Both the viscosity and rotation are found to suppress the instability of the system

Prem Kumar Bhatia and Ravi Prakash Mathur

Department of Mathematics, M. B. M. Engineering College, Faculty of Engineering, Jai Narain Vyas University, Jodhpur, India

Reprint requests to Prof. P. K. B.; E-mail: pkbhatia123@yahoo.com Z. Naturforsch. 61a, 258 – 262 (2006); received January 23, 2006

We have studied the stability of two superposed viscous compressible gravitating streams rotating about an axis perpendicular to the direction of a horizontal magnetic field. For wave propagation par- allel to the direction of the magnetic field the dispersion relation is derived by solving the linearized perturbation equations. Both the viscosity and rotation are found to suppress the instability of the system.

Key words: Rotation; Magnetic Field; Gravitating Streams; Perturbations; Instability.

1. Introduction

The study of the instability of gravitating media is important because of its relevance in the fragmentation and collapse of interstellar media.

Among others, Mouschovias [1, 2] and Mestel and Paris [3] have demonstrated the importance of the in- stability in a gravitating homogeneous static medium in the context of collapse and fragmentation in mag- netic molecular clouds. Sengar [4] examined the effect of a magnetic field on superposed gravitating homoge- neous streams while Sorker and Sarazin [5] have drawn attention to the importance of this instability problem of gravitating media in gravitational plasma filaments in cooling flows in galaxies and clusters of galaxies.

Vranjes and Cadez [6] have studied the effect of radia- tive processes on the gravitational instability in a static medium.

Singh and Khare [7] investigated the instability in two gravitating inviscid streams under a uniform hori- zontal magnetic field and a uniform rotation about the vertical. Agarwal and Bhatia [8] studied this problem for inviscid streams rotating uniformly about the axis of the magnetic field. Shrivastava and Vaghela [9] have examined the combined influence of variable streams and radiation on the magnetogravitational instability in an interstellar medium. The velocity shear instability in hydrodynamics and plasmas under varying assump- tions has been studied in recent years by several re- searchers. Among others, Benjamin and Bridges [10]

have shown that the velocity shear instability in hydro-

0932–0784 / 06 / 0500–0258 $ 06.00 c2006 Verlag der Zeitschrift f ¨ur Naturforschung, T ¨ubingen·http://znaturforsch.com

dynamics admits of a canonical Hamiltonian formula- tion. Allah [11] has studied the instability of streams in the presence of the effects of heat and mass transfer, Luo et al. [12] have examined the effect of negatively charged dust on the parallel velocity shear instability in magnetized plasmas.

In astrophysical situations the instability of gravitat- ing streams in a uniform horizontal magnetic field is interesting when the streams rotate about an axis in the horizontal plane which is perpendicular to the direction of the magnetic field. The authors [13] have recently studied this problem for inviscid streams. For more re- alistic situations the effect of viscosity of the streams on the growth rate of the unstable modes should be ex- amined. This aspect forms the subject of the present investigation where we study the problem for streams of equal kinematic viscosities.

2. Perturbation Equations

Consider two superposed gravitating viscous streams occupying, respectively, the regions z >0 and z<0. The streams are assumed to be ideally conducting and permeated by a horizontal magnetic field along the x-axis. The streams are assumed to be of uniform densities,ρ1andρ2, and uniform viscosities, µ1 and µ2, moving with uniform speeds V₁ and V₂ along the direction of the magnetic field. The whole system rotates with a uniform angular velocity Ω about an axis which is perpendicular to the magnetic field.

The relevant linearized perturbation equations are:

ρs

∂

∂^tus+ Vs·

us

=

− δ^ps+

×h

×H+ρs δφs+µs 2us

+1

3µs ( ·u_s) +2ρs

u_s×Ω,

(1)

∂

∂^ths+ V·

h= × u_s×H

, (2)

2δφs=−Gδρs, (3)

∂

∂^tδ^ps+ V_s·

δ^ps=C²_s ∂

∂^tδρs+ V_s·

δρs

, (4)

∂

∂^tδρs+ V_s·

δρs+ρs( ·u_s) =0, (5)

·h=0. (6)

In these equationsh= (h_x,h_y,h_z),δρ^,δφ ^andδ^p are the perturbations, respectively, in the magnetic field H, densityρ, gravitational potentialφ and pressure p due to the small disturbance given to the system which produces the velocity fieldu= (u,v,w)in the system.

Equations (1) to (6) are the same for the two streams, and the subscript ‘s’ distinguishes the two streams; s= 1 corresponds to the upper region z>0 and s=2 to the lower region z<0.

