J. Astrophys. Astr. (December 2017) 38:64 © Indian Academy of Sciences https://doi.org/10.1007/s12036-017-9480-7

Jeans Instability of the Self-Gravitating Viscoelastic Ferromagnetic Cylinder with Axial Nonuniform Rotation and Magnetic Field

JOGINDER SINGH DHIMAN and RAJNI SHARMA∗

Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla 171 005, India. ∗Corresponding author. E-mail: [email protected]

MS received 15 March 2017; accepted 24 May 2017; published online 27 November 2017

Abstract. The effects of nonuniform rotation and magnetic field on the instability of a self gravitating infinitely extending axisymmetric cylinder of viscoelastic ferromagnetic medium have been studied using the Generalised Hydrodynamic (GH) model. The non-uniform magnetic field and rotation are acting along the axial direction of the cylinder and the propagation of the wave is considered along the radial direction, while the ferrofluid magnetization is taken collinear with the magnetic field. A general dispersion relation representing magnetization, magnetic permeability and viscoelastic relaxation time parameters is obtained using the normal mode analysis method in the linearized perturbation equation system. Jeans criteria which represent the onset of instability of self gravitating medium are obtained under the limits; when the medium behaves like a viscous liquid (strongly coupled limit) and a Newtonian liquid (weakly coupled limit). The effects of various parameters on the Jeans instability criteria and on the growth rate of self gravitating viscoelastic ferromagnetic medium have been discussed. It is found that the magnetic polarizability due to ferromagnetization of medium marginalizes the effect of non-uniform magnetic field on the Jeans instability, whereas the viscoelasticity of the medium has the usual stabilizing effect on the instability of the system. Further, it is found that the cylindrical geometry is more stable than the Cartesian one. The variation of growth rate against the wave number and radial distance has been depicted graphically.

Keywords. Gravitational instability—viscoelastic medium—non uniform magnetic field—non-uniform rota- tion—ferromagnetic medium—strongly/weakly coupling limit—magnetic polarizability.

1. Introduction (Lorentz force) and observed that the Jeans criterion remains unaffected in the presence of these effects. Fur- The study of gravitational instability of interstellar ther, Argal et al. (2014) revealed in the study of Jeans medium plays a crucial role in understanding the star instability of rotating self-gravitating viscoelastic fluid formation and its evolution, gravitational collapse, for- that the rotation plays a significant role in structure mation and evolution of interstellar molecular clouds processes of compact systems such as neutron stars, and galactic structure. The gravitational instability white dwarf stars and supernovae. The instability prob- occurs when an object’s self gravity exceeds the oppos- lems in astrophysical domain are generally investigated ing forces such as internal gas pressure or material by considering the uniform character of magnetic field rigidity, and the object collapses. Sir James Jeans was and rotation. Larson (2003) in the study related to the the first who gave the criterion for the onset of grav- star formation revealed that this idealization of uniform itational instability in 1902 and now known as Jeans character of magnetic field and rotation in theoretical criterion. Jeans (1929) considered an infinite homoge- studies is valid only for laboratory purposes, however neous self gravitating medium and derived the criterion in the interstellar interior and atmosphere, the mag- in terms of wave number. The self gravitational insta- netic field and rotation may be non-uniform/variable bility was later investigated by many authors including and may alter the nature of the instability. He also Chandrasekhar (1961), Miyama et al. (1984), Larson observed that the magnetic field is an important com- (1985)andBinney & Tremaine (1987) under the effects ponent in supporting molecular clouds against gravity. of uniform rotation (Coriolis force) and