Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 53, 4, pp. 1067-1081, Warsaw 2015 DOI: 10.15632/jtam-pl.53.4.1067 LINEAR STABILITY ANALYSIS FOR FERROMAGNETIC FLUIDS IN THE PRESENCE OF MAGNETIC FIELD, COMPRESSIBILITY, INTERNAL HEAT SOURCE AND ROTATION THROUGH A POROUS MEDIUM Kapil Kumar Department of Mathematics and Statistics, Gurukula Kangri Vishwavidyalaya, Haridwar, India e-mail: kkchaudhary000@gmail.com V. Singh Department of Applied Sciences, Moradabad Institute of Technology, Moradabad, India Seema Sharma Department of Mathematics and Statistics, Gurukula Kangri Vishwavidyalaya, Haridwar, India The effects of magnetic field and heat source strength on thermal convection of a compres- sible rotating ferromagnetic fluid through a porous medium are investigated theoretically using linear stability theory. A normal mode analysis method is employed to find solutions for the fluid layer confined between parallel planes with free boundaries. The cases of sta- tionary and oscillatory instabilities are discussed. For the stationary state, compressibility, medium porosity and temperature gradient due to heat source have destabilizing effects, whereas rotation and ratio ofmagnetic permeability delay the onset of convection. Thema- gnetic field and medium permeability have both stabilizing and destabilizing effects under certain conditions. The variations in the stationary critical thermal Rayleigh number and neutral instability curves in (Ra1,x)-plane for various values of physical parameters are shown graphically to depict the stability characteristics. The sufficient conditions for the non-existence of overstability are obtained and the principle of exchange of stabilities holds true in the absence of magnetic field and rotation under certain conditions. Keywords: ferrofluids, rotation,magnetic field, porousmedium, heat source, compressibility 1. Introduction Ferrofluids (also known asmagnetic fluids) are electrically non-conducting colloidal suspensions of fine solid ferromagnetic particles or nanoparticles (iron, nickel, cobalt etc.) and their study opens a wide range of attractive and futuristic applications in various engineering and medi- cal science purposes like vacuum technology, instrumentation, lubrication mechanism, acoustics theory, recovery of metals, detection of tumours, drug delivery to a target site, magnetic flu- id bearings, non-destructive testing, sensors and actuators, sorting of industrial scrap metals. They also serve as a challenging subject for scientists interested in the basics of fluidmechanics. The ferromagnetic nanoparticles are coated with a surfactant to prevent their agglomeration. Rosensweig (1985, 1987) discussed the fundamental concepts related to the use of ferrofluids and provides a comprehensive and detailed application of ferrohydrodynamics (also known as FHD) in various commercial usages such as novel zero-leakage rotary shaft seals used in compu- ter disk drives (Bailey, 1983); semiconductor manufacturing (Moskowitz, 1975); pressure seals for compressor and blowers (Rosensweig, 1985); tracer of blood flow in non-invasive circulatory measurements (Newbower, 1972) and in loudspeakers to conduct heat away from the speakers coil (Hathaway, 1979). The thermal instability problemof ferrofluids is a current topic of frontier research and also attractive from a theoretical point of view. Thus, the overall field of ferroflu- id research has a highly interdisciplinary character bringing physicists, engineers, chemists and 1068 K. Kumar et al. mathematicians together. Finlayson (1970) discussed the convective instability problemof a fer- romagnetic fluid layer heated from below when under the effect of a uniform vertical magnetic field with or without considering the effect of body force (gravity force). He quantified that the magnetization of a ferromagnetic fluid depends upon the magnetic field strength, temperature gradient and density of fluid, and is known as ferroconvection (which is very similar to Bénard convection as noted byChandrasekhar, 1981). Lalas