Thermo-Convective Instability in a Rotating Ferromagnetic Fluid Layer

Open Phys. 2018; 16:868–888

Research Article

Precious Sibanda* and Osman Adam Ibrahim Noreldin Thermo-convective instability in a rotating ferromagnetic fluid layer with temperature modulation https://doi.org/10.1515/phys-2018-0109 Received Dec 18, 2017; accepted Sep 27, 2018 1 Introduction

Abstract: We study the thermoconvective instability in a Ferromagnetic fluids are colloids consisting of nanometer- rotating ferromagnetic fluid confined between two parallel sized magnetic particles suspended in a fluid carrier. The infinite plates with temperature modulation at the bound- magnetization of a ferromagnetic fluid depends on the aries. We use weakly nonlinear stability theory to ana- temperature, the magnetic field, and the density of the lyze the stationary convection in terms of critical Rayleigh fluid. The magnetic force and the thermal state of the fluid numbers. The influence of parameters such as the Taylor may give rise to convection currents. Studies on the flow number, the ratio of the magnetic force to the buoyancy of ferromagnetic fluids include, for example, Finalyson [1] force and the magnetization on the flow behaviour and who studied instabilities in a ferromagnetic fluid using structure are investigated. The heat transfer coefficient is free-free and rigid-rigid boundaries conditions. He used analyzed for both the in-phase and the out-of-phase mod- the linear stability theory to predict the critical Rayleigh ulations. A truncated Fourier series is used to obtain a number for the onset of instability when both a magnetic set of ordinary differential equations for the time evolu- and a buoyancy force are present. The generalization of tion of the amplitude of convection for the ferromagnetic Rayleigh Benard convection under various assumption is fluid flow. The system of differential equations is solved reported by Chandrasekhar [2]. In the last few decades the using a recent multi-domain spectral collocation method study of heat transfer in ferromagnetic fluids has attracted that has not been fully tested on such systems before. The many researchers due to the potential application of these solutions sets are presented as sets of trajectories in the fluids in industry, such as in the sealing of rotating shafts, phase plane. For some supercritical values of the Rayleigh ink, and so on. An authoritative introduction to research number, spiralling trajectories that transition to chaotic on magnetic fluids is given by Rosensweig [3]. solutions are obtained. Additional results are presented in Schwab et al. [4] studied the Finlayson problem exper- terms of streamlines and isotherms for various Rayleigh imentally in the case of a strong magnetic field and deter- numbers. mined the parameters for the onset of convection. Their results were shown to be in good agreement with those Keywords: Thermal instability, Ferromagnetic Fluids, of Finlayson [1]. Stiles and Kagan [5] extended the experi- Weakly nonlinear stability, Rotation, Multi-domain spec- mental problem reported by Schwab et al. [4] by introduc- tral collocation method ing a strong magnetic field. A weakly nonlinear stability PACS: 47.11.Kb; 47.20.Bp analysis was used by Russell et al. [6] for magnetized ferrofluids heated from above with the Rayleigh number as the control parameter for the onset of convection. They showed that heat transfer depends on the temperature dif- ference between the bounding surfaces. The rotation of fluids is an interesting topic that has *Corresponding Author: Precious Sibanda: School of Mathemat- been studied by, for example, Greenspan [7]. The classical ics, Statistics and Computer Science, University of KwaZulu-Natal, Rayleigh-Benard problem when the fluid layer is rotating Private Bag X01, Scottsville, Pietermaritzburg 3209, South Africa; is well known in the case of ordinary viscous fluids and Email: [email protected] Osman Adam Ibrahim Noreldin: School of Mathematics, Statistics has been reported by Chandrasekhar [2]. However, ferro- and Computer Science, University of KwaZulu-Natal, Private Bag magnetic fluids are known to exhibit very peculiar char- X01, Scottsville, Pietermaritzburg 3209, South Africa; acteristics when set to rotate. Demonstrating the effect of Email: [email protected]

Open Access. © 2018 P. Sibanda and O. A. I. Noreldin, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License Thermo-convective in rotating ferromagnetic fluid Ë 869 rotation on convection in ferromagnetic fluids is scientif- 2 Mathematical formulation ically important to researchers. Gupta and Gupta [8] ex- amined the onset of convection in a ferromagnetic fluid Consider a ferromagnetic fluid confined between two infi- heated from below and rotating about a vertical axis sub- nite horizontal plates at z = −h/2 and z = h/2. The layer ject to a uniform magnetic field. They concluded that over- is heated from below and cooled from above, and is rotat- stability may not occur for Prandtl numbers smaller than ing uniformly about the vertical axis with constant angular unity. The thermo-convective instability in a rotating fer- velocity Ω. The lower and upper plates are subjected to an rofluid was further analyzed by Venkatasubramanian and oscillatory temperature T +∆T[1+ϵ2 cos(ωt+φ)] where ω Kaloni [9]. They presented both analytical and numerical 0 is the modulation frequency and φ is the phase angle. The results for free and rigid boundary conditions. Their re- Oberbeck-Boussinesq approximation is assumed to be ap- sults were in good agreement with those of Finlayson [1] plicable. The magnetization M of the ferrofluid is assumed and Chandrasekhar [2] for some limiting cases. Thermo- to be parallel to the magnetic field H. The equations de- convection in a ferromagnetic fluid has been studied by scribing the fluid motion under these assumptions are the other researchers, for instance, [10, 12]. continuity equation, modified momentum equation, en- The problem associated with convection in ferromag- ergy equation and Maxwell’s equations (Finlayson [1] and netic fluids is both relevant and mathematically challeng- Gupta and Gupta [8]): ing. The unmodulated Rayleigh Benard problem of a ferromagnetic fluid has been extensively studied. The effect ofa ∇ · V = 0, (1) magnetic modulation on the stability of a magnetic liquid DV ρ = −∇P′ + µ∇2V + ρg layer heated from above was studied by Aniss et al. [13]. 0 Dt They used the Floquet theory for their study of the onset of convection. The study showed the possibility of a com- ρ + ∇ · (HB) + 2ρ V × Ω) + 0 ∇(|Ω × r|), (2) petitive interaction between harmonic and subharmonic 0 2 [︃ ]︃ modes at the onset of convection. Convective instability in (︂ ∂M)︂ DT ρ C − µ H · a ferromagnetic fluid layer with time-periodic modulation 0 V.H 0 ∂T V,H Dt in the temperature field was investigated by Singh and Ba- jaj [14] using the linear stability theory and the classical (︂ ∂M)︂ DH Floquet theory. Their result agrees with those of Aniss et +µ T · = κ∇2T + Φ, (3) 0 ∂T Dt al. [13]. V,H Convection in a rotating horizontal fluid layer confined in a porous medium with temperature modulation ∇ · B = 0; ∇ × H = 0, (4) at the boundary was studied by Bhadauria [19]. He investigated the stability of the flow using the Galerkin method where V is the velocity field, ρ0 is the density at the am- ′ µ0 2 and the Flouquet theory. In this study we analyze ther- bient temperature, P = P + 2 H is the pressure, µ is the moconvective instability in a rotating ferromagnetic fluid viscosity, g is the gravitational body force, B is the mag- layer with time periodic temperature boundary conditions. netic induction, µ0 is the magnetic permeability, T is the The fluid layer is heated from below and rotates about the temperature, κ is the thermal conductivity, CV,H is the heat vertical axis subject to a uniform magnetic field. We as- capacity at constant volume and magnetic field, α is the sume two stress free and two rigid boundary conditions. thermal expansion coefficient and Φ is the viscous dissi- The Ginzburg Landau equation is obtained, see [20] for de- pation. The magnetization and magnetic field are related tails on the relevance of the Ginzburg Landau equation. by the formula Nonlinear ordinary differential equations of the Lorenz B = µ (H + M). (5) type are obtained and solved numerically using the multi- 0 domain spectral collocation method [16–18]. This method The magnetization is dependent on the temperature and has not been fully tested before on evolution equations of magnitude of magnetic field, so that this nature, hence the accuracy of solutions obtained us- H ing this method is also a matter of concern in this study. M = M(H, T). (6) H Heat transfer in the rotating horizontal fluid layer is discussed. Equation (6) is linearized using the Taylor expansion