magnetic field The effect of non-uniform rotation on the onset of 64 Page 2 of 10 J. Astrophys. Astr. (December 2017) 38:64 self-gravitational instability has been investigated by the pulsars were identified, in the year 1968, as highly Bel & Schatzman (1958), Anand & Kushwaha (1962), compact neutron stars in rotation formed by the grav- Simon (1962) and recently Dhiman & Dadwal (2010, itational collapse of ordinary stars at the end of their 2011) have studied the combined effects of non-uniform thermonuclear evolution, possibly after a supernova rotation and magnetic field on the self-gravitating explosion. Ray & Banerji (1980) further reported that gaseous medium and observed that the Jeans criteria iron has the largest binding energy per nucleon, it is the gets modified due to the non-uniform rotation. end product of thermonuclear evolution. If super mas- Janaki et al. (2011) in the study of Jeans instabil- sive objects exist as suggested by some astrophysicists ity in a viscoelastic fluid supposed that the transitions (cf. Baade & Zwicky 1934), then they are expected to between elastic solid state and viscous liquid state give contain a large proportion of iron at the end and the the characteristics of viscoelastic plasma in which both interior of evolved objects will therefore contain a large properties exist together. Kaw & Sen (1998), Janaki proportion of ferrous material and may play an impor- et al. (2011), Rosenberg & Shukla (2011)andPrajapati tant role in gravitational collapse. and Chhajlani (2013) have also studied the behavior of Laroze et al. (2013) defined the ferrofluids as the viscoelastic fluid by considering the different problems magnetic stable colloidal suspensions of magnetic in the domain of Generalized Hydrodynamic model nanoparticles dispersed in a carrier liquid. They reported (GH) as proposed by Frenkel (1946). In GH model, that in the absence of an external magnetic field, the the generalized hydrodynamical equations of momen- magnetic moments of the particle are randomly orien- tum transfer for viscoelastic medium incorporates  the tated and there is no net macroscopic magnetization. + τ ∂ Frenkel’s term or viscoelastic operator 1 ∂t (cf. In an external magnetic field, however, the particle’s Janaki et al. 2011) which accounts for the relaxation magnetic moments easily orient and a large (induced) effects arising out of growing correlations among the magnetization is present. They investigated the Bénard– particles. Further, we note that the GH model is identi- Marangoni instability in a viscoelastic ferrofluid and cal to the well-known MHD model with this difference emphasize the effects of the viscoelasticity and the in the equation of momentum transfer. Kelvin force on the instability thresholds. The various Recently, Dhiman and Sharma (2014) studied the aspects of ferrohydrodynamics have been discussed in onset of gravitational instability of a magnetized vis- detail by Rosensweig (1985) in his book. coelastic medium in the longitudinal and transverse Mamun and Shukla (2002) also supported the exis- modes of wave propagation under the strongly and tence of ferromagnetic dust particles in a magnetically weakly coupling limits in the presence of rotation. They supported dark interstellar self-gravitating interstellar observed that the instability criteria get modified due molecular cloud. They theoretically studied the propa- to the viscoelastic effects under the strongly coupling gation characteristics of magnetized waves, as well as limit, whereas rotation has stabilizing effect on the the instabilities of sound waves in a self-gravitating dark growth rate of instability. Sharma et al. (2015) in the interstellar molecular cloud containing ferromagnetic study of Jeans instability of rotating viscoelastic fluid dust grains and baryonic gas clouds. In addition to the in the presence of uniform magnetic fluid, reported that usual Jeans instability, they also observed that the sound the strongly coupled plasmas has importance in the inte- waves suffer a new type of instability, which is due to rior of large planets, white dwarfs, neutron star, highly the combined effects of the baryonic