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 fer- rofluids 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 ferro- magnetic fluid has been extensively studied. The effect ofa r · V = 0, (1) magnetic modulation on the stability of a magnetic liquid DV ρ = −rP0 + µr2V + 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 + r · (HB) + 2ρ V × Ω) + 0 r(jΩ × rj), (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 · = κr2T + Φ, (3) 0 ∂T Dt al. [13]. V,H Convection in a rotating horizontal fluid layer con- fined in a porous medium with temperature modulation r · B = 0; r × H = 0, (4) at the boundary was studied by Bhadauria [19]. He investi- gated the stability of the flow using the Galerkin method where V is the velocity field, ρ0 is the density at the am- 0 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 dis- cussed. 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 0 0 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 (︂ )︂ 0 0 M0 0 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 0 0 0 0 H 3 + M 3 = (1 + χ)H − KT . (15) ρ = ρ0(1 − α(T − T1)). (8) We assume that K∆T << (1 + χ)H0.