑛+(𝑛+1)𝑖(𝑛+1)−2/(𝑛+1)𝑖𝛾𝜋𝐴𝐺𝐶𝑈0𝑡𝑛02/(𝑛+1),(35) and in the nongravitating and nonmagnetic case 𝐺=0,𝑀2=∞.(36) For the purpose of numerical integration, the values of the constant parameters are taken to be [8, 10, 33]𝛾=5/3, 𝑖=2, 𝐺0=0,10,100,𝑀𝐴−2=0.0,0.01,0.02,0.05, 𝑛=−1/4,−1/6. For fully ionized gas 𝛾=5/3, and; therefore, it is applicable to stellar medium. Rosenau and Frankenthal [8] have shown that the effects of magnetic field on the flow field behind the shock are significant when 𝑀𝐴−2≥0.01; therefore, the above values of 𝑀𝐴−2 are taken for calculations in the present problem. The value 𝑀𝐴−2=0 corresponds to the nonmagnetic case. The value 𝐺0=0 corresponds to the solution in nongravitating case, the solution obtained by Nath [10] in non-rotating case when 𝛼=1/2,𝑖=1.

Table 1 shows the variation of density ratio 𝛽(=𝜌1/𝜌2) across the shock front and the position of the piston 𝜂𝑝 for different values 𝑀𝐴−2 with 𝛾=5/3, 𝐺0=0,10,100, 𝑛=−1/4,−1/6 in both the gravitating and nongravitating cases. Also, Table 1 shows that the distance of the piston from the shock front is less in the case of gravitating medium in comparison with that in the case of nongravitating medium. Physically, it means that the gas behind the shock is compressed in gravitating medium; that is, the shock strength is increased in gravitating medium.


𝑀 𝐴 − 2 𝑛 𝑤 = − 1 / ( 𝑛 + 1 ) Gravitating caseNongravitating case
𝛽 ( 𝑀 = 1 0 ) Position of the piston 𝜂 𝑝

0.0 − 1 / 4 − 4 / 3 0.2575 𝐺 0 = 1 0 𝐺 0 = 1 0 0 𝛽 ( 𝑀 = ∞ ) 𝜂 𝑝
0.7929610.8589790.25000.786644
− 1 / 6 − 6 / 5 0.25750.8283410.8719490.25000.826576

𝑀 𝐴 − 2 𝑛 α    𝑤 𝛽 Position of the piston 𝜂 𝑝
Gravitating caseNongravitating case
𝐺 0 = 1 0 𝐺 0 = 1 0 0

0.01 − 1 / 4 1 / 3   − 4 / 3 0.2733230.7629360.8501650.74148
− 1 / 6 1 / 5   − 6 / 5 0.2856260.8107850.8887470.78455

0.02 − 1 / 4 1 / 3   − 4 / 3 0.2959470.7530160.86225040.705453
− 1 / 6 1 / 5   − 6 / 5 0.3203050.7997560.8986640.747313

0.05 − 1 / 4 1 / 3   − 4 / 3 0.361060.7264030.8701990.606753
− 1 / 6 1 / 5   − 6 / 5 0.4210920.7771620.9048180.63214