M = M0 + χ(H − H0) − K(T − T1), (7) 870 Ë P. Sibanda and O. A. I. Noreldin

′ ′ where χ ≡ (∂M/∂H)H0 ,T1 is the magnetic susceptibility H3 = Hb + H 3, M3 = Mb + M 3, and K ≡ −(∂M/∂T)H0 ,T1 is pryomagnetic coefficient, H0 is where the prime represents a perturbed quantity. The lin- the uniform magnetic field and T1 = (T∞ + T0)/2, T∞ and earization of Eqs. (6) and (7) gives T0 are the temperatures at h/2 and −h/2, respectively. The study is restricted to the case when magnetization induced (︂ )︂ ′ ′ M0 ′ H i + M i = 1 + H i , i = 1, 2. (14) by the temperature variation is much smaller than that in- H0 duced by the external magnetic field. The density varies linearly with temperature as ′ ′ ′ ′ H 3 + M 3 = (1 + χ)H − KT . (15)

ρ = ρ0(1 − α(T − T1)). (8) We assume that K∆T << (1 + χ)H0. For more details see [1]. Substituting Eq. (13) and using the curl operator on Eq. (9) we obtain the vorticity equation 3 Weakly nonlinear stability ∂ζ ′ ∂w′ ρ = µ∇2ζ ′ + ρ Ω . (16) 0 ∂t 0 ∂z In this section we use weakly nonlinear stability analysis ′ to study the evolution of disturbances in a ferromagnetic Substituting Eqs. (12), (14) and (15) in Eq. (9) and using H = ′ fluid with two free and two rigid boundary conditions and ∇ϕ where ϕ is the magnetic potential, the z−component temperature modulation. Using Eqs. (4) and (5) and as- of the resulting equation can be written as, suming that the magnetic field H is collinear with the mag- ∂ ρ ∇2w′ − µ∇4w′ = ρ gα∇2T′ (17) netic induction B, Eq. (2) reduces to 0 ∂t 0 l µ K2β ∂ DV 2 0 2 ′ 2 ′ ρ = −∇P + µ∇ V + ρg + µ M · ∇H (9) + ∇l T − µ0Kβ ∇l ϕ 0 Dt 0 1 + χ ∂z ρ0 ∂ζ ′ + 2ρ0V × Ω) + ∇(|Ω × r|). − 2ρ Ω . 2 0 ∂z The basic state solution of Eqs. (1)–(4) with (9) is obtained On using Eq. (13) in Eq. (3) and linearizing we obtain as ∂T′ ∂T′ (︂ ∂ (︂ ∂ϕ′ )︂ ρ C + ρ Cw′ − µ T K (18) V = 0, (10) 0 ∂t 0 ∂z 0 0 ∂t ∂z 2 ′ )︂ (︂ 2 )︂ (︂ )︂ ∆T 2 ′ ∂ ϕ µ0T0K β ∂F T (z, t) = T1 + − βz + ϵ (F(z, t)), +w + ρ Cβ − −1 + ϵ w b 2 ∂z2 0 1 + χ ∂z where = ∇2T′. {︂ ∆T F(z, t) = Re [︀eiφ sinh λ(1/2 − z/h) Finally, substituting Eqs. (14) and (15) into Eq. (4) we have sinh λ }︂ 2 ′ (︂ )︂ ′ ]︀ iωt ∂ ϕ M0 2 ′ ∂T + sinh λ(1/2 − z/h) e , (1 + χ) 2 − 1 + ∇l ϕ − K = 0, (19) ∂z H0 ∂z ∆T λ2 = iωρ Ch2/κ, β = . 2 2 2 2 2 0 h where ρo C = ρ0CV,H + µ0KH0 and ∇l = ∂ /∂x + ∂ /∂y is the Laplace operator in two dimension. For the clarity Following [1, 3] we define we drop the prime from the perturbed quantities and introduce the following dimensionless variables Mb + Hb = constant, (11) * * * * ′ and (x , y , z ) = (x, y, x)/h, w = w h/ν, (20) * 2 (︂ )︂ t = µt/ρ0h , K(Tb − T1)) Hb = H0 − ^ez , (12) * ′ 2 * 1 ′ 1 + χ ζ = ζ h /ν, θ = (κRa 2 T )/(ρ0Cβνh), (︂ )︂ 1 K(T − T1)) * ′ 2 M = M + b ^e . ϕ = ((1 + χ)κRa 2 ϕ )/(ρ0Cβνh ). b 0 1 + χ z The linearized perturbed Eqs. (16)–(19) in the dimension- We superimpose small perturbations on the basic state. less form can be written as, The perturbed quantities are defined as * ∂ 2 * 4 * 1 ∂ζ T = T + T′, H = H′ , M = M′ for i =1,2; (13) ∇ w − ∇ w + Ta 2 (21) b i i i i ∂t* ∂z Thermo-convective in rotating ferromagnetic fluid Ë 871