gas dynamics and compressed solids, semiconductor devices, electrical self-gravitational field in both weakly and highly colli- discharge, nuclear matter, dusty and laser-plasma. sional regimes. Jones & Spitzer (1967) gave a model for the exis- Odenbach (2003) studied the magnetoviscous and tence of gas-dust interstellar medium with a highly viscoelastic effects in ferrofluids and discussed vari- pronounced property of magnetic polarizability. They ous effects due to the influence of magnetic fields on reported that, the existence of such a type of medium the rotation of single magnetic nanoparticles as well may be due to a super paramagnetic dispersion of the as cooperative phenomena and their importance for fine ferromagnetic grains suspended in a gaseous cloud viscous effects in ferrofluids. Mathew et al. (2013) of molecular hydrogen. Ray & Banerji (1980)inthe studied the gravitational instability in a ferromagnetic study of perfectly conducting ferrofluid in general rel- fluid saturated porous medium with non-classical heat ativity reported that the pulsars are the astronomical conduction and investigated the influence of porous, objects which are formed after a supernova explosion magnetic and non-magnetic parameters on the onset of and occurs in massive stars collapsing at the end of ferroconvection. They found that the Bénard problem their thermonuclear evolution. They also opined that for a MaxwellÐCattaneo ferromagnetic fluid is always J. Astrophys. Astr. (December 2017) 38:64 Page 3 of 10 64 less stable than the classical ferroconvection problem. the analysis of Janaki et al. (2011), Dhiman & Dadwal Ram & Sharma (2014) also studied the effect of rota- (2012)andDhiman & Sharma (2016) to investigate the tion and MFD viscosity on ferrofluid flow with rotating effects of viscoelasticity and magnetic polarizability on disk and calculated the displacement thickness of the the onset of gravitational instability of ferromagnetic boundary layer, angle of rotation and an expression for medium. total volume flowing outward the z-axis. Cylindrical structures of astronomical objects are of great importance in many ways as reported by Dhi- 2. Mathematical formulation of the problem man & Sharma (2016) in their work related to the viscoelastic effects on the onset of gravitational instabil- Consider an infinitely extending homogeneous, self ity in the simultaneous presence of non-uniform rotation gravitating, infinitely, electrically conducting viscoelas- and magnetic field in cylindrical geometry under the tic ferromagnetic medium permeated with non-uniform strongly and weakly coupling limits. They observed that magnetic field and rotation. the viscoelasticity of the medium modifies the instabil- We shall study the onset of gravitational instability ity criterion under the strongly coupling limit. Dhiman of this physical system by considering the GH model &Dadwal(2012) have studied the effect of ferromag- for viscoelastic fluid and the set of ferrohydrodynamic netic dust cloud on the gravitational instability of a equations given by Shliomis (1972, 2001, 2002)forfer- gaseous axisymmetric cylinder in the presence of non- romagnetic medium. Further, it is assumed that the fluid uniform rotation and magnetic field and observed that and the ferrous particles have the same velocity while the effect of non-uniform magnetic field on the grav- writing the modified momentum transfer equation. itational instability is marginalized by the magnetic Rosensweig (1985) in his study of ferrohyrody- polarizability of ferrofluid. namics showed that due to the presence of ferrofluid, On the basis of the above discussions and motivated an additional force (magnetic force) comes into play by various studies related to the possibility of presence in the momentum transfer equation, which is given of ferromagnetic medium with strongly coupled effects by in some astrophysical objects; (in atmosphere of neu- ⎡ ⎤ H   tron star and supernova) our aim here is to investigate ∂ M υ 1 f =−∇⎣μ H + μ H 2⎦ the Jeans instability of an axisymmetric cylinder of a m 0 ∂υ d 0 , 2 ferromagnetic viscoelastic medium in the presence of 0 H T non-uniform rotation and magnetic field. Further, the + B ∇ H considered physical model is justified as the neutron star    is supposed to be composed of viscoelastic fluid with which upon utilizing the relation; B = μ0(H + M) −1 magnetic field H0 ∼ 1013G and rotation  ∼ 100 s yields (cf. Cumming et al. 2004; Rudiger et al. 2009; Potekhin ⎡ ⎤ H   & Chabrier 2000) shall be useful to study the instability ∂ M υ f =−∇⎣μ H⎦ + μ M ·∇H. in the self-gravitating neutron star incorporating effects m 0 ∂υ d 0 , of both elastic, viscosity and ferromagnetism. 0 H T