The stability analysis for the considered configura- tion must be investigated for the modes of propaga- tion along and perpendicular to the direction of the magnetic field, as has been studied by the authors re- cently [13] for inviscid streams. In the case of viscous superposed gravitating streams rotating about an axis perpendicular to the magnetic field, the analysis for the mode of propagation along the axis about which the system rotates, is not tractable mathematically.

For the considered configuration we, therefore, analyse mathematically the stability problem for the

mode of wave propagation having a finite component parallel to the magnetic field. We assume that all the perturbed quantities vary in space and time as

F(z)exp(ik_xx+nt), (7) where F(z)is some function of z, k_xis the wave num- ber of the perturbation along the x-axis, and n (may be complex) is the rate at which the system departs away from the equilibrium. For perturbations of the form (7), equations (1) to (6) become, on writing D≡ _dz^d:

ρs

σs−γs

D²−4

3k_x²

u_s=

−ik_xδ^ps+ik_xρsδφs−2ρsΩ^ws+1

3ik_xρsγsDw_s, (8)

ρs

σs−γs(D²−k²_x)

v_s=ik_xh_yH_s, (9) ρs

σs−γs

4 3D²−k_x²

w_s=

−Dδ^ps+ρsDδφs+H_s(ik_xh_z−Dh_x) +2ρsΩ^us+1

3ρsγsD(ik_xu_s),

(10)

σs(h_x,h_y,h_z) =H_s(−Dw_s,ik_xv_s,ik_xw_s), (11) (D²−k²_x)δφs=−Gδρs, (12) σsδ^ps=C_s²σsδρs, (13) σsδρs=−ρs(ik_xu_s+Dw_s), (14)

ik_xh_x+Dh_z=0, (15)

where we have written

σs=n+ik_xV_s. (16) On eliminating some of the variables from the above equations we finally obtain the sixth-order differential equation inδφ^:

(D²−k²_x)(D²−αs²)(D²−βs²)δφs=0, (17) where

αs²+βs²= 1 N_s

σs²(M_s²+C_s²) +M_s²C_s²k²_x+γsσs

7 3σs²+8

3σsγsk²_x + 7

3M_s²+2C_s²

k²_x−Gρs

, (18)

αs²βs²= 1 N_s



σs⁴+σs²

(M_s²+C_s²)k²_x−Gρs+4Ω²+M_s²k²_x(C²_sk²_x−Gρs)+

σsγsk²_x 7 3σs²+4

3σsγsk²_x+ 4

3M²_s+C²_s

k²_x−Gρs



, (19)

N_s=σsγs

M_s²+C²_s+4 3σsγs

. (20)

3. The Superposed Streams

Now we seek the solutions of (17) which remain bounded in the regions occupied by the streams. The solutions appropriate to the two streams are therefore

δφ1=A₁exp(−α1z) +B₁exp(−β1z)

+E₁exp(−k_xz) (z>0), (21) δφ2=A₂exp(α2z) +B₂exp(β2z)

+E₂exp(k_xz)(z<0), (22) where A₁, B₁, E₁, A₂, B₂, E₂are constants of integra- tion to be determined by applying appropriate bound- ary conditions to the solutions. In writing the solutions

forδφ1andδφ2 it is assumed thatα1,β1,α2,β2are so defined that their real parts are positive. The six boundary conditions which must be satisfied at the in- terface z=0 are:

(i) continuity of the perturbed gravitational poten- tial, i.e.δφ1=δφ2;

(ii) continuity of the normal derivative of the per- turbed gravitational potential, i. e. D(δφ1) =D(δφ2);

(iii) continuity of the total perturbed pressure, i.e.δ^p1+H(h_x)1=δ^p2+H(h_x)2;

(iv) uniqueness of the normal displacement at any point (fluid element) on the interface, i. e. ^w_σ¹

1 =^w_σ²

2; (v) continuity of the normal derivative of the dis- placement, i. e. D

w₁ σ1

=D w₂

σ2

;

(vi) continuity of the tangential component of viscous stresses, i. e. µ1(D²+k²_x)

w1

σ1

=µ2(D²+ k²_x)

w₂ σ2

. Now, on eliminating u_s,δ^ps, h_x, and h_z, (10) can be written as

w_s

σs =ρsk_x²G+

C_s²k²_x+σs²+1

3γsσsk²_x−γsσs(D²−k²_x)

(D²−k²_x)δφs

ρsσsG[σsD−γs(D²−k²_x)D−2ik_xΩ] . (23) On applying the solutions (21) and (22) and using (23) the six boundary conditions lead to the following six relations:

A₁+B₁+E₁−A₂−B₂−E₂=0, (24)