2 3 1 2 * 1 ∂ 2 * − Ra 2 (1 + M )∇ θ + Ra 2 M ∇ ϕ = 0, w = ϵw1 + ϵ w2 + ϵ w3 + ··· , (28) 1 l 1 ∂z l

* * 2 3 1 ∂w ∂ζ 2 * ζ = ϵζ1 + ϵ ζ2 + ϵ ζ3 + ··· , (29) −Ta 2 + − ∇ ζ = 0, (22) ∂z* ∂t* 2 3 (︂ )︂ * θ = ϵθ1 + ϵ θ2 + ϵ θ3 + ··· , (30) 1 1 2 ∂F * ∂θ (Ra 2 − Ra 2 M ) −1 + ϵ w + Pr (23) 2 ∂z* ∂t* 2 3 ∂θ* ∂ (︂ ∂ϕ* )︂ ϕ = ϵϕ1 + ϵ ϕ2 + ϵ ϕ3 + ··· . (31) + Prw* − ∇2θ* − PrM ∂z* 2 ∂t* ∂z* Substituting Eqs. (27)–(31) into Eqs. (21)–(24), at the lowest ∂2ϕ* − PrM w* = 0, order of ϵ we obtain 2 ∂z*2 BZ = R , (32) * 2 * 1 1 ∂θ ∂ ϕ 2 * − + + M3∇l ϕ = 0, (24) ∂z* ∂z2 where 4 ρ Cβαgh4 √ √ √ 4Ωh 0 ⎛ −∇4 Ta ∂ − Raα(1 + M )∇2 Raα M ∂ ∇2 ⎞ where Ta = ν2 is the Taylor number, Ra = νκ is the √ ∂z 1 l 1 ∂z l 2 ⎜ − Ta ∂ −∇2 0 0 ⎟ µ0 K β B = ⎜ √ ∂z ⎟, Rayleigh number, M1 = is the ratio of the mag- ⎜− Raα(1 − M ) 0 −∇2 0 ⎟ (1+χ)ρ0 αg ⎝ 2 ⎠ 2 ∂ ∂2 2 µ0 T0 K 0 0 − ∂z ∂z2 + M3∇l netic force to the buoyancy force, M2 = is a nondi- (1+χ)ρ0 C µC ⎛ ⎞ ⎛ ⎞ mensional parameter, Pr = κ is the Prandtl number and w1 0 M0 M3 = (1 + )/(1 + χ) is a measure of nonlinearity of the ⎜ ζ ⎟ ⎜0⎟ H0 ⎜ 1 ⎟ ⎜ ⎟ Z1 = ⎜ ⎟ and R1 = ⎜ ⎟ . This equation corresponds magnetization. The magnetic Rayleigh number can be ob- ⎝θ1 ⎠ ⎝0⎠ tained from the formula N = RaM1. Hereafter the asterisk ϕ1 0 will be dropped from Eqs. (21)-(24). to the linear equations in [1, 9]. Solving Eq. (32) we obtain The associated boundary conditions for the system of the solution Eqs. (21)-(24) are w1 = A(τ) sin ax cos πz, (33) – Free boundary conditions

2 w = 0, D w = 0, Dζ = 0, θ = 0, Dϕ = 0, (25) 1 Ta 2 π 1 ζ1 = − 2 2 A(τ) sin ax sin πz, (34) at z = ± . π + a 2

– Rigid boundary conditions α 1 (Ra ) 2 (1 − M2) θ1 = A(τ) sin ax cos πz, (35) w = 0, Dw = 0, ζ = 0, θ = 0, (26) π2 + a2 1 at z = ± , 2 (︃ α 1 )︃ (Ra ) 2 (1 − M2)π ϕ1 = − 2 2 2 2 (36) where (π + a )(π + a M3) D = ∂/∂z. A(τ) sin ax sin πz,

where a is a dimensionless wave number. Thus, the sta- 3.1 The solution for stress free boundaries tionary Rayleigh number is given as

2 2 3 2 2 2 The solution for stress free boundaries has been discussed α ((π + a ) + π Ta)(π + a M3) Ra = 2 2 2 2 2 . in [1, 3, 9]. Here we only emphasize the solution aspects (1 − M2)(a π M1 + a (M1 + 1)(π + a M3)) which have not been discussed before. We solve the eigen- (37) value problem with two stress free boundaries to study the To find the critical wave number and the corresponding onset of instability in the ferromagnetic fluid. We consider critical Rayleigh number we set a2 = π2x. Then the sta- a small variation in time scale τ = ϵ2t such that stationary tionary Rayleigh number can be written as convection occurs at lower orders of ϵ and introduce the following asymptotic expansions (π4(1 + x)3 + Ta)(1 + xM ) Raα = 3 . (38) 1 1 1 1 (1 − M2)x(1 + xM3(M1 + 1)) 2 α 2 2 2 4 2 Ra = (Ra ) + ϵ Ra2 + ϵ Ra4 + ··· , (27) 872 Ë P. Sibanda and O. A. I. Noreldin

This result agrees with [1, 3, 9]. Since M2 is very small as in- Table 1: Comparison of the critical wave number and corresponding dicated in [1, 9] it can be neglected in the subsequent anal- Rayleigh number for M1 = 1 and M3 = 5. ysis. Present study Ref [9] The critical wave number and the corresponding criti- α α cal Rayleigh number are obtained from Ta ac Rac ac Rac 1 2.11212 269.6382 2.7348 461.5368 ∂Raα = 0, 10 2.15849 275.0998 3.7827 883.4022 ∂x 100 2.49953 322.9328 5.7215 2727.2022 then we have 1000 3.65242 608.4028 8.6174 10619.856 10000 5.66466 1862.582 12.8255 45606.430 − Ta − 2Ta (xM3 + xM1M3) (39) (︁ )︁ − Ta x2M2 + x2M M2 3 1 3 Table 2: Comparison of the critical wave number and corresponding (︂ Magnetic Rayleigh number for M → ∞ and M = 3. 4 2 2 1 3 + π (1 + x) − 1 + 2x − 2xM3 − 2xM1M3 − 4x M3 )︂ Present study Ref [9] 2 2 3 2 3 2 + x M M1 + 2x M + 2x M1M = 0. α α 3 3 3 Ta ac Nc ac Nc 1 2.6084 880.7959 2.7348 1037.8896 For the case Ta = 0, M = 0 and M = 0, the classical 1 3 10 2.62069 897.4541 3.8770 1885.6592 critical wave number is 100 2.72574 1050.072 5.7736 5617.3574 π ac = √ 1000 3.2225 2083.258 8.6487 21520.978 2 10000 4.28305 7315.093 12.8455 91759.273 with corresponding classical critical Rayleigh number