andCarmi (1971) studied a thermoconvec- tive instability problemof ferrofluidswithout considering buoyancy effects, whereas the problem of thermal convection in a ferromagnetic fluid saturating a porous medium under the influence of rotation and/or suspended dust particles was simulated by Sunil et al. (2005a,b). Copious literatures (Odenbach, 2002; Neuringer and Rosensweig, 1964; Berkovsky and Bashtovoy, 1996; Sherman and Sutton, 1962) are available to deal with the hydrodynamic and hydromagnetic instability problems of ferrofluids and forcing further investigation in the whole research area. The thermo-convective transport phenomenon in a rotating porousmedium is of significant importance in modern science and engineering problems such as rotating machinery, crystal growth, foodprocessingengineering, centrifugal filtrationprocesses, biomechanics and in thermal power plants (to generate electricity by rotation of turbine blades). Magneto-hydrodynamics (MHD) theory of electrically conducting fluids has several scientific and practical applications in atmospheric physics, astronomy and astrophysics, space sciences, etc. Magnetic field is also used in several clinical areas such as neurology and orthopaedics for probing and curing the internal organs of the body in several diseases like tumours detection, heart and brain diseases, stroke damage, etc. Aggarwal andMakhija (2014) studied the effect of Hall current on thermal instability of ferromagnetic fluid in the presence of horizontal magnetic field through a porous medium. Spiegel and Veronis (1960) simplified the set of equations for compressible fluids by assuming that the vertical height of the fluid ismuch smaller than the scale height as defined by them, and the fluctuations in density, temperature and pressure did not exceed their total static variations. The thermal instability problem for a compressible fluid in the presence of rotation andmagnetic field was studied by Sharma (1997). Detailed investigations related with the problem of convection through various porous me- diums were supplied and very well defined by Nield and Bejan (2006). The fluid flow problems saturating a porous medium plays a key role in petroleum and chemical industry, geophysical fluid dynamics, filtering technology, recovery of crude oil from Earth’s interior, etc. Kumar et al. (2014a,b, 2015) addressed theoretically the thermal instability problems of couple-stress and ferromagnetic fluids by considering the effects of various parameters such as rotation, suspen- ded particles, compressibility, heat source and variable gravity throughDarcy and/or Brinkman porousmedium. The physical properties of comets, meteorites and interplanetary dust strongly suggest the importance of porosity in astrophysical situations (McDonnel, 1978). The governing hydrodynamic equations ofmotion are solved using a regular perturbation technique.The objec- tive of the present study is to discuss the influence of rotation, compressibility and heat source on thermal stability of a ferromagnetic fluid layer heated from below through a porousmedium using linear stability analysis.Theunderstandingof rotating ferrofluid instability problemsplays a key role in microgravity environmental applications. Some existing results are recovered as a particular case of the present study. 2. Governing equations Consider an infinite horizontal porous layer saturatedwith a non-conducting compressible ferro- magnetic fluid confinedbetween the parallel planes z =0and z = d subject to a uniformvertical magnetic field of intensity H(0,0,H) and uniform vertical rotation Ω(0,0,Ω). A Cartesian fra- me of reference is chosen with the z-axis directed vertically upwards and the x- and y-axes at the lower boundary plane. It is also assumed that the flow in the porousmedium is governed by Linear stability analysis for ferromagnetic fluids... 