In the present paper, the self-gravitational instabil-  Also, we can have μ M ∇ H =μ M 1 ∇ (H · H)=μ ity of an infinitely extending axisymmetric cylinder 0 0 H 2 0 of viscoelastic ferromagnetic medium permeated with   (M ·∇)H,whereμ0 is the permeability of free space non-uniform magnetic field and rotation is studied −7 −1 and μ0 = 4π × 10 Henry m . under both the strong and weakly coupling limits. The In view of above discussion, it is to note that due to the physical problem is mathematically formulated using presence of ferrofluid the momentum transfer equation the GH model. The non-uniform magnetic field and given by Janaki et al. (2011) using the GH model gets rotation are considered to act along the axial direction of modified. Therefore, the equation of continuity, modi- the cylinder. The dispersion relation is obtained by using fied momentum transfer equation, Poisson equation and the normal mode analysis and further analyzed under the ferrofluid magnetization that govern the above physical strongly and weakly coupling limits. The effects of wave model are now given by (cf. Janaki et al. 2011; Rosen- number and radial distance on the growth rate of Jeans berg & Shukla 2011; Shliomis 1972, 2001, 2002) instability under both strongly and weakly coupling lim- ∂ρ its have been calculated and the obtained values are + (u · grad ) ρ + ρ (∇·u) = 0, (1) depicted graphically. The present paper thus extends to ∂t 64 Page 4 of 10 J. Astrophys. Astr. (December 2017) 38:64   ∂ ∂u Moreover, in deriving equation (2) we have used the 1 + τ ρ + (u · grad) u − ρ grad φ ∂ ∂ Maxwell constitutive relation given by t t        + ∗ + μ  ·∇  − ρ  ×  ∂ ∂u grad p 0 M H 2 u 1+τ ρ +(u. grad ) u ∂t ∂t μ  = μ∇2u + ξ + ∇(∇·u), (2) 3 − ρ grad φ + grad p = grad σij, ∇2φ =− π (ρ − ρ ), 4 G 0 (3) σ    where ij represents the strain tensor. dM 1    Following Spiegel and Thiffeault (2003), equation = (∇×u) × M − α M − M0 dt 2   (3) representing the gravitational potential above is − β M × M × H . (4) considered so as to avoid Jeans Swindle. If we con- sider M collinear with H at any moment and deter- In Chu formation of electrodynamics (cf. Penfield & mined by its instaneous value, and consequently in Haus 1967), which gives the relation amongst the mag- the phenomenological magnetization equation given by netic field H, ferrofluid magnetization M and magnetic  Shliomis (2002), both the magnetization relaxation time induction B, (α) and M × H vanishes. Also, (M ·∇)H vanishes      = μ  +  . due to collinearty of M and H. Hence, initially when B 0 H M (5)  = ,ρ = ρ , = ,φ = φ , ∂ =  ,  u 0 0 p p0 0 ∂t 0andM H In addition to the above equations, we have the fol- are collinear equations (1)Ð(4) are identically satisfied, lowing set of magnetic induction equations derived thus avoiding the Jeans Swindle in the present case. from Maxwell’s equations. In the present paper, we shall consider an infinitely extending axisymmetric cylinder of a homogeneous, ∇×H = 0, (6) self gravitating, infinitely electrically conducting vis- ∇·B = 0, (7) coelastic ferromagnetic medium permeated with non- uniform magnetic field and rotation and investigate the we have onset of instability of this otherwise, stable configura- tion. In order to study the gravitational instability of   ∇·H =−∇·M. (8) axis-ymmetric configurations, the system is described , ,  , ,   by means of cylindrical coordinates. The cylinder is In the above equations, u H M B and r respec- assumed to be rotating about its axis (z-axis) with non- tively represent the velocity, magnetic field, ferrofluid uniform angular velocity ω. The propagation of wave is