(1) 𝑀 𝐴 − 2 = 0 , 𝑛 = − 1 / 4 , 𝐺 0 = 0
(2) 𝑀 𝐴 − 2 = 0 , 𝑛 = − 1 / 6 , 𝐺 0 = 0
(3) 𝑀 𝐴 − 2 = 0 , 𝑛 = − 1 / 4 , 𝐺 0 = 1 0
(4) 𝑀 𝐴 − 2 = 0 , 𝑛 = − 1 / 6 , 𝐺 0 = 1 0
(5) 𝑀 𝐴 − 2 = 0 , 𝑛 = − 1 / 4 , 𝐺 0 = 1 0 0
(6) 𝑀 𝐴 − 2 = 0 , 𝑛 = − 1 / 6 , 𝐺 0 = 1 0 0
(7) 𝑀 𝐴 − 2 = 0 . 0 1 , 𝑛 = − 1 / 4 , 𝐺 0 = 0
(8) 𝑀 𝐴 − 2 = 0 . 0 1 , 𝑛 = − 1 / 6 , 𝐺 0 = 0
(9) 𝑀 𝐴 − 2 = 0 . 0 1 , 𝑛 = − 1 / 4 , 𝐺 0 = 1 0
(10) 𝑀 𝐴 − 2 = 0 . 0 1 , 𝑛 = − 1 / 6 , 𝐺 0 = 1 0
(11) 𝑀 𝐴 − 2 = 0 . 0 1 , 𝑛 = − 1 / 4 , 𝐺 0 = 1 0 0
(12) 𝑀 𝐴 − 2 = 0 . 0 1 , 𝑛 = − 1 / 6 , 𝐺 0 = 1 0 0
(13) 𝑀 𝐴 − 2 = 0 . 0 2 , 𝑛 = − 1 / 4 , 𝐺 0 = 0
(14) 𝑀 𝐴 − 2 = 0 . 0 2 , 𝑛 = − 1 / 6 , 𝐺 0 = 0
(15) 𝑀 𝐴 − 2 = 0 . 0 2 , 𝑛 = − 1 / 4 , 𝐺 0 = 1 0
(16) 𝑀 𝐴 − 2 = 0 . 0 2 , 𝑛 = − 1 / 6 , 𝐺 0 = 1 0
(17) 𝑀 𝐴 − 2 = 0 . 0 2 , 𝑛 = − 1 / 4 , 𝐺 0 = 1 0 0
(18) 𝑀 𝐴 − 2 = 0 . 0 2 , 𝑛 = − 1 / 6 , 𝐺 0 = 1 0 0


(1) 𝑀 𝐴 − 2 = 0 . 0 1 , 𝑛 = − 1 / 4 , 𝐺 0 = 0
(2) 𝑀 𝐴 − 2 = 0 . 0 1 , 𝑛 = − 1 / 6 , 𝐺 0 = 0
(3) 𝑀 𝐴 − 2 = 0 . 0 1 , 𝑛 = − 1 / 4 , 𝐺 0 = 1 0
(4) 𝑀 𝐴 − 2 = 0 . 0 1 , 𝑛 = − 1 / 6 , 𝐺 0 = 1 0
(5) 𝑀 𝐴 − 2 = 0 . 0 1 , 𝑛 = − 1 / 4 , 𝐺 0 = 1 0 0
(6) 𝑀 𝐴 − 2 = 0 . 0 1 , 𝑛 = − 1 / 6 , 𝐺 0 = 1 0 0
(7) 𝑀 𝐴 − 2 = 0 . 0 2 , 𝑛 = − 1 / 4 , 𝐺 0 = 0
(8) 𝑀 𝐴 − 2 = 0 . 0 2 , 𝑛 = − 1 / 6 , 𝐺 0 = 0
(9) 𝑀 𝐴 − 2 = 0 . 0 2 , 𝑛 = − 1 / 4 , 𝐺 0 = 1 0
(10) 𝑀 𝐴 − 2 = 0 . 0 2 , 𝑛 = − 1 / 6 , 𝐺 0 = 1 0
(11) 𝑀 𝐴 − 2 = 0 . 0 2 , 𝑛 = − 1 / 4 , 𝐺 0 = 1 0 0
(12) 𝑀 𝐴 − 2 = 0 . 0 2 , 𝑛 = − 1 / 6 , 𝐺 0 = 1 0 0

Figures 1(a) to 1(c) and 2 show the variation of the flow variables 𝑢/𝑢2, 𝜌/𝜌2=𝑝/𝑝2, 𝑚/𝑚2, and ℎ/ℎ2, with 𝜂 at various values of the parameters 𝑀𝐴−2, 𝐺0, and 𝑛.

Figure 1(a) shows that the reduced fluid velocity 𝑢/𝑢2 increases from the shock front to the piston whereas it decreases when 𝑀𝐴−2=0, 𝑛=−1/4 and 𝑀𝐴−2=0.01or 0.02, 𝑛=−1/4or −1/6 in the case of nongravitating medium and 𝑀𝐴−2=0.01 or 0.02, 𝑛=−1/4or −1/6 in the case of gravitating medium for 𝐺0=10.

Figure 1(c) shows that the reduced mass 𝑚/𝑚2 decreases from the shock front to the piston, and Figure 2 shows that the reduced azimuthal magnetic field ℎ/ℎ2 increases from the shock front to the piston.