1069 Darcy’s law in the equation of motion withmedium porosity ε and permeability k1 for the case of free and perfect conducting boundaries. The geometrical configuration of the present problem is shown in Fig. 1. Fig. 1. Geometrical sketch of the physical problem The basic governing equations of motion, continuity, energy and Maxwell equations for a magnetized ferrofluid saturating a homogenous porous medium with constant viscosity under Boussinesq approximation are given as follows (Finlayson, 1970; Rosensweig, 1985; Sunil et al., 2005a,b) ρ ε [∂q ∂t + 1 ε (q ·∇)q ] =−∇p+ρXi +µ0(M ·∇)H− ( µ k1 ) q+ 2ρ ε (q×Ω) + µe 4π [(∇×H)×H] ε ∂ρ ∂t +∇· (ρq) = 0 [ερcv +ρscs(1−ε)] ∂T ∂t +ρcv(q ·∇)T = kT∇2T +Φ ε (∂H ∂t ) = [∇× (q×H)]+εη(∇2H ∇·H=0 (2.1) where the symbols ρ, t, µ, q, ∇p, µe, µ0, H, Xi = −gλi, ρs, cs, cv, T , kT , Φ and η denote, respectively, density of the compressible fluid, time, co-efficient of viscosity, fluid velocity, pres- sure gradient term, magnetic permeability of the medium, magnetic permeability of vacuum 4π ·10−7H/m (H – Henry), magnetic field intensity, gravitational acceleration term, density of the solid material, heat capacity of the solid material, specific heat at constant volume, tempe- rature, effective thermal conductivity, internal heat source strength and electrical resistivity. The rotational effect induces two terms in the equation of motion, namely, the Centrifugal force (−0.5grad |Ω×r|2) and theCoriolis force 2(q×Ω). InEq. (2.1)1, p =(pf−0.5ρ|Ω×r|2) is the reduced pressure, where pf stands for the fluid pressure andΩ denotes the angular velocity. Maxwell’s equations for an electrically non-conducting fluid with no displacement currents are ∇·B=0 ∇×H=0 (2.2) The magnetic induction B, magnetization M and the intensity of magnetic field H are related by (Penfield and Haus, 1967) B= µ0(H+M) (2.3) In general, themagnetizationM of a ferrofluid dependsupon themagnitude ofmagnetic fieldH and temperature T , but in the present study it is assumed that the magnetization does not 1070 K. Kumar et al. dependupon themagnetic field strength and is a function of temperature only. So, themagnetic equation of state takes the form M=M0[1+χ(T0−T)] (2.4) where T0 andM0 are the reference temperature and referencemagnetization, respectively, with M0 =M(T0). χ =−(1/M0)(∂M/∂T)H0 stands for the pyromagnetic co-efficient andH0 is the uniformmagnetic field of thefluid layerwhenplaced in an externalmagnetic fieldH=Hext0 ×λi, where λi is the unit vector in the vertical direction. According to Spiegel andVeronis (1960), the equations for compressible fluids are equivalent to those for incompressible fluids if the static temperature gradient β is replaced by the term (β −g/cp) and f is defined as any of the state variable (p,ρ,T) and is expressed in the form f(x,y,z,t)= fm +f0(z)+f ′(x,y,z,t) (2.5) where fm is the constant space distribution of f, f0 is the variation in the absence of motion, f ′(x,y,z,t) stands for the fluctuations in f resulting frommotion of the fluid and cp stands for the specific heat at constant pressure. The quantities of the basic state are given by q=qb = [0,0,0] p = pb(z) ρ = ρb(z) = ρ0(1+αβz) H=Hb(0,0,Hz) M=Mb(z) β = T0−T1 d T = Tb(z)= T0− ( β − g cp ) z+ Φ 2κ (zd−z2) H0+M0 =Hext0 (2.6) and ρ = ρm[1−αm(T −Tm)+Km(p−pm)] αm =− (1 ρ ∂ρ ∂T ) m Km = (1 ρ ∂ρ ∂p ) m p(z)= pm −g z∫ 0 (ρ0+ρm) dz (2.7) where ρ0 and T0 stands for the density and temperature of the fluid at the lower boundary,whe- reas pm and ρm stand for a constant space distribution of pressure p and density ρ, respectively. The subscript b denotes the basic state, α is the coefficient of thermal expansion and β denotes the basic temperature gradient. Now, to analyze the stability of thebasic state using theperturbation technique, infinitesimal perturbations are assumed around the basic state solutions of the following form q=qb +q ′ p = pb(z)+p ′ T = Tb(z)+θ ρ = ρb(z)+ρ ′ M=Mb(z)+M ′(mx,my,mz) H=Hb +h(hx,hy,hz) (2.8) where q′(u,v,w), p′, θ, ρ′, M′, h(hx,hy,hz) are the perturbations in velocity q, pressure p, temperature T , density ρ, magnetization M and magnetic field intensity H, respectively. The changes in density andmagnetizationM′ caused by perturbation θ in temperature T are defined as ρ′ =−αρmθ M′ =−χM0θ (2.9) Linear stability analysis for ferromagnetic fluids... 