α1A₁+β1B₁+kxE₁+α2A₂+β2B₂+kxE₂=0, (25) T₁A₁+T₂B₁−ρ1GM₁²k_x⁵E₁+T₃A₂+T₄B₂+ρ2GM₂²k⁵_xE₂=0, (26) Q₁A₁+Q₂B₁+S₁E₁+Q₃A₂+Q₄B₂+S₂E₂=0, (27) α1Q₁A₁+β1Q₂B₁+k_xS₁E₁+α2Q₃A₂+β2Q₄B₂+k_xS₂E₂=0, (28) ρ1γ1[(α1²+k²_x)Q₁A₁+(β1²+k²_x)Q₂B₁+2k_x²S₁E₁]+ρ2γ2[(α2²+k²_x)Q₃A₂+ (β2²+k²_x)Q₄B₂+2k_x²S₂E₂] =0, (29) where

T₁= (α1²−k²_x)

σ1²α1²C₁²k²_x+M₁²C₁²k⁴_x+α1γ1σ1k⁴_x

C₁²−1 3α1²M₁²

−2iα1²k_xσ1Ω(M₁²+C₁²)

−Gρ1M₁²k²_xα1³, (30)

Q₁=ρ2σ2[−σ2α1+γ2α1(α1²−k²_x)−2ik_xΩ]

ρ1Gk²_x+ σ1²+

C²₁+1 3γ1σ1

k_x²

(α1²−k_x²)−σ1γ1(α1²−k²_x)² , (31)

S1=σ2ρ1ρ2Gk³_x(−σ2−2iΩ). (32)

The coefficient T₂ is obtained by changingα1 toβ1

in T₁. T₃ is obtained from T₁ by changing σ1 to σ2, α1toα2, C₁to C₂, M₁to M₂,ρ1toρ2,γ1toγ2and G

to−G. The coefficient T₄is obtained from T₃by chang- ingα2toβ2. The coefficient Q₂is obtained from Q₁by changingα1toα2, i to−i and interchangingρ1andρ2,

Fig. 1. Plot of the growth rate against the wave number.

σ1 andσ2,γ1 andγ2. On changing α2 to β2 in Q₃, we get Q₄. The coefficient S₂ is obtained from S₁by changingσ2toσ1and i to−i.

4. The Dispersion Relation

For non-trivial solution the determinant of the ma- trix of the coefficients of A₁, B₁, E₁, A₂, B₂, E₂in (24) to (29) must vanish. This gives the dispersion relation.

Since the dispersion relation is quite complicated, it is not possible to solve it analytically. One has therefore to solve it numerically to ascertain the influence of the effects of rotation and viscosity on the instability of the system. As our objective is to determine qualitatively the effect of viscosity and rotation on the growth rate of the unstable mode of disturbance, and as the disper- sion relation is quite complicated, we consider a sim- ple model of the superposed streams. We consider the case of two streams of the same density and the same kinematic viscosity flowing past each other with the same velocity in opposite directions. We also assume that the Alfven velocities and the sound speeds in the two streams are the same. We, therefore have

ρ1=ρ2, γ1=γ2, M₁²=M₂², C₁²=C₂², V₁=V, V₂=−V. (33)

The dispersion relation simplifies considerably for these values of the parameters. Although the model of the streams considered is highly idealized, it is nev- ertheless hoped that it will reveal the essential features of the effects of viscosity and rotation on the instability of the system. The same model was earlier considered by Singh and Khare [7] for inviscid streams, i. e. when γ1=γ2=0.

5. Conclusions

For several values of the parameters characterizing the viscosity and rotation, the dispersion relation has been solved numerically for fixed values of the Alfven velocity and the speed of sound. These calculations are presented in Figs. 1 and 2, where we plot the growth rate (positive root of n) against the wave number k_xfor γ(viscosity) = 1.0, 2.0, 3.0, 4.0 and rotation (Ω^{) = 1.0,} 2.0, 3.0 for fixed values of the other parameters.

From Fig. 1 we see that the growth rate decreases with the increase inΩ, showing thereby that rotation has a stabilizing influence on the unstable mode of dis- turbance. We also see from Fig. 2 that the growth rate decreases with the increase inγ. The viscosity, there- fore, suppresses the instability of the system.

Fig. 2. Plot of the growth rate against the wave number.

We may thus conclude that both the viscosity and ro- tation suppress the instability of the superposed grav- itating streams when the streams rotate about an axis in the horizontal plane which is perpendicular to the direction of the horizontal magnetic field. The results obtained agree with the earlier observations of Agar- wal and Bhatia [8].

Acknowledgements

This work was started during the tenure of a UGC Emeritus Fellowship to P.K.B. and was completed un- der the AICTE Emeritus Fellowship to him. P. K. B.

gratefully acknowledges the financial support provided under these fellowships.

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