magnetization, magnetic induction, rotation and posi- r τ,ρ, φ taken along the radial direction of the cylinder, hence tion vectors respectively; and denote the ∂/∂r is the only non zero component of the gradient. viscoelastic relaxation time, density of fluid and gravita- et al. μ, ξ, μ , Now, proceeding as in Janaki (2011), Dhiman tional potential; 0 G and cs respectively denote &Dadwal(2012)andDhiman & Sharma (2016), the the coefficient of viscosity, coefficient of bulk viscos- system of generalized basic hydrodynamic equations ity, magnetic permeability, the universal gravitational (1)Ð(4), (6)and(8) that governs this physical configura- constant and the speed of sound in isothermal medium. η tion representing the equation of continuity, equations Further, α = τ = 3V is the Brownian time (magne- B KbT of motion, Poisson equation, ferrofluid magnetization tization relaxation time) of rotational particle diffusion, and magnetic induction under these assumptions in β = 1 , η 6ηϕ where is the dynamic viscosity of the carrier cylindrical coordinates are respectively given by fluid, ϕ = nV is the volume fraction of magnetic grains in the liquid. Here, n is the number density and V the ∂ρ + (u · grad) ρ + ρ (∇·u) = 0, (9) volume of a single particle. ∂t     Also,inequation(2), ∂ ∂ u2 τ ρ ur θ     1 + + (u · grad) ur − H ∂  H ∂t ∂t r ∗ = +μ υ M +μ  ,  p p 0 dM 0 MdH ∂ ∂ φ ∂ ρ 0 ∂υ 0 Hr Mθ H θ 2 H,T − μ M − − ρ + c 0 ∂r r r ∂r s ∂r ∗  where p = p + ps + pm is the composite pressure ρ and p, ps and pm are the hydrostatic pressure, mag- − 2 (uθωz −uzωθ) netostrictive and fluid magnetic pressure, respectively. J. Astrophys. Astr. (December 2017) 38:64 Page 5 of 10 64  ∂2 ∂ = μ ur + 1 ur − ur In view of the above physical configuration, the ∂r 2 r ∂r r 2 equilibrium (basic) state under discussion is clearly   2 characterized as μ ∂ ur 1 ∂ur ur + ξ + + − , (10)   ∂ 2 ∂ 2 u = (0, rω,0); H = (0, 0, H ); M = (0, 0, M ); 3 r r r r  z z ∂ ∂uθ u uθ ω = (0, 0,ω); φ = φ0 and ρ = ρ0. (20) + τ ρ + (u. ) u − r 1 ∂ ∂ grad θ t t r Using the above basic state described in equation (20) μ0 ∂ Hθ ∂ Mθ ∂ Hr in equations (9)Ð(19), we have the following equations; − M + H + Mθ    ∂r r r ∂r ∂r ∂ ∂φ ∂ρ 2  + τ ρ ω2 − − ω + 2 = , 1 r 2uθ cs 0 ∂ Mr 2Mθ Hr ∂t ∂r ∂r − Hθ + ∂ (21) r r     1 ρ∂φ 1 ∂ρ ∂ ∂rω ∂2rω 1 ∂rω rω − + c2 −2 (u ω − u ω ) 1 + τ ρ = μ + − , r ∂θ s r ∂θ z r r z ∂t ∂t ∂r 2 r ∂r r 2  2 ∂ uθ 1 ∂uθ uθ  (22) = μ + − , (11) ∂ 2 ∂ 2 ∂ r r r r  1 + τ (−2ρuθ ω) = 0, (23) ∂t ∂ ∂uz  1 + τ ρ + (u · grad) u ∂ ∂ z 1 ∂ ∂φ0 t t r = 0, (24) μ ∂ H ∂ M ∂ H r ∂r ∂r − 0 z M + H z + M r ∂ r r ∂ z ∂ ∂ Hz 2 r r  r = 0. (25) ∂ − ∂φ ∂ρ ∂r Mr Mz Hr Mr Hz 2 − Hz + − + c From equation (25), we get ∂r r ∂z s ∂z   2 H = constant. (26) ∂ uz 1 ∂uz z − 2 ρ (u ωθ − uθω ) = μ + , r z ∂r 2 r ∂r In order to investigate the gravitational instability of the considered system, let the infinitesimally small per-  (12) turbations are added to the initial state of the system 1 ∂ ∂φ0 r =−4πG (ρ−ρ0) (13) represented in equation (20). So, we have the following r ∂r ∂r   perturbed quantities: ∂ ∂ ∂ ∂ Mr + Mr + 1 uθ + uθ + uz ur Mθ Mz u = (0 + u , rω + uθ , 0 + u ) ; ∂t ∂r 2 ∂r r ∂r r z     H = (0 + hr , 0 + hθ , Hz + hz) ; + β M M H − H Mθ θ r θ r    M = (0 + m , 0 + mθ , M + m ) , − − = , r z z Mz Mz Hr Hz Mr 0 (14)  φ = φ0 + δφ and ρ = ρ0 + δρ . (27) ∂ Mθ ∂ Mθ 1 ∂uθ uθ + ur − Mr + Here, (u , uθ , u ) , (h , hθ , h ) , (m , mθ , m ) ,δφand ∂t ∂r 2 ∂r r r z r z r z    δρ are the perturbations in the basic velocity, magnetic + β − Mz Mθ Hz Hθ M z field, magnetization, gravitational potential and density,   respectively. − M M Hθ − H M = 0, (15) r r r θ  Using these perturbed quantities defined in equa- ∂ Mz ∂ Mz 1 ∂uz tion (27) in equations (9)Ð(19), using the equations + ur − Mr ∂t ∂r 2 ∂r (21)Ð(25) and then linearizing the resulting equations    by neglecting the second and higher order perturbed + β M M H − H M r z r z r quantities, we get the following linearized perturbation − − = , Mθ Mθ Hz Hθ M z 0 (16) equations: ∂ Hz ∂δρ ∂ = 0, (17) + ρ ur + ρ ur = , ∂r 0 0 0 (28)  ∂t ∂r r   ∂ Hθ Hθ ∂ ∂ ∂δφ ∂δρ + = , ur 2 ∂ 0 (18) 1 + τ ρ −2uθ ω − + c r r ∂t ∂t ∂r s ∂r     ∂ Hr Hr ∂ Mr Mr 2 + =− + . 