1071 Using equation (2.8) in equations (2.1) and assuming the perturbation quantities to be very small, the following linearized perturbation equations are obtained as follows 1 ε (∂q′ ∂t ) =− 1 ρm (∇p′)−g ( ρ′ ρm ) λi − µ0χM0(∇H)θ ρm + µ0(M ·∇)h ρm − υ k1 q′ + 2 ε (q′×Ω)+ µe 4πρm [(∇×h)×H] ∇·q′ =0 E (∂θ ∂t ) =− (∂Tb ∂z ) w+κ(∇2θ) ∇·h=0 ε (∂h ∂t ) =(∇H)q′+εη(∇2h) (2.10) where E = ε+(1− ε)[ρscs/(ρmcv)], λi = [0,0,1] and w stands for the vertical fluid velocity. Eliminating u, v and ∇p′ from the momentum equation and retaining the vertical component of fluid velocity, the following perturbation equations are obtained 1 ε ∂ ∂t (∇2w)= ( gα− µ0χM0∇H ρm ] ∇21θ+ µ0M0(1+χ∆T) ρm ∇21 (∂hz ∂z ) − υ k1 (∇2w) − 2Ω ε (∂ζ ∂z ) + µeH 4πρm [ ∂ ∂z (∇2hz) ] 1 ε (∂ζ ∂t ) =− υ k1 ζ + 2Ω ε (∂w ∂z ) + µeH 4πρm (∂ξ ∂z ) ( E ∂ ∂t −κ∇2 ) θ = βLh(z)w ε (∂hz ∂t ) =H (∂w ∂z ) +εη(∇2hz) ε (∂ξ ∂t ) =H (∂ζ ∂z ) +εη(∇2ξ) (2.11) where ξ = ∂hy ∂x − ∂hx ∂y (z-components of current density), ζ = ∂v ∂x − ∂u ∂y (z-component of vorticity), ∇2 = ∂2 ∂x2 + ∂ 2 ∂y2 + ∂ 2 ∂z2 (three dimensional Laplacian operator),∇21 = ∂ 2 ∂x2 + ∂ 2 ∂y2 (two dimensional horizontal Laplacian operator), L = 1− 1 G = 1− βcp g (modified dimensionless compressibility parameter),S = Φd 2βκL (dimensionless heat source parameter),h(z) = 1−S ( 1−2z d ) (non-uniform temperature gradient) and κ = kT ρmcv (thermal diffusivity of the fluid). 3. Normal modes and linear stability analysis The system of equations (2.11) can be solved by using themethod of normalmodes inwhich the perturbation quantities have solutions with dependence upon x, y and t of the following form [w,θ,ζ,hz,ξ] = [W(z),Θ(z),Z(z),K(z),X(z)]exp[i(kxx+kyy)+nt] (3.1) where kx and ky are the horizontal wave numbers along the x and y directions, respectively, k2 = k2x +k 2 y is a dimensionless resultant wave number and n is the growth rate of harmonic disturbance. Infinitesimal perturbations of the state may either grow or damp depending upon the growth raten. Substituting expression (3.1) into linearized differential equations (2.11) along with z = z∗d, a = kd, σ = nd2/υ, D = ∂/∂z∗, the following non-dimensional form is obtained (after ignoring the asterisk) (σ ε + 1 Pl ) (D2−a2)W(z)=− ( g− µ0χM0∇H ρmα )αa2d2Θ υ − [µ0M0(1+χ∆T) ρm a2d υ − µeHd 4πρmυ (D2−a2) ] DK − 2Ωd3 ευ DZ (3.2) 1072 K. Kumar et al. and (σ ε + 1 Pl ) Z = 2Ωd ευ DW + µeHd 4πρmυ DX [(D2−a2)−Ep1σ]Θ =− βd2 κ Lh(z)W [(D2−a2)−p2σ]K =− Hd εη DW [(D2−a2)−p2σ]X =− Hd εη DZ (3.3) The dimensionless parameters in equations (3.2) and (3.3) are the thermal Prandtl number Pr1 = υ/κ, themagneticPrandtl numberPr2 = υ/η and thedimensionlessmediumpermeability Pl = k1/d 2. The boundary conditions appropriate for the case of two free boundaries are defined as { W = D2W = DZ = Θ =0 at z =0 and z =1 hx,hy,hz are continuous at the boundaries (3.4) The solution to equations (3.2) and (3.3) satisfying boundary conditions (3.4) can be taken in the form W = W0 sin(lπz) l =1,2,3, . . . (3.5) where W0 is a constant. The most suitable mode corresponds to l = 1 (fundamental mode). Therefore, using solution (3.5) with l =1 into equations (3.2) and (3.3), the dispersion relation is obtained as follows (after eliminating Θ, X, Z and K) (1+x)(1+x+ iEPr1σi)(1+x+ iPr2σi)=Ra1xεPLh(z) 1+x+ iσiPr2 ε+ iσiP − PQ1 ε+ iσiP [xΓ +(1+x)](1+x+ iσiEPr1) −TA1P2(1+x+ iσiEPr1)(1+x+ iσiPr2)2 1 1+x+ iσiPr2+Q1P (3.6) where RaF is the thermal Rayleigh number for ferromagnetic fluids, Q – Chandrasekhar num- ber, QM – modified Chandrasekhar number for ferromagnetic fluids, Γ – ratio of magnetic permeability with magnetization to magnetic strength and TA – Taylor number RaF = ( g− µ0χM0∇H ρmα )αβd4 υκ Q = µeH 2d2 4πρmυη QM = µ0M0(1+χ∆T) ρm Hd2 υη Γ = QM1 QM = 4πµ0M µeH TA = 4Ω2d4 υ2 and Ra1 = RaF π4 x = a2 π2 iσi = σ π2 P = π2Pl Q1 = Q π2 QM1 = QM π2 TA1 = TA π4 Equation (3.6) is the required dispersion relationship that accounts for the effects of rotation, mediumpermeability, mediumporosity, compressibility, uniform heat source andmagnetic field on thermal instability of the ferromagnetic fluid in a porousmedium. From equation (3.6), the thermal Rayleigh number Ra1 can be separated into the real and imaginary parts as Ra1 = X1+ iσiX2 (3.7) Linear stability analysis for ferromagnetic fluids... 