α 27 4 α 1 Rac = π . Prπ(Ra ) 2 4 R = − A2(τ) sin2 ax sin 2πz. 23 2(π2 + a2) The magnetic Rayleigh number is also of an interest and can be expressed as The solution at the second order is 4 3 α α (π (1 + x) + Ta)M1(1 + xM3) w = 0, ζ = 0, (45) N = Ra M1 = . (40) 2 2 x(1 + xM3(M1 + 1)) For large values of M the magnetic Rayleigh number in 1 (︃ α 1 )︃ Prπ(Ra ) 2 the absence of buoyancy effects is obtained as θ = − (46) 2 4(π2 + a2)(2π2 + a2) 4 3 α (π (1 + x) + Ta)(1 + xM3) 2 2 N = 2 . (41) A (τ) sin ax sin 2πz, x M3 The critical wave number and corresponding critical mag- (︃ 2 α 1 )︃ netic Rayleigh number are obtained from solving the equa- Prπ (Ra ) 2 ϕ2 = − 2 2 2 2 2 2 (47) tion 4(2π + a )(π + a )(2π + a M3) 2 2 −2Ta + M3x + π4(1 + x)2(−2 + x − M3x + 2M3x2) = 0. (42) A (τ) sin ax cos 2πz.

The critical wave number and corresponding Rayleigh At the third order, we obtain numbers are given for different values of the Taylor number Ta in Tables 1 and 2. BZ3 = R3, (48) At second order O(ϵ2) we obtain the following equa- ⎛w ⎞ ⎛R ⎞ tions: 3 31 ⎜ ⎟ ⎜ ⎟ ⎜ ζ3 ⎟ ⎜R32⎟ where Z3 = ⎜ ⎟ and R2 = ⎜ ⎟ with BZ2 = R2 (43) ⎝θ3 ⎠ ⎝R33⎠ ⎛ ⎞ ⎛ ⎞ ϕ3 R34 w2 R21 1 1 ⎜ ζ2 ⎟ ⎜R22⎟ {︂ 2 α 2 where Z = ⎜ ⎟ and R = ⎜ ⎟ with dA a (Ra ) 2 R (1 + M1) 2 ⎜ ⎟ 2 ⎜ ⎟ 훾2 a2 θ2 R23 R31 = − + 2 2 2 (49) ⎝ ⎠ ⎝ ⎠ dτ 훾 (π + a M3) ϕ2 R24 }︂ 2 2 ((1 + M1)π + a M3)A(τ) sin ax cos πz, R21 = 0, R22 = 0, R24 = 0, (44) Thermo-convective in rotating ferromagnetic fluid Ë 873

1 2 Ta 2 π dA of y. Using the stream function defined by R = − sin ax sin πz, (50) 32 훾2 dτ ∂ψ ∂ψ u = and w = − (︂ ∂z ∂x α 1 ∂F 1 R = − (︀(Ra ) 2 − Ra 2 )︀A(τ) (51) 33 ∂z 2 equations (1)–(4) reduce to α 1 )︂ 2 2 α 1 2 Pr(Ra ) 2 dA Pr π (Ra ) 2 ∂ 2 2 4 α 1 ∂ ϕ + sin ax cos πz + ∇ ψ − |J(ψ, ∇ ψ)| = ∇ ψ − M Ra 2 (57) 훾2 dτ 2훾2(2π2 + a2) ∂t L L L 1 ∂x∂z 3 3 α 1 ∂θ 1 ∂ζ A (τ) sin ax cos πz cos 2πz, − Ra 2 (1 + M ) + Ta 2 , 1 ∂x ∂z

R34 = 0. (52) ∂ζ 2 1 ∂ψ + |J(ψ, ζ )| = ∇L ζ − Ta 2 , (58) Here, 훾2 = a2 + π2. To obtain the Ginzburg Landau equa- ∂t ∂z tion we applied the Fredholm solvability condition [11, 18] (︂ )︂ ∂θ 2 α 1 ∂ψ 1 2π Pr + |J(ψ, θ)| = ∇L θ − Ra 2 , (59) ∫︁2 ∫︁a [︂ ]︂ ∂t ∂x w^ 1R31 + ^ζ1R32 + θR^ 33 dxdz = 0, (53) 0 0 ∂2ϕ ∂2ϕ ∂θ + M = , (60) ∂z2 3 ∂x2 ∂z where w^ 1, ^ζ1 and θ^1 are the solutions of the adjoint system of the first order. This gives where J is the Jacobian matrix. The solution of Eqs. (57)–

1 1 (60) represented as a minimal double Fourier series of [︂ 2 α 4 ]︂ {︂ α 2 2 π훾 PrRa π Taπ dA πRa Ra2 modes (1,1) for the stream function and magnetic potential + 4 − 4 = 2 (54) 4a 4a훾 4a훾 dτ 4a훾 and modes (0,2) and (1,1) for temperature and vorticity of 1 α 1 α 2 2 the finite amplitude convection of the ferromagnetic fluid πRa aπRa Ra2 (1 + M1) − 2 I + 2 2 2 a훾 4훾 (π + a M3) flows as (︂ )︂}︂ 3Pr2π3Raα (1 + M )π2 + a2M A − A3. ψ = sin ax sin πz, (61) 1 3 64a훾4(2π2 + a2) 11 The above equation reduces to θ = B11 cos ax sin πz + B02 sin 2πz, (62) dA = ∆ A − ∆ A3, (55) dτ 1 2 ζ = C11 sin ax cos πz + C02 sin 2πz, (63) 2 6 α 3 where ∆1 = 훾 /(훾 + PrRa − Taπ ) (︁ 2 * 2 2 )︁ * α a Ra (1+M1)((1+M1)π +a M3) Ra − 4Ra I + 2 2 , π +a M3 ϕ = D cos ax cos πz, (64) 1 1 11 * α 2 2 2 3 α (︀ (︀ 2 2)︀)︀ Ra = Ra Ra2 , ∆2 = 3 Pr π Ra / 16 2π + a , and where A11, B11, B02, C11, C02 and D11 are time t depen- 1 2 dent amplitudes. This is equivalent to a truncated Galerkin ∫︁ dF I = cos2(πz)dz. method. Substituting and integrating over the domain, we dz 0 obtain a set of four ordinary differential equations for the In this study we are also interested in heat transfer time evolution of the amplitudes of convection of a ferro- in ferromagnetic fluids. The Nusselt number for ferromag- magnetic fluid in the form netic fluids is defined as α 1 dA aπRa 2 11 = −훾2A − B (65) Heat transfer by conduction+convection 11 2 2 M 11 Nu(τ) = (56) dt 훾 (π + a3 ) Heat transfer by conduction 1 1 α 2 2 2 α 1 aRa (1 + M1) πTa Prπ (Ra ) 2 − B + C , = 1 + A2(τ). 훾2 11 훾2 11 4훾2(2π2 + a2)