4μ ∂ ur 1 ∂ur ur ∂ ∂ (19) = ξ + + − , (29) r r r r 3 ∂r2 r ∂r r 2 64 Page 6 of 10 J. Astrophys. Astr. (December 2017) 38:64     ∂ ∂ ∂ ω 2 uθ r 4μ ∂ ur 1 ∂ur ur 1 + τ ρ + ur + ur ω = ξ + + − , (41) ∂t ∂t ∂r ∂ 2 ∂ 2   3 r r r  r ∂2uθ 1 ∂uθ uθ d(rω) = μ + − , (30) (1 + τσ) σuθ + ur + ur ω ∂r 2 r ∂r r 2 dr   2 ∂ ∂uz μ0 ∂ Mz 1 ∂ (rhr ) ∂ uθ 1 ∂uθ uθ 1 + τ ρ − hr + Mz = ν + − , (42) ∂t ∂t 2 ∂r r ∂r ∂ 2 ∂ 2   r r r r 2 1 ∂ (rmr ) ∂ uz 1 ∂uz μ0 dMz 1 d (rhr ) + Hz = μ + , (31) (1 + τσ) σuz − hr + Mz r ∂r ∂r2 r ∂r 2ρ dr r dr    1 ∂ ∂δφ 1 d (rm ) ∂2u 1 ∂u r =−4πGδρ, (32) + H r = ν z + z , (43) r ∂r ∂r z ∂ 2 ∂ r dr r r r ∂hz 1 d dδφ = 0, (33) r =−4πGδρ, (44) ∂r  r dr dr ∂hθ hθ + = , dhz ∂ 0 (34) = 0, (45) r r dr 1 ∂ (rmr ) 1 ∂ (rhr ) 1 d (rhθ ) =− , (35) = 0, (46) r ∂r r ∂r   r dr ∂mr 1 ∂ (rω) ∂uz 1 d (rmr ) 1 d (rhr ) + mθ + ω + M =− , ∂ ∂ z ∂ (47) t  2  r  r r dr r dr   1 d (rω) du + β −Mz Mzhr − Hzm = 0, (36) z  r σmr + mθ + ω + Mz ∂mθ 1 ∂rω  2  dr  dr − mr + ω ∂ ∂ + β −Mz Mzhr − Hzm = 0, (48) t  2  r  r 1 drω + β Mz mθ Hz − hθ M = 0, (37) z σmθ − mr + ω ∂ ∂ 2 dr mz + Mz = .    ur 0 (38) + β − = , ∂t ∂r Mz mθ Hz hθ M z 0 (49) dM σm + u z = 0. (50) z r r 3. Gravitational instability d We shall now investigate the local stability of the In order to investigate the stability of the above station- above system in the neighborhood of r = r0,since ary state, we shall apply the normal mode method and the above equations involve variable coefficients. So, thus considering the dependence of the perturbations on following the analysis adopted in Dhiman & Dadwal r and t of the form; (2012), let us assume that the perturbations have the ∗ following periodic form in the neighborhood of r = r0, ψ (r) exp (σt). (39) exp (−ikr), Here σ is the frequency of the perturbation. Using the above dependence (39) of the perturbations where k is the wave number. For this type of depen- ∗ representing one of the perturbed quantities; ψ (r) ≡ dence, we have u , uθ , u , m , mθ , m ,ω ,ωθ ,ω , h ,hθ,h , δρ, δφ , r z r z r z r z / ≡− . we have d dr ik ∂ ∂ d ≡ σ ψ∗(r) ≡ ψ∗(r). Now, using this dependence in equations (40)Ð(50) ∂ and ∂ t r dr and then substituting the values of δρ, F, T and ur as So, in view of the above dependence, the linearized defined by Dhiman & Sharma (2016) in equations (60)Ð perturbation equations (28)Ð(38) have the following (64) in the resulting equations, we have the following forms after simplifications; equations, obtained after some simplifications;    ρ d (ru ) 2 2 2 0 r (1 + τσ) σ −4πGρ + c k T −2ωuθ σδρ + = 0, (40) s r  dr  4μ  δφ 2 δρ ξ + σ d cs d 3 2 1 (1 + τσ) σur −2uθ ω − + =− k + T, (51) dr ρ dr ρ r 2 J. Astrophys. Astr. (December 2017) 38:64 Page 7 of 10 64     σ T 2 1 2 1 (1 + τσ) σuθ + F =−ν k + uθ , A = (1 + τσ) σ + ν k + , ρ r r 2 r 2 (52) = + , = 2 2 − π ρ.   Bz Mz H z Q cs k 4 G μ ( + τσ) σu − 0 ( ιkm M + ιkH m ) It is to note here that dispersion relation equation (59) 1 z ρ 2 r z z r 2 includes the effect of shear, bulk viscosity, magnetic 2 1 permeability, non-uniform rotation, ferro fluid magne- =−ν k + uz, (53) r 2 tization of the medium and magnetic field respectively   ιkMzuz 1 represented by ν, ξ, μ0,ω,M and H. σmr − − mθ F  2 2  ⎧ It is clear from relation (59) that, either μ  + β M M m + H m = 0, (54) ⎨ ξ + 4 σ 2 2 z z r z r   3 1 (1+τσ) σ 3 +Qσ A+ k2 + A 1 ⎩ ρ 2 σmθ + mr F + βmθ Hz Mz = 0, (55) r 2 ⎫ ιkσ TM ⎬ σm + z = 0. (56) z ρ + ( + τσ)2 ωσ 2 = r 1 2 F⎭ 0(60) From equation (52), we have or  (1 + τσ) σTF 1 =  . ( + τσ) σ + ν k2 + uθ (57) 1 2 ρr ( + τσ) σ + ν k2 + 1  r  1 r2 1 2 × (σ +Mzβ Bz)(σ + Mzβ Hz) + F Further, equation (51) upon using the value of uθ from  4  the equation (57), yields the following equation; μ0 ιk − (1 + τσ) 2ιkBz MZ (σ +Mzβ Hz) = 0.   