1073 where X1, X2 and σi are real numbers defined as X1 = 1 xεPLh(z) ( [(1+x)2ε−σ2i PPr1E(1+x)] + PQ1[xΓ +(1+x)][(1+x)2+σ2iPr1Pr2E] (1+x)2+σ2iPr 2 2 + TA1P 2 [(1+x)+Q1P]2+σ2iPr 2 2 { [(1+x)2−σ2iPr1Pr2E][(1+x+Q1P)ε+σ2i PPr2] −σ2i [(1+x)(Pr2+Pr1E)][(1+x)P +Q1P2−Pr2ε] }) X2 = 1 xεPLh(z) ( [(1+x)2P +Pr1Eε(1+x)]+ PQ1[xΓ +(1+x)](1+x)(Pr1E −Pr2) (1+x)2+σ2iPr 2 2 + TA1P 2 [(1+x)+Q1P]2+σ2iPr 2 2 { [(1+x)2−σ2iPr1Pr2E][(1+x+Q1P)P −Pr2ε] + [(1+x+Q1P)ε+σ 2 i PPr2](1+x)(Pr2+Pr1E) }) (3.8) SinceRa1 is a physical quantity, itmust be real. Hence, fromequation (3.7) it follows that either σi =0 (stationary state) or X2 =0, σi 6=0 (oscillatory state). It should also be noted that when µ0 =0 (i.e. Γ =0) then from equation (3.8)2 X2 cannot vanish and therefore, σi must be zero. This implies that for an ordinary viscous fluid, the principle of exchange of stabilities is valid even in the presence of a porousmedium, and this statement is verified in Section 6. 3.1. The stationary state For real σi, themarginal instability (or neutral instability) occurs when σi =0. Substituting σi = 0 into equations (3.7) and (3.8)1, the modified thermal Rayleigh number is obtained for the onset of stationary convection in the following form Rastat1 = 1 xεPLh(z) { (1+x)2ε+PQ1[xΓ +(1+x)]+TA1P 2 (1+x) 2ε Q1P +(1+x) } (3.9) Equation (3.9) leads to the marginal instability curves in stationary conditions. For higher values of permeability (P → ∞) which correspond to the case of pure fluids, equation (3.9) gives Rastat1 = 1 Lh(z) {Q1[xΓ +(1+x)] εx + TA1(1+x) 2 Q1x } (3.10) Minimizing equation (3.9) with respect to x yields an equation of degree four in x as x4+A1x 3+A2x 2+A3x+A4 =0 (3.11) where A1 =2(1+Q1P) A2 = Q 2 1P 2+2Q1P − Q1P ε +TA1Q1P 3−TA1P2 A3 =− [ 2(1+Q1P)+2TA1P 2+ 2Q1P(1+Q1P) ε ] A4 =−1+Q21P2+2Q1P − Q1P ε −TA1Q1P3−TA1P2− (Q21P 2+2Q1P)Q1P ε 1074 K. Kumar et al. In the absence of heat source parameter (i.e. h(z) = 1), equation (3.9) gives Rastat1 = 1 xεPL { (1+x)2ε+PQ1[xΓ +(1+x)]+TA1P 2 (1+x) 2ε Q1P +(1+x) } (3.12) which agrees with the previous publishedwork byAggarwal andMakhija (2014) in the absence of the Hall effect but in the presence of rotation and compressibility. The classical results for Newtonian fluids can be obtained as a particular case of the present study. For an incompressible (L = 1), non-rotatory and non-magnetized system, equation (3.12) reduces to Rastat1 = (1+x)2 Px (3.13) This coincides with the classical Rayleigh-Bénard result for a Newtonian fluid in a porous me- dium. To analyze the effects of various parameters such asmodified compressibility, mediumporo- sity, temperature gradient due to internal heating, rotation, magnetic field andmedium perme- ability, the behaviour of dRastat1 /dL, dRa stat 1 /dε, dRa stat 1 /dh(z), dRa stat 1 /dTA1, dRa stat 1 /dQ1 and dRastat1 /dP is examined analytically. Differentiating equation (3.9) with respect to various parameters, i.e. L, ε, h(z), TA1, Q1, P, leads to following expressions dRastat1 dL =− R ⊕ L2h(z) dRastat1 dε =− 1 Lh(z) {[xΓ +(1+x)]Q1 ε2x } dRastat1 dh(z) =    1 L(1−S)2 R⊕ at z =0 − 1 L(1+S)2 R⊕ at z = d (3.14) This shows that themodifiedcompressibility,mediumporosity and temperature gradient (except for the lower boundary) have a destabilizing effect dRastat1 dTA1 = 1 Lh(z) { P(1+x)2 x[Q1P +(1+x)] } (3.15) which is positive, thereby implying the stabilizing effect of the rotational parameter dRastat1 dQ1 = 1 Lh(z) {xΓ +(1+x) εx − TA1P 2(1+x)2 x[Q1P +(1+x)]2 } dRastat1 dP = 1 Lh(z) { TA1(1+x) 2 x[Q1P +(1+x)] − PQTA1(1+x) 2 x[Q1P +(1+x)]2 − (1+x)2 P2x } (3.16) Equations (3.16) show that the magnetic field and medium permeability have dual effects. In a non-rotating frame, the magnetic field has a stabilizing effect, whereas permeability has a destabilizing effect where R⊕ = (1+x)2 Px + [xΓ +(1+x)]Q1 εx + TA1P(1+x) 2 x[Q1P +(1+x)] Linear stability analysis for ferromagnetic fluids... 