dC11 2 1 = −훾 − πTa 2 A11, (66) 3.2 The general Lorentz type equations dt

α 1 2 We restrict the analysis to the case of two-dimensional dis- dB aRa 2 훾 11 = A − B − aπA B , (67) turbances so that all physical quantities are independent dt Pr 11 Pr 11 11 02 874 Ë P. Sibanda and O. A. I. Noreldin

2 dB02 aπ 4π – The steady solution represented by the point = A B − B . (68) dt 2 11 11 Pr 02 √︃ (︂ 2b(PrKR − (Ta + 1)) To simplify the equations we introduce new variables (x* , x* , x* , x* ) = ± , 1 2 3 4 Pr2(1 + Ta) 2 α 1 aπ −a πRa 2 2 √︃ X1 = A11, X2 = B11, τ = 훾 t, (69) Ta + 1 2b(PrKR − (Ta + 1)) kPrR − (Ta + 1) 훾2 훾6 ± , , K Pr2(Ta + 1) kPr √︃ )︂ 2 α 1 2 1 2b(PrKR − (Ta + 1)) −a πRa 2 aπ Ta 2 ∓ . X = B and X = C . (70) 2 2 3 훾6 02 4 훾6 11 Ta Pr (1 + Ta)

This reduces Eqs. (65)-(68) to general Lorentz type equa- The stability of the stationary point associated with the * tions motionless solution X = (0, 0, 0, 0) is determined by roots of the following characteristic polynomial equation dX1 = −X1 + KX2 + X4, (71) dτ 3 2 P(ξ) = ξ + d1ξ + d2ξ + d3 = 0, (78)

dX2 −1 where = RX1 − Pr X2 − X1X3, (72) −1 dτ d1 = Pr , d2 = Ta − 1 − KR and dX 1 Ta − PrKR − 1 3 = X X − bPr−1X , (73) d = dτ 2 1 2 3 3 Pr b for the eigenvalues ξ , (i = 2, 3, 4) and ξ = − . It is clear dX i 1 Pr 4 = −TaX − X , (74) dτ 1 4 that ξ1 is always negative as Pr > 0. The remaining eigenvalues are obtained from Eq. (78), and using the Routh- a2 α 4π2 where R = 훾6 Ra , b = 훾2 , Hurwitz criteria [21], the polynomial Eq. (78) has negative 2 2 2 2 (2π +a M3)+M1(π +a M3) real roots if and only if and K = 2 2 . π +a M3 Ta − 1 R < KPr 3.3 Stability of Lorentz equations and Pr > 1. This implies that the stationary solution is a stable node. Hence the critical value of R where the sta- In this section we discuss the stability of the nonlinear tionary solution of ferromagnetic fluid flow loses stability systems of differential equations that describe the evolu- and steady convective flow takes over is tion of the convection amplitudes for a ferromagnetic fluid Ta − 1 flow. Firstly, we note that the nonlinear Eqs. (71)–(74) are R = . invariant under the transformation KPr The stability of the stationary point corresponding to the S(X1, X2, X3, X4) = (−X1, −X2, −X3, −X4). (75) steady convective flow is determined by the roots ofthe characteristic equation These equations are also uniformly bounded and dissipa- tive in the phase space 4 3 2 p(ξ) = ξ + c1ξ + c2ξ + c3ξ + c4 = 0, (79) 4 ∑︁ ∂X˙ [︂ ]︂ i = − 1 + Pr−1 + bPr−1 < 0 (76) where Xi i=1 c1 = 2Pr + 2bPr, Thus the volume of the phase space moving with the flow 2 2 2 c2 = 2b − 2Pr − 2KPr R + 2Pr Ta for time τ > 0 is given by + Pr2x*2 + 2KPr2x* , (︁ )︁ 1 3 [︀ −1 −1]︀ 2 V(t) = V(0) exp − 1 + Pr + bPr τ . (77) c3 = −2Pr − 2bPr − 2bKPrR − 2KPr R + 2PrTa + 2bPrTa + KPr3x* x* + 2bKPrx* + 2KPr2x* , We find that the stationary points of the system of nonlin- 1 2 3 3 2 *2 2 *2 ear Eqs. (71)–(74) are: c4 = −2b − 2bKPrR + 2bTa − Pr x1 + Pr Tax1 2 * * * – The motionless conduction solutions (0, 0, 0, 0). + KPr x1x2 + 2bKPrx3. Thermo-convective in rotating ferromagnetic fluid Ë 875

Applying the Routh-Hurwitz criteria to Eq. (79), it is clear 3.4 The method of solution that c1 > 0, and c3 > 0 if and only if In this section, we describe the multi-domain spectral col- Ta + 1 < PrKR location method [15–18] used to obtain the solutions to and Eqs. (71)–(74). The multi-domain technique assumes that (︂ b + 1 )︂ R < − . the interval Λ = [0, T] can be decomposed into p non over- bK + Pr lapping sub-intervals. The sub-intervals are defined as