2ρ 2 1 (1 + τσ) σ + ν k2 + (1 + τσ) (61) r 2    The relations (60)and(61) respectively represent the σ 2−4πGρ+c2k2 T + (1 + τσ)2 2ωσ FT s self-gravitating mode and non self-gravitating mode for   ξ+ 4μ σ 1 3 viscoelastic ferromagnetic medium. + (1 + τσ) σ + ν k2 + As discussed by Prajapati and Chhajlani (2013)and r 2 ρ  Dhiman & Sharma (2016) in their studies, under the 1 × k2 + T = 0. (58) assumption of the strongly coupling limit, the wave r 2 frequency is much greater than the inverse of the vis- coelastic relaxation time and is represented as στ  1, Hence, equations (53)Ð(56)and(58) yields the fol- and the medium behaves like a viscous liquid (SCP) or lowing dispersion relation in view of the existence of solid and hence τ can be retained in the equation of non-trivial solution of these equations; motion. However, under the weakly coupling limit, the    (1 + τσ) σ 3 + Qσ A wave frequency is much lower than the inverse of vis- ⎫ coelastic relaxation time τ and is represented as στ  1 μ  ξ + 4 σ 2 2 ⎬ and the medium behaves like a liquid and the effect of 3 2 1 2 2 + k + A+(1+τσ) 2 ω σ F τ can be ignored in this limit. ρ r 2 ⎭   We shall now discuss the gravitational instability of 1 the system under both the strongly and weakly cou- × (1 + τσ) σ + ν k2 + r 2 pling limits and shall derive the instability criteria for   the present problem from the relation (60). 1 2 × (σ +Mzβ Bz)(σ +Mzβ Hz) + F Under the strongly coupling limit (στ  1), the dis-  4  persion relation (60) reduces to μ ι     0 k ν2 + ν − (1+τσ) 2ιkBz M (σ +M β Hz) = 0. 2 c 1 2ρ 2 Z z σ 4 + σ 2ωF + Q + k2 + τ r 2 (59)    1 ν 1 − Q + ν2 k2 + k2 + = 0. (62) Here, c r 2 τ r 2 64 Page 8 of 10 J. Astrophys. Astr. (December 2017) 38:64

The constant term of the above equation yields the In order to study the effects of these parameters on criterion for the onset of gravitational instability, as the growth rate, under the strongly and weakly coupling   limits respectively, writing equations (62)and(65)in ν 1 1 k2 + Q+ν2 k2 + = 0. (63) the following dimensionless forms; τ 2 c 2 r r  ∗ ∗  ∗ ∗ ξ + ν ∗ 1 After simplification of equation (63), we get the follow- γ 4 + γ 2 −4ω 2+k 2 − 1 + κ 2 + τ ∗ r ∗2 ing instability criterion:     ν∗ − ∗2 − + ξ ∗ κ∗2 + 1 κ∗2 + 1 1 k 1 ∗ ∗ ∗ c2k2 + ν2 k2 + < πGρ . r 2 τ r 2 s c 2 4 0 (64) r = 0, (67) ∗  The same criterion obtained by Dhiman & Sharma 2 (ς+ν ) γ 3 + γ + k∗2ς + k∗2ν∗ (2016) for viscoelastic medium. ∗2  r  Further, when the effect of viscoelasticity of the ςν∗   + γ ω∗2+ k∗2 − + + κ∗2 + ςν∗k∗2 medium is absent, the relation (62) yields the same insta- 4 1 ∗2 1 2  r  bility criterion obtained by Dhiman & Dadwal (2012)   ∗ ∗ ∗ 1 for non-viscous gaseous medium, which is given by − k 2 − 1 ν κ 2 + = 0, (68) r ∗2 d   c2k2 + 2ω r 2ω < 4πGρ . s dr 0 where the dimensionless parameters used are (cf. Dhi- man and Sharma 2014) Thus, from the above inequality, we can conclude that ∗ 2 the viscoelasticity of the medium eliminates the effect σ ∗ kc ∗ ν ω j ∗ ν ∗ ω γ = , k = s ,ν = ,ξ = c ,ω = , of non-uniform rotation in the self gravitating infinitely ω ω c2 c2 ω j j s  s j extending axisymmetric cylinder. ∗ rω j ω j 4 ∗ Under weakly coupling limit (στ  1), the disper- r = ,ς= ξ + μ ,τ = τωj . c c2ρ 3 sion relation (60) reduces to the following form; s s 0 ⎛ ⎞ The values of growth rate of the self-gravitational insta-  ξ+ 4μ 1 3 bility for different values of wave numbers under the σ 3 + σ 2 k2 + ⎝ν + ⎠ r 2 ρ strongly and weakly coupling limits have been calcu- ⎛ ⎞ lated from the dimensionless equations (67)and(68). μ  ξ+ 4 2 The variation of growth rate with respect to these 3 1 + σ ⎝2ωF + ν k2 + − Q⎠ obtained values of wave number is shown graphi- ρ r 2 cally in Fig. 1. The fixed values of other parame-  ters used in calculating the growth rates are as fol- 1 − Qν k2 + = . lows: 2 0 (65) r ∗ ∗ ∗ ∗ ω = 0.5, r = 0.28,ξ = 0.3,ς= 0.3,τ = 1.5, The constant term of the above equation yields the ν∗ = . following instability criterion; 0 5  1 The effect of radial distance on the growth rate of Qν k2 + = 0 (66) Jeans instability has been calculated under the strongly r 2 and weakly coupling limits from the dimensionless which is the same criterion as obtained by Dhiman & equations (67)and(68) respectively. Figures 2 and 3 Sharma (2016) for the non-ferromagnetic viscoelastic represent the variation of the growth rate with the nor- medium under the weakly coupling limit. malized wave number under the strongly and weakly coupling limits respectively, for some fixed values of radial distance ( r ∗ = 0.28, 2.83, 28.3) (cf. Nagasawa 4. Growth rate of instability 1987).