1075 3.2. The oscillatory state For an oscillatory state, setting X2 = 0, σi 6= 0 in equation (3.8)2 gives a polynomial in σ2i of degree two in the form a0σ 4 i +a1σ 2 i +a2 =0 (3.17) Solving equation (3.17) for σ2i , one gets σ2i = −a1± √ a21−4a0a2 2a0 (3.18) For simplicity, the values of coefficients a0, a1 and a2 are not mentioned here to save spaces. With σ2i determined from equation (3.18), the Rayleigh number for an oscillatory instability can be obtained with the help of equations (3.7) and (3.8)1 as Raosc1 = 1 xεPLh(z) ( (1+x)2ε−σ2i PPr1E(1+x) + PQ1[xΓ +(1+x)][(1+x)2+σ2iPr1Pr2E] (1+x)2+σ2iPr 2 2 + TA1P 2 [(1+x)+Q1P]2+σ2iPr 2 2 { [(1+x)2−σ2iPr1Pr2E ] [(1+x+Q1P)ε+σ 2 i PPr2] −σ2i (1+x)(Pr2+Pr1E)[(1+x)P +Q1P2−Pr2ε] }) (3.19) The values of the critical wave number xc for the oscillatory case can be obtained from equation (3.19) with the condition dRaosc1 /dx =0 and then substituting this critical wave number xc into equation (3.19) yields the critical Rayleigh number Raosc1c for the oscillatory instability. Further, substituting these criticalwavenumberand the criticalRayleighnumberof oscillatory instability into equation (3.18) gives the critical frequency for the oscillatory case. 4. Results and discussion In the present Section, we mainly focused on the determination of critical wave numbers and critical thermal Rayleigh numbers for the stationary case. The values of the critical wave num- ber xc for the onset of stationary instability are determined numerically from equation (3.11) with the condition dRastat1 /dx =0, and then equation (3.9) will give the critical thermal Rayle- igh number for the stationary state. The variations in critical thermal Rayleigh numbers Rastat1c for various values of physical parameters are depicted graphically in Fig. 2. Also, the variations of marginal (neutral) instability curves in the (Ra1 −x) plane for different parametric values (L,h(z),ε,TA1,Q1,P) are shown in Fig. 3.We fixed the values of the parameters except for the varying parameter. 1076 K. Kumar et al. Fig. 2. Variation of Ra1c verrsus L (a), TA1 (b), Q1 (c), P (d), ε (e), h(z) (f)various valuesof physical parameters; curve 1: Q1 =0, QM1 =1, ε =1, P =1, h(z)=5, L =5, TA1 =0, curve 2: Q1 =1, QM1 =3, ε =2, P =2, h(z)=10, L =10, TA1 =2, curve 3: Q1 =3, QM1 =5, ε =3, P =3, h(z)= 15, L =15, TA1 =4, curve 4: Q1 =5, QM1 =7, ε =4, P =4, h(z)= 20, L =20, TA1 =6, curve 5: Q1 =7, QM1 =9, ε =5, P =5, h(z)= 25, L =25, TA1 =8 Linear stability analysis for ferromagnetic fluids... 1077 Fig. 3. Neural instability curve for different values of: (a) comprssibility parameter, h(z)= 5, P =5, ε =5, Q1 =10, QM1 =10, TA1 =500; (b) temperature gradient, L =10, P =2, ε =3, Q1 =10, QM1 =10, TA1 =500; (c) porosity, L =5, h(z)= 5, P =2, Q1 =20, QM1 =20, TA1 =1000; (d) rotation parameter, L =3, h(z)=3, P =3, ε =2, Q1 =50, QM1 =50; (e) permeability, L =2, h(z)= 2, P =5, ε =3, Q1 =10, QM1 =10, TA1 =50; (f) magnetic field parameter, L =5, h(z)=5, P =5, ε =5, QM1 =5, TA1 =50 1078 K. Kumar et al. 5. The overstable case Now, the possibility whether the instability may occur as overstability is examined. Equating the real and imaginary parts of equation (3.6) leads to [(1+x)2−σ2i EPr1Pr2](1+x)=Ra1xεPLh(z) (1+x)ε+σ2i PPr2 ε2+σ2i P 2 −P[(1+x)ε+σ2iPr1PE] (1+x)Q1+QM1x ε2+σ2i P 2 −TA1P2 { [(1+x)2+σ2iPr2(Pr2−Pr1E)](1+x)2 +[(1+x)2−σ2iPr2(Pr2+2Pr1E)]Q1P(1+x)−σ4i EPr1Pr32 } σi(Pr2+Pr1E)(1+x) 2 =Ra1xεPLh(z) σi[Pr2ε−P(1+x)] ε2+σ2i P 2 − σiP[Pr1Eε−P(1+x)][(1+x)Q1+QM1x] ε2+σ2i P 2 −TA1P2σi [ (Pr2+Pr1E)(1+x) 3 +Q1P(2Pr2+Pr1E)(1+x) 2+σ2iPr 2 2(Pr2+Pr1E)(1+x)−σ2iPr1Pr22Q1PE ] (5.1) Eliminating Ra1 between equations (5.1) and assuming σ2i = y, a four degree polynomial in y is obtained as follows b0y 4+b1y 3+ b2y 2+ b3y+ b4 =0 (5.2) where b0 = TA1P 6Pr32[PPr2(1+x)−Pr1E(Pr2ε+Q1P2)] b4 = TA1P 3ε4(1+x)5+[Pε4+TA1P 2ε4(Pr1Eε+Q1P 2)](1+x)4 +[Pr1Eε 5+TA1Q1P 3ε5(Pr1E +Pr2)](1+x) 3+[Q1Pε 4(Pr1E −Pr2)](1+x)2 +[QM1Pε 