Also, c4 > 0 if and only if Λi = [τi−1, τi], i = 1, 2, ··· , p. (81) (︂2b(1 + PrKR))︂ x*2 < − 1 Pr2 In each sub-interval, the system of Eqs. (71)–(74) is written in the form with Ta + 1 < PrKR. i i i i X˙ 1,s+1 + α1,1X1,s+1 + α1,2X2,s + α1,3X3,s (82) Hence the fixed point is stable if the condition i i i i + α1,4X4,s + f1(X2,s , X3,s , X4,s) = g1 3 * 2 * * * i i i i Pr c3 + Pr c2 + Prc1 + c0 (80) X˙ 2,s+1 + α2,1X1,s+1 + α2,2X2,s+1 + α2,3X3,s 3 * 2 * * * i i i i > Pr d3 + Pr d2 + Prd1 + d0, + α2,4X4,s + f2(X1,s+1, X3,s , X4,s) = g2 ˙ i i i i is satisfied where X3,s+1 + α3,1X1,s+1 + α3,2X2,s+1 + α3,3X3,s+1 i i i i (︂ + α3,4X4,s + f3(X1,s+1, X2,s+1, X4,s) = g3 c* = 8KR + 8K2R2 + 4KTax* x* 3 1 2 i i i i X˙ 4,s+1 + α4,1X1,s+1 + α4,2X2,s+1 + α4,3X3,s+ *3 * *2 * 2 * * * + 2Kx1 x2 + 4Kx1 x3 + 4K x1x2x3 i i i i + α4,4X4,s+1 + f4(X1,s+1, X2,s+1, X3,s+1) = g4, )︂(︂ )︂ + 8K2x*2 + 8KTax* 1 + b , 3 3 with initial conditions * 2 c2 = 8(KR + 1) + 16b + 8b + 8KR(3 + 2b) i i−1 Xn,s+1(τi−1) = Xn (τi−1), n = 1, 2, 3, 4. (83) + 8K2R2(b + 1) + 8Ta2(b2 + 2b + 1) * 2 *2 * Here αn,k and gn (n, k = 1, 2, 3, 4) are constants while fn + 8KTax3(2b + 3b + 1) + (4Kx1 x3 2 *2 2 2 *2 * * is the nonlinear component of each equation. Each sub- + 8K x3 )(b + b ) + 4K R(x1 x2 + 2x3), interval Λi is transformed to [−1, 1] using the transforma- * 3 2 *2 * 2 c1 = 8KR(b + b ) + 4Kx1 x2(2b + b ) tion + (8Kx* + 8KRTa + 4K2Rx*2x* )(1 + b) 3 1 2 τi−1 − τi * τi−1 + τi * 2 * * 2 2 τ = t + , t ∈ [−1, 1]. (84) + 16bK Rx3, c0 = 8b + 4b + 8Ta(b 2 2 * 3 2 + 2b + 1) + 8Kx3(b + 2b + b) The Chebyshev-Gauss-Lobatto collocation points are used 2 2 2 * to discretize the unknown functions + 8KRTa(b + b) + 8b K Rx3, (︂ πj )︂ and t*i = cos , j = 0, 1, ··· , N. (85) j N (︂ * *2 2 *2 * d3 = 8KRTa + 4KRx1 + 4(K + K )x1 x2 The derivative of the unknown function at the collocation )︂(︂ )︂ point is given by * 2 * + 8Kx3 + 16K Rx3 1 + b , i N (︁ )︁ (︁ )︁ dXn,s+1 i ∑︁ i *i i * * * 2 (τ ) = Djk Xn,s+1(t ) = DXn,s+1 (86) d2 = 16Ta + 4Kx1x2 b + 2b + 1 dτ k k=0 (︁ )︁ (︁ )︁ + 8KRTa + 8Kx* 2b2 + 3b + 1 3 where 2 * (︁ 2 )︁ 2 *2 *2 * * * * D = 2D/δτ + 16K Rx3 b + b + K (x1 x2 + 4x1x2x3 + 4x3), i * * 3 2 (︀ * * * with δτ = τ − τ and D is Chebyshev differentiation ma- d1 = 8Kx3(b + b ) + * KTax3 + 4KTax1x2 i i−1 i 2 2)︀ 2 * * * *2 trix. The vector functions at the collocation points are + 8K R (b + 1) + 4bK (x1x2x3 + 2x3 ), * 3 2 2 (︁ )︁T d0 = 8KR(b + 2b + b) + 4Ta(b + 2b + 1) i i *i i *i Xn,s+1 = Xn,s+1(t0 ), ··· , Xn,s+1(tN ) . * 2 2 2 *2 + 8KTax3(b + b) + 4b K x3 . 876 Ë P. Sibanda and O. A. I. Noreldin

Substituting Eq. (86) into Eq. (82) and reducing to matrix form we obtain the system

i i AnXn,s+1 = Rn (87) with An = D + αn,nI and [︂ i i i i R1 = g1 − α1,2X2,s + α1,3X3,s + α1,4X4,s (88) ]︂ i i i + f1(X2,s , X3,s , X4,s) (a)

[︂ i i i i R2 = g2 − α2,1X1,s+1 + α2,3X3,s + α2,4X4,s (89) ]︂ i i i + f2(X1,s+1, X3,s , X4,s)

[︂ i i i i R3 = g3 − α3,1X1,s+ + α3,2X2,s+1 + α3,4X4,s (90) ]︂ i i i + f3(X1,s+1, X2,s+1, X4,s) (b) [︂ i i i i R4 = g4 − α4,1X1,s+1 + α4,2X2,s+1 + α4,3X3,s+1 (91) ]︂ i i i + f4(X1,s+1, X2,s+1, X3,s+1) where gn is gn multiplied by a vector of ones of size (N + 1) × 1 and I is an identity matrix of size (N + 1) × (N + 1).

4 Results and discussion (c)

We have presented a weakly nonlinear stability analysis of a rotating layer of a ferromagnetic fluid with temperature modulation at the boundary. We have obtained mathematical expressions for the stationary Rayleigh number Raα and the magnetic Rayleigh number Nα . Our results agree qualitatively with the results in [1, 9]. To provide a measure of validation of our results we give a comparison with [9] in Tables 1 and 2 of the influence of the Taylor number on the critical wave number and the corresponding Rayleigh numbers. Although the results in the two studies are not (d) directly comparable, of interest is the general trend ob- α served, namely that in both cases, increasing Ta increases Figure 1: The effect of stationary Rayleigh number Ra versus wave number a for various values of (a) the Taylor number Ta, (b)the the critical wave number and the Rayleigh numbers sug- ratio of magnetic force to the buoyancy force parameter M1 with gesting that the influence of Taylor number is to stabilize Ta = 0 (c) the ratio magnetic force to the buoyancy force parameter the system. M1 with Ta = 10 (d) the nonlinearity of magnetization parameter The instability curves are given in Figures 1–3. Fig- M3 ure 1(a)–1(d) shows the influence of various parameters Thermo-convective in rotating ferromagnetic fluid Ë 877

(a) (b)