Figure 1(b) shows that the reduced density (pressure) (𝜌/𝜌2)(=𝑝/𝑝2) decreases; but it increases in the cases when (i) 𝑛=−1/6, 𝑀𝐴−2≠0,𝐺0≠0 and (ii) 𝑛=−1/6, 𝑀𝐴−2=0,𝐺0≠0 and starts to decrease after attaining a maximum in the cases (i) and the formation of maxima is absent in the cases (ii). As can be seen from (26) for “𝐷,” there is a singularity at the piston where 𝑈(𝜂𝑝)=(𝑛+1) because this equation becomes singular there. This singularity is removable, and a finite solution for “𝐷” is obtained as shown in Figure 1(b) (see curves 1–6) in nonmagnetic case (𝑀𝐴−2=0). In the magnetic case in Figure 1(b), the curves 7–18 show that the singularity is nonremovable, and the derivative of density tends to negative infinity.

This singularity can be physically interpreted as follows [10, 16, 18]: the path of the piston diverges from the path of the particle immediately ahead rarifying the gas.

The piston position 𝜂𝑝 at which 𝑈=(𝑛+1), is obtained after numerical integration of (26) to (29) in the self-gravitating case and from (26), (27), and (29) in the nongravitating case with the boundary conditions (30) and is tabulated in Table 1 for different values of 𝑀𝐴−2, 𝑛, and 𝐺0. The piston position 𝜂𝑝 is related with the velocity ratio of shock and piston, from (22), (23), and (24) as follows: 𝜂𝑝=1𝐶=𝑉𝑢𝑝−1.(37)

From Table 1 and Figures 1(a)–1(c) and 2, it is found that the effects of an increase in the value of 𝑀𝐴−2 (i.e., the effect of an increase in the strength of ambient magnetic field) are the following.(i)To decrease 𝜂𝑝 when 𝐺0=0 or 10 (i.e., to increase the distance of the piston from the shock front), whereas to increase it when 𝐺0=100 (i.e., to decrease the distance of the piston from the shock front). Physically it means that the gas behind the shock is less compressed when 𝐺0=0 or 10, whereas it is more compressed when 𝐺0=100, that is, to decrease the shock strength in the case when 𝐺0=0 or 10 and to increase it when 𝐺0=100 (see Table 1). Thus, the effect of an increase in the strength of initial magnetic field is reduced by an increase in the strength of self-gravitation.(ii)To increase the value of 𝛽 (i.e., to decrease the shock strength) (see Table 1).(iii)To increase the reduced fluid velocity 𝑢/𝑢2 and the reduced azimuthal magnetic field ℎ/ℎ2 in the case of self-gravitating medium but to decrease these flow variables in the case of nongravitating medium (see Figures 1(a) and 2).(iv)To increase the reduced mass 𝑚/𝑚2 but to decrease it when 𝑛=−1/6 and 𝐺0=10 or 100 (see Figure 1(c)).(v)To increase the reduced density (pressure) (see Figure 1(b)).

The effects of an increase in the value of the piston velocity exponent 𝑛 or the exponent 𝑤 for the variation of the initial density (i.e., the effects of the decrease in the value of the exponent 𝛼 for the variation of the initial magnetic field) of the ambient medium are the following.(i)To increase 𝜂𝑝, that is, to decrease the distance of the piston from the shock front. Physically, it means that the gas behind the shock is more compressed; that is, the shock strength is increased (see Table 1).(ii)To increase the value of 𝛽, when 𝑀𝐴−2≠0 (i.e., to decrease the shock strength), which is same as in (i) (see Table 1).(iii)To increase the reduced fluid velocity 𝑢/𝑢2 and the reduced azimuthal magnetic field ℎ/ℎ2 (see Figures 1(a) and 2).(iv)To decrease the reduced mass 𝑚/𝑚2 (see Figure 1(c)). (v)To increase the reduced density (pressure) in general (see Figure 1(b)).

Effects of an increase in the value of the gravitational parameter 𝐺0 are(i)to increase 𝜂𝑝, that is, to decrease the distance of the piston from the shock front (see Table 1),(ii)to increase the flow variables the reduced fluid velocity 𝑢/𝑢2, the reduced density (pressure) (𝜌/𝜌2)(=𝑝/𝑝2), and the reduced azimuthal magnetic field ℎ/ℎ2 (see Figures 1(a), 1(b) and 2), (iii)to decrease the reduced mass 𝑚/𝑚2 (see Figure 1(c)).

Acknowledgment

The authors are grateful to Dr. J. P. Vishwakarma, Professor of mathematics, DDU Gorakhpur University Gorakhpur-273009, India, for many useful discussions.

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Copyright © 2011 G. Nath and A. K. Sinha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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