We shall now analyze the effects of wave number and radial distance on the growth rate of self-gravitational 5. Results and conclusions instability of viscoelastic ferromagnetic cylinder with axial non-uniform rotation and magnetic field under In the present analysis, we have studied the self gravitat- both the strongly and weakly coupling limits. ing instability of an infinitely extending axisymmetric J. Astrophys. Astr. (December 2017) 38:64 Page 9 of 10 64

Figure 3. Variation of normalized growth rate against the Figure 1. Variation of normalized growth rate against the normalized wave number (k∗) in weakly coupling limit for normalized wave number (k∗) in strongly and weakly cou- fixed values of radial distance r ∗ = 0.28, 2.83, 28.3. pling limits. The instability criteria and the effects of some physi- cal parameters on the growth rate are discussed for this general configuration under both strongly and weakly coupling limits. From the present investigations, we observed that the presence of ferro magnetization and non-uniform rotation and magnetic field have no effect on the gravita- tional instability as the Jeans criteria remains unaltered, whereas the shear and bulk viscosities have the stabi- lizing effect on the stability of the system. Further, it is found that the cylindrical geometry in the strongly coupling limit is more stable than the Cartesian case. Also, the criterion obtained under the weakly coupling limit are same as reported by Dhiman & Sharma (2016) in their analysis of viscoelastic axisymmetric cylinder in the presence of non-uniform magnetic field and rota- tion. Further, it is clear from the above analysis that under the strongly and weakly coupling limits the Figure 2. Variation of normalized growth rate against the magnetic field (uniform or nonuniform) has no effect normalized wave number (k∗) in strongly coupling limit for on the instability of the system. This may be due to fixed values of radial distance r ∗ = 0.28, 2.83, 28.3. the reason that the effect of magnetic field on the gravitational instability has been marginalized by the magnetic polarizability of the ferro medium and thus cylinder of viscoelastic ferromagnetic medium in the instability criteria remain unchanged. Thus, we can con- presence of non-uniform rotation and magnetic field. A clude that the magnetic field applied in ferromagnetized general dispersion relation, which includes the effect of medium has no effect on the self-gravitational instabil- ferro magnetization, magnetic permeability, viscoelas- ity. tic relaxation time, non-uniform rotation and magnetic Also, the effects of wave number and radial distance field, is obtained using the normal mode analysis on the growth rate of self gravitational instability of vis- method on the perturbation equations of the problem. coelastic ferromagnetic cylinder with axial non-uniform 64 Page 10 of 10 J. Astrophys. Astr. (December 2017) 38:64 magnetic field and rotation have been studied. From Dhiman, J. S., Sharma, R. 2016, J. Astrophys Astron., 37,5. Fig. 1, it is observed that as the wave number increases Dhiman, J. S., Sharma, R. 2014, Phys. Scr., 89, 125001. the growth rate of instability increases and takes large Dhiman, J. S., Dadwal, R. 2012, International Scholarly values under the strongly coupling limit. Further, from Research Network, ISRN Astronomy and Astrophysics, Figs. 2 and 3, one can observe that as the values of 104941. radial distance increases the growth rate decreases under Frenkel, Y. I. 1946, Kinetic Theory of Liquids, Clarendon, Oxford. both the strongly and weakly coupling limits, and hence Janaki, M. S., Chakrabarti, N., Benerjee, D. 2011, J. Phys. have stabilizing effect on the self-gravitational instabil- Plasmas, 18, 012901. ity. Thus, we can say that as the radial distance increases, Jeans, J. H. 1929, Astronomy and Cosmogony, Cambridge the system becomes more stable. University Press, Cambridge. Rosenberg & Shukla (2011), reported that the vis- Jones, R. V., Spitzer, L. 1967, Astrophys. J., 147, 943. coelastic coefficients are the function of coupling Kaw, P. K., Sen, A. 1998, Phys. Plasma, 5, 3552. parameter j , which characterizes the ratio of the elec- Krishan, S., Kushwaha, R. S. 1963, Pub. Astron. Soc. Jpn., trostatic Coulomb interaction between neighbouring 15, 253. plasma particles to the thermal (kinetic) energy of the Laroze, D., Martinez-Mardones, J., Pleiner, H. 2013, Eur. particles. Thus, the investigation about these coeffi- Phys. J. Spec. Top., 219, 71. cients provides the vital information about the coupling Larson, R. B. 1985, Mon. Not. R. Astron. Soc., 214, 379. strength of the viscosity and elasticity of the material. Larson, R. B. 2003, Rep. Prog. Phys., 66, 1651. Lyford, N. D., Baumgarte, T. W., Shapiro S. L. 2003, Astro- Further, by calculating the Jeans wave number one can phys. J., 583, 410. get information about the gravitational instability and Mamun, A. A., Shukla, P. K. 2002, JETP Lett., 75(5), can have an idea about the collapse of the interstellar 213. cloud. In other words, the state in which wave num- Mathew, S., Maruthamanikandan, S., Nagouda, S. S. 2013, ber is less than the Jeans wave number, the instability IOSR J. Math., 6(1), 7. occurs. Similarly, by calculating the Jeans mass we can Miyama, S. M., Hayashi, C., Narita, S. 1984, Astrophys. J., identify that the forming star will be a neutron star or 279, 621. any other astrophysical object, i.e. if the Jeans Mass Nagasawa, M. 1987, Prog. Theor. Phys., 77(3), 635. is smaller than the Chandrasekhar limit of 1.44 solar Odenbach, S. 2003, Ferrofluids: Magnetically Controllable masses then the end product is a neutron star. The neu- Fluids and Their Applications, Springer, Berlin. tron stars in binaries have masses close to 1.4 times Ostriker, J. P., Hartwick, F. D. A. 1968, Astrophys. J., 153, than that of the solar mass (cf. Krishan & Kushwaha 797. Penfield, P., Haus, H. A. 1967, Electrodynamics of Moving 1963; Lyford et al. 2003; Ostriker & Hartwick 1968). 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