Figure 2: The effect of magnetic Rayleigh number Nα versus wave number a for various values of (a) the Taylor number Ta (b) the nonlinearity of magnetization parameter M3

(a) (b)

Figure 3: The variation of Nusselt number Nu with time τ in in-phase modulation (ϕ = 0) for different values of : (a) the Prandtl number Pr, (b)the Taylor number Ta, (c) the ratio of magnetic force to buoyancy force M1 and (d) the nonlinearity of magnetization M3 on the stationary Rayleigh number. It can be seen in Fig- stabilizing effect on the system. This result is similar to ures 1(a)and 2(a) that as Ta increases from 0 to 50 the val- that of an ordinary viscous fluid. Rotation has a stabiliz- ues of the stationary Rayleigh and the magnetic Rayleigh ing influence on ferromagnetic fluid flow. Figures 1(b)and numbers both increase. This shows that rotation has a 1(c) show the relative influence of the size of the magnetic 878 Ë P. Sibanda and O. A. I. Noreldin

(a) (b)

Figure 4: The variation of Nusselt number Nu with time τ in out-phase modulation (ϕ = π) for different values of : (a) the Prandtl number Pr, (b)the Taylor number Ta, (c) the ratio of magnetic force to buoyancy force M1 and (d) the nonlinearity of magnetization M3

force to the buoyancy force parameter M1. As M1 increases creasing Ta and M1 increases the Nusselt number, thus from 0 to 5 the stationary Rayleigh number is reduced. the rate of heat transfer increases. Figures 4(a)–4(d) show This suggest the magnetic and the buoyancy force are both changes in the Nusselt number with respect to time τ due destabilizing to the ferromagnetic fluid flow. Further, as to the influence of various parameters in the case ofout M1 increases with Ta = 10 fixed, the stationary Rayleigh of phase modulation. It can be observed that the Nusselt number decreases, suggesting M1 has a destabilizing ef- number for in-phase modulation is less than for out of fect for both low and high Taylor numbers. From Figures phase modulation. 1(d) and 2(b) it is observed that increasing M3 from 1 to A multi-domain spectral collocation method was used 20 reduces both the Rayleigh number and the magnetic to find the nonlinear amplitudes in ferromagnetic fluid Rayleigh number, this is destabilization to the system. convection equations for various values of R. The solu- The Ginzburg-Landau equation is obtained using tion sets were obtained using initial conditions selected the nonlinear stability analysis at the third order of ϵ. in the neighborhood of the stationary points correspond- The equation was solved using a multi-domain spectral ing to the motionless solutions. The simulations were method. The heat transfer coefficient, represented by the done to a maximum time τmax = 20. For a sense of Nusselt number, is presented graphically for in-phase and the accuracy of the method, the solutions were com- out-phase modulation in Figures 3–4. Figure 3(a)–3(d) pared with solutions obtained using the Runge-Kutta show the effect of Pr, Ta, M1 and M3 on the Nusselt num- based ode45 routine. Figures 5(a)–5(d) show the time se- ber with time τ. It can be observed that on increasing the ries solution of X1(τ) for different supercritical values of Pr and M3, the Nusselt number decreases. Hence increas- R. As R increases, periodic solutions are obtained. Here ing these parameters reduces the rate of heat transfer. In- a comparison between the multi-domain spectral collo- Thermo-convective in rotating ferromagnetic fluid Ë 879

(a) (b)

(e)

Figure 5: Comparison between a Multi-domain spectral collocation methods and ode45 methods for the solution of X1(τ) for different values of R (a) R = 2, (b) R = 4, (c) R = 10, (d) R = 20 and (e)R = 100 cation method and the ode45 is given. In Figures 6– rium points that correspond to the motionless solution are 11 we present a projection of the trajectories onto the stable spirals. The solutions are presented in part c and d (X1, X2), (X1, X3), (X1, X4), (X2, X3), (X2, X4) and (X3, X4) of each figure when R = 20 and R = 25, respectively. For phase planes, respectively. The initial supercritical con- these Rayleigh numbers, chaotic solutions are obtained. vective solution R = 2 is presented in part a in each fig- These changes in solutions are further presented in Fig- ure. We observe that the trajectories attracted to equilib- ures 6(c)–6(d), 7(c)–7(d) and 9(c)–9(d). The results pre- 880 Ë P. Sibanda and O. A. I. Noreldin

(a) (b)

(e)

Figure 6: The evolution of trajectory over times in phase space for increasing values of Rayleigh number.( in term of R) corresponding to (a)R = 2, (b)R = 4, (c)R = 10, (d) R = 20 and (e)R = 100. The graph represented the projection of the solution into X1X2 plane sented in Figures 8(c)–8(d) and 10(c)–10(d) show a tran- The clockwise and anti-clockwise flows are shown via neg- sition to a limit cycle. Increasing the values of R, for exam- ative and positive stream function values, respectively. ple, when R = 100 the results are complex with a signifi- With the Rayleigh number increasing from 2 to 200, the cant level of unpredictability. magnitude of the stream function values increase. The Figure 12 shows the streamlines patterns for the flow sense of motion in the subsequent cells is opposite that of a ferromagnetic fluid. Two different eddies are observed. of an adjoining cell, indicating symmetry in the forma- Thermo-convective in rotating ferromagnetic fluid Ë 881

(a) (b)

(e)

Figure 7: The evolution of trajectory over times in phase space for increasing values of Rayleigh number. (in term of R) corresponding to (a) R = 2, (b) R = 4, (c) R = 10, (d) R = 20 and (e) R = 100. The graph represented the projection of the solution into X1X3 plane tion of ferromagnetic convective cells. Figure 13 shows the Also, increasing R reduces the density of the isotherms im- isotherm patterns as the Rayleigh number changes from 2 plying a delay of the onset of instability. to 200. Three different eddies are observed. The small eddy at the left corner diminishes as R increases from 2 to 200. 882 Ë P. Sibanda and O. A. I. Noreldin

(a) (b)

(e)

Figure 8: The evolution of trajectory over times in phase space for increasing values of Rayleigh number. (in term of R) corresponding to (a) R = 2, (b) R = 4, (c) R = 10, (d) R = 20 and (e) R = 100. The graph represented the projection of the solution into X1X4 plane Thermo-convective in rotating ferromagnetic fluid Ë 883

(a) (b)

(e)