4x(Pr1E −Pr2)](1+x) (5.3) The coefficients b1, b2 and b3 involving the large number of terms are not written here as they do not play any role in determining the overstability. Since σi is real for overstability to occur, therefore all the roots of y should be positive. So, from equation (5.2), the product of roots equals b4/b0 must be positive. b0 is negative if PPr2(1+x) < Pr1E(Pr2ε+Q1P 2) i.e. Pκ(1+x) < ηE(Pr2ε+Q1P 2) (5.4) and b4 is positive if Pr1E > Pr2 i.e. ηE > κ (5.5) Thus, for inequalities (5.4) and (5.5), theoverstability cannot occur and theprinciple of exchange of stabilities holds true. Therefore, the aforementioned inequalities are the sufficient conditions for thenon-existence of overstability, violation ofwhichdoesnotnecessarily imply theoccurrence of overstability. 6. Principal of exchange of stabilities and oscillatory modes Here, the conditions have beenderived, if any, underwhich theprinciple of exchange of stabilities is satisfiedand thepossibility of oscillatorymodes for the ferromagnetic fluid takes place.For this Linear stability analysis for ferromagnetic fluids... 1079 purpose, equation (3.2) is multiplied by W∗ (the complex conjugate of W) and then integrated over the range of z using equations (3.3). With the help of boundary conditions (3.4), it gives (σ ε + 1 Pl ) I1− ( g− µ0χM0∇H ρmα ) καa2 βυLh(z) (I2+EPr1σ ∗I3) + µ0M0(1+χ∆T) ρm a2ε Pr2H (I4+Pr2σ ∗I5) + µeε 4πρmPr2 (I6+Pr2σ ∗I4)+d 2 [(σ∗ ε + 1 Pl ) I7+ µeε 4πρmPr2 (I8+Pr2σ ∗I9) ] =0 (6.1) where the integrals I1-I9 are positive definite and defined as I1 = 1∫ 0 (|DW |2+a2|W |2) dz I2 = 1∫ 0 (|DΘ|2+a2|Θ|2) dz I3 = 1∫ 0 (|Θ|2) dz I4 = 1∫ 0 (|DK|2+a2|K|2) dz I5 = 1∫ 0 |K|2 dz I6 = 1∫ 0 (|D2K|2+a4|K|2+2a2|DK|2) dz I7 = 1∫ 0 (|Z|2) dz I8 = 1∫ 0 (|DX|2+a2|X|2) dz I9 = 1∫ 0 (|X|2) dz (6.2) Putting σ = iσi in equation (6.1) and equating the imaginary part leads to σi [ I1 ε + ( g− µ0χM0∇H ρmα ) καa2 βυLh(z) Pr1EI3− µ0M0(1+χ∆T) ρm a2ε Pr2H Pr2I5 − µeεPr2I4 4πρmPr2 − d 2I7 ε + µeεd 2Pr2I9 4πρmPr2 ] =0 (6.3) From equation (6.3), it is concluded that either σi = 0 or σi 6= 0, i.e. the modes may be non-oscillatory or oscillatory, respectively. For a non-magneto-rotatory system (i.e. I4 = I5 = I7 =0), equation (6.2) reduces to σi [I1 ε + ( g− µ0χM0∇H ρmα ) καa2 βυLh(z) Pr1EI3 ] =0 (6.4) It is obvious from equation (6.4) that if g > µ0χM0∇H/(ρmα) then the term inside the square bracket will surely be positive, which leads to σi =0. Therefore, the modes are non-oscillatory and the principle of exchange of stabilities is satisfied. The oscillatorymodes are introduced due to the presence of magnetic field and rotation. Thus the sufficient condition for the oscillatory modes to appear in the system is that the inequality g < µ0χM0∇H/(ρmα) is satisfied. Further, for an ordinary viscous fluid µ0 =0 (i.e. Γ =0), equation (6.3) reduces to σi [I1 ε + gκαa2 βυLh(z) Pr1EI3 ] =0 (6.5) which implies that σi =0 and the principles of exchange of stabilities is found to hold good. 1080 K. Kumar et al. 7. Conclusions In this study, linear stability theory is used to find the critical Rayleigh number for the onset of both stationary andoscillatory thermal instabilities. The effects of various embeddedparameters (rotation, magnetic field, compressibility, heat source, permeability and porosity) on thermal instability of a ferrofluid have been analyzed for the stationary state. The main conclusions drawn are presented as: • For the case of stationary convection, compressibility, medium porosity and temperature gradient due to heat source (except at the lower boundary) accelerate the onset of convec- tion, whereas rotation and ratio of magnetic permeability delay the onset of convection. The magnetic field and medium permeability have dual effects on thermal instability of the system, whereas in the absence of rotation, the stabilizing effect of the magnetic field and the destabilizing effect of the medium permeability is obvious from equations (3.16). • The conditions Pκ(1+x) < ηE(Pr2ε+Q1P2) and ηE > κ are the sufficient conditions for the non-existence of overstability. The principle of exchange of stabilities holds good for an ordinary viscous fluid and also in the absence of magnetic field and rotation for g > µ0χM0∇H/(ρmα). Hence, the oscillatory modes are due to the presence of magnetic field and rotation only. • Finally, from the present study, it is concluded that the compressibility, porosity, perme- ability, rotation, magnetic field and heat source parameter have profound effects on the onset of ferroconvection saturating a porousmedium.The present work will also be useful for understandingmore complex problems under different physical parameters mentioned above, and it is also possible to suppress the convective instability in a ferromagnetic fluid layer by controlling the magnitude of heat source, compressibility andmedium porosity. References 1. Aggarwal A.K., Makhija S., 2014, Hall effect on thermal stability of ferromagnetic fluid in porousmedium in presence of horizontalmagnetic field,Thermal Science, 18, 2, 503-514 2. Bailey R.L., 1983, Lesser known applications of ferrofluids, Journal of Magnetism and Magnetic Materials, 39, 178-182 3. Berkovsky B.M., Bashtovoy V.G., 1996,Magnetic Fluids and Applications Handbook, Begell House, NewYork 4. Chandrasekhar S.C., 1981,Hydrodynamic and Hydromagnetic Stability, Dover, NewYork 5. FINLAYSONB.A., 1970, Convective instability in ferromagnetic fluids, Journal of Fluid Mecha- nics, 40, 4, 753-767 6. Hathaway D.B., 1979, Use of ferrofluid in moving coil loudspeakers, dB Sound Engineering Ma- gazine, 13, 42-44 7. Kumar K., Singh V., Sharma S., 2014a, Thermo-magnetic convection in a rotating couple- stress fluid through a Brinkman porous medium, International Journal of Applied Mathematics and Mechanics, 10, 8, 78-93 8. KumarK., SinghV., Sharma S., 2014b,Magneto-rotational convection for ferromagnetic fluids in the presence of compressibility and heat source through a porous medium, Special Topics and Reviews in Porous Media, 5, 4, 311-323 9. Kumar K., Singh V., Sharma S., 2015, On the onset of convection in a dusty couple-stress fluid with variable gravity through a porous medium in hydromagnetics, Journal of Applied Fluid Mechanics, 8, 1, 55-63 Linear stability analysis for ferromagnetic fluids... 1081 10. Lalas D.P., Carmi S., 1971, Thermoconvective stability of ferrofluids,Physics of Fluids, 14, 2, 436-437 11. McDonnel J.A.M., 1978,Cosmic Dust, JohnWiley and Sons, Toronto 12. Moskowitz R., 1975, Dynamic sealing with magnetic fluids,ASLE Transactions, 18, 2, 135-143 13. Neuringer J.L., Rosensweig R.E., 1964, Ferrohydrodynamics, Physics of Fluids, 7, 12, 1927-1937 14. Newbower R.S., 1972, A new technique for circulatorymeasurements employing magnetic fluid tracers,Proceedings Biomedical Symposium, San Diego 15. Nield D.A., Bejan A., 2006,Convection in Porous Media, Springer, New-York 16. Odenbach S., 2002,Magnetoviscous Effects in Ferrofluids, Springer-Verlag, Berlin 17. Penfield P., HausH.A., 1967,Electrodynamics ofMovingMedia, Institute of TechnologyPress, Cambridge 18. Rosensweig R.E., 1985,Ferrohydrodynamics, Cambridge University Press, Cambridge 19. Rosensweig R.E., 1987,Magnetic fluids,Review of Fluid Mechanics, 19, 437-463 20. Sharma R.C., 1997, Thermal instability of compressible fluid in the presence of rotation and magnetic field, Journal of Mathematical Analysis and Applications, 60, 227-235 21. ShermanA., SuttonG.W., 1962,Magnetohydrodynamic, NorthwesternUniversityPress, Illinois 22. Spiegal E.A., Veronis G., 1960, On the Boussinesq approximation for a compressible fluid, Astrophysical Journal, 131, 442-447 23. Sunil, Sharma D., Sharma R.C., 2005a, Effect of dust particles on thermal convection in ferromagnetic fluid saturating a porous medium, Journal of Magnetism and Magnetic Materials, 288, 183-195 24. Sunil, Sharma D., Sharma V., 2005b, Effect of dust particles on rotating ferromagnetic fluid heated from below saturating a porous medium, Journal of Colloid and Interface Science, 291, 152-161 Manuscript received January 28, 2014; accepted for print July 7, 2015