Figure 9: The evolution of trajectory over times in phase space for increasing values of Rayleigh number. (in term of R) corresponding to (a) R = 2, (b) R = 4, (c) R = 10, (d) R = 20 and (e) R = 100. The graph represented the projection of the solution into X2X3 plane 884 Ë P. Sibanda and O. A. I. Noreldin

(a) (b)

(e)

Figure 10: The evolution of trajectory over times in phase space for increasing values of Rayleigh number. (in term of R) corresponding to (a) R = 2, (b) R = 4, (c) R = 10, (d) R = 20 and (e) R = 100. The graph represented the projection of the solution into X2X4 plane Thermo-convective in rotating ferromagnetic fluid Ë 885

(a) (b)

(e)

Figure 11: The evolution of trajectory over times in phase space for increasing values of Rayleigh number. (in term of R) corresponding to (a) R = 2, (b) R = 4, (c) R = 10, (d) R = 20 and (e) R = 100. The graph represented the projection of the solution into X3X4 plane 886 Ë P. Sibanda and O. A. I. Noreldin

(a) (b)

(e) (f)

Figure 12: The streamlines for different values of the Rayleigh number R e.g. R = 2, 4, 10, 20, 100, 200 Thermo-convective in rotating ferromagnetic fluid Ë 887

(a) (b)

-1 -2 -3

-3

-3 -1 -2 -2 -1 -1 0 0 0 1 1 1 2 2 3 3 2

1 3 2

0 2 1 1 0

-1 0

(e) (f)

Figure 13: The isotherms for different values of the Rayleigh number R e.g. R = 2, 4, 10, 20, 100, 200 888 Ë P. Sibanda and O. A. I. Noreldin

[7] Greenspan H.P., The theory of rotating fluids, 1968, Cambridge 5 Conclusion University press, London. [8] Gupta M.D., Gupta A.S., Convective instability of a layer of a fer- We investigated the thermoconvective instability in a ro- romagnetic fluid rotating about a vertical axis, Int. J. Eng, Sci., tating ferromagnetic fluid layer with time periodic temper- 1979, 17(3), 271–277. ature boundary conditions. The influence of flow parame- [9] Venkatasubramanian S., Kaloni P.N., Effects of rotation on the thermo-convective instability of a layer of a ferrofluids, Int. J. ters such as the Rayleigh number on the onset of instability Eng. Sci., 1994, 32(2), 237–256. was determined using a weakly nonlinear stability anal- [10] Gotoh K., Yamada M., Thermal convective in a horizontal layer ysis. The results are broadly in line with the earlier find- of magnetic fluid, J. Phys. Soc. Japan, 1982, 51(9), 3042–3048. ings in [1, 3, 8]. The heat transport has been analyzed for [11] Bhasauria B.S., Siddheshwar P.G.,Kumar J., Suthar O.P.,Weakly both the in-phase and out of phase temperature modula- nonlinear stability analysis of temerature/gravity modulated tions. The influence of the parameters such as the Prandtl stationary Rayleigh-Benard convection in a rotating porous medium, Transp. Porous Med., 2012,92(3), 633–647. number, Taylor number and the magnetization parameter [12] Rudraiah N., Sekhar N.G., Convective on magnetic fluids with in- on the Nusselt number for the in-phase modulation was ternal heat generation, ASME J. Heat Transfer, 1991, 113(1), 122– found to be less significant compared to the case of out of 127. phase modulation. [13] Aniss S., Belhaq M., Souhar M., Effects of amagnetic modula- The set of nonlinear differential equations for con- tion on the stability of a magnetic liquid layer heated from above, J. Heat Transfer, 2001,123(3) 428–432. vection amplitude was solved using a multi-domain spec- [14] Singh J., Bajaj R., convective instability ina ferrofluids layer with tral collocation method. The accuracy of the solutions was temperature modulation rigid boundaries, Fluid Dyn. Res., 2011, determined by comparison with solutions using a differ- 43(2), 025502. ent independent method, namely the Runge-Kutta based [15] Trefethen L.N., Spectral methods in MATLAB, SIAM, Philadel- ode45 Matlab solver. The stability of the equilibrium solu- phia, 2000, 26(2), 199. tions of the nonlinear differential equations has been ana- [16] Motsa S.S., Dlamini P., Khumalo M., A new multistage spectral relaxation method for solving chaotic initial value systems, lyzed. Transitions from different states have been demon- Nonlin. Dyn., 2013, 72(1-2) 265–83. strated for different parameter values, for example from [17] Motsa S.S., Magagula V.M., Goqo P.S.,Oyelakin I.S., Sibanda P., steady convection to chaotic solutions at high Rayleigh A multi-domain spectral collocation approach for solving Lane- numbers. Emden type equations, Chaper 7, Numerical Simulation - From Brain Imaging to Turbulent flows, 2016, 10, 5772-63016. [18] Noreldin O.A.I., Sibanda P., Mondal S., Weakly Nonlinear Sta- bility in a Horizontal Porous Layer Using a Multi-domain Spec- References tral Collocation Method, chapter 9, Complexity in Biological and Physical Systems, 2018, 71066. [19] Bhadauria B.S., Fluid convection in a rotating porous layer un- [1] Finlayson, B.A., Convective instability of ferromgnetic fluids, J. der modulated temperature on the boundaries, Transp. Porous Fluid Mech., 1970, 40(4), 755–767. Med., 2007, 67 297–315. [2] Chandrasekhar, S., Hydrodynamic and hydromagnetic stability, [20] Mielke A., The Ginzburg-Landau Equation in Its Role as a Mod- 1961, Oxford University Press, Oxford. ulation Equation, Chapter 15, Handbook of Dynamical Systems, [3] Rosensweig, R.E.,Ferrohydrodynamics, 1985, Cambridge Uni- 2002, 2, 759–834. versity Press. [21] Routh-Hurwitz criterion, Encyclopedia of Mathematics, http: [4] Schwab L., Hildebrandt U., Stierstadt K., Magnetic Benard con- //www.encyclopediaofmath.org/index.php?title=Routh-Hurwit vection, J. Magn. Magn. Mat., 1983, 39(1-2), 113–114. z_criterion&oldid=33371 [5] Stiles P.J., Kagan M., Thermoconvective instability of a horizontal layer of ferrofluid in a strong magnetic field, J. Magn. Magn. Mat., 1990, 85(1-3), 196–198. [6] Ruessll C.L., Blnnerhassett P.J., Stiles P.J., Large wave number convection in magnetized ferrofluids, J. Magn. Magn. Mat., 1995, 149(1-2), 196–121.