Ferromagnetic fluids are colloids consisting of nanometer-sized magnetic particles suspended in a fluid carrier. The magnetization of a ferromagnetic fluid depends on the temperature, the magnetic field, and the density of the fluid. The magnetic force and the thermal state of the fluid may give rise to convection currents. Studies on the flow of ferromagnetic fluids include, for example, Finalyson  who studied instabilities in a ferromagnetic fluid using free-free and rigid-rigid boundaries conditions. He used the linear stability theory to predict the critical Rayleigh number for the onset of instability when both a magnetic and a buoyancy force are present. The generalization of Rayleigh Benard convection under various assumption is reported by Chandrasekhar . In the last few decades the study of heat transfer in ferromagnetic fluids has attracted many researchers due to the potential application of these fluids in industry, such as in the sealing of rotating shafts, ink, and so on. An authoritative introduction to research on magnetic fluids is given by Rosensweig .
Schwab et al.  studied the Finlayson problem experimentally in the case of a strong magnetic field and determined the parameters for the onset of convection. Their results were shown to be in good agreement with those of Finlayson . Stiles and Kagan  extended the experimental problem reported by Schwab et al.  by introducing a strong magnetic field. A weakly nonlinear stability analysis was used by Russell et al.  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 difference between the bounding surfaces.
The rotation of fluids is an interesting topic that has been studied by, for example, Greenspan . The classical Rayleigh-Benard problem when the fluid layer is rotating is well known in the case of ordinary viscous fluids and has been reported by Chandrasekhar . However, ferromagnetic fluids are known to exhibit very peculiar characteristics when set to rotate. Demonstrating the effect of rotation on convection in ferromagnetic fluids is scientifically important to researchers. Gupta and Gupta  examined the onset of convection in a ferromagnetic fluid heated from below and rotating about a vertical axis subject to a uniform magnetic field. They concluded that over-stability may not occur for Prandtl numbers smaller than unity. The thermo-convective instability in a rotating ferrofluid was further analyzed by Venkatasubramanian and Kaloni . They presented both analytical and numerical results for free and rigid boundary conditions. Their results were in good agreement with those of Finlayson  and Chandrasekhar  for some limiting cases. Thermo-convection in a ferromagnetic fluid has been studied by other researchers, for instance, [10, 12].
The problem associated with convection in ferromagnetic fluids is both relevant and mathematically challenging. The unmodulated Rayleigh Benard problem of a ferromagnetic fluid has been extensively studied. The effect of a magnetic modulation on the stability of a magnetic liquid layer heated from above was studied by Aniss et al. . They used the Floquet theory for their study of the onset of convection. The study showed the possibility of a competitive interaction between harmonic and subharmonic modes at the onset of convection. Convective instability in a ferromagnetic fluid layer with time-periodic modulation in the temperature field was investigated by Singh and Bajaj  using the linear stability theory and the classical Floquet theory. Their result agrees with those of Aniss et al. .
Convection in a rotating horizontal fluid layer confined in a porous medium with temperature modulation at the boundary was studied by Bhadauria . He investigated the stability of the flow using the Galerkin method and the Flouquet theory. In this study we analyze thermoconvective instability in a rotating ferromagnetic fluid layer with time periodic temperature boundary conditions. The fluid layer is heated from below and rotates about the vertical axis subject to a uniform magnetic field. We assume two stress free and two rigid boundary conditions. The Ginzburg Landau equation is obtained, see  for details on the relevance of the Ginzburg Landau equation. Nonlinear ordinary differential equations of the Lorenz type are obtained and solved numerically using the multi-domain spectral collocation method [16, 17, 18]. This method has not been fully tested before on evolution equations of this nature, hence the accuracy of solutions obtained using this method is also a matter of concern in this study. Heat transfer in the rotating horizontal fluid layer is discussed.
2 Mathematical formulation
Consider a ferromagnetic fluid confined between two infinite horizontal plates at z = −h/2 and z = h /2. The layer is heated from below and cooled from above, and is rotating uniformly about the vertical axis with constant angular velocity Ω. The lower and upper plates are subjected to an oscillatory temperature T0+ ΔT[1+ϵ2 cos(ωt+φ)] where ω is the modulation frequency and φ is the phase angle. The Oberbeck-Boussinesq approximation is assumed to be applicable. The magnetization Mof the ferrofluid is assumed to be parallel to the magnetic field H. The equations describing the fluid motion under these assumptions are the continuity equation, modified momentum equation, energy equation and Maxwell’s equations (Finlayson  and Gupta and Gupta ):(1)(2)(3)(4)
where V is the velocity field, ρ0 is the density at the ambient temperature, is the pressure, μ is the viscosity, g is the gravitational body force, B is the magnetic induction, μ0 is the magnetic permeability, T is the temperature, κ is the thermal conductivity, C V,H is the heat capacity at constant volume and magnetic field, α is the thermal expansion coefficient and Φ is the viscous dissipation. The magnetization and magnetic field are related by the formula(5)
The magnetization is dependent on the temperature and magnitude of magnetic field, so that(6)
Equation (6) is linearized using the Taylor expansion(7)
where is the magnetic susceptibility and is pryomagnetic coefficient, H0 is the uniform magnetic field and T1 = (T∞ + T0)/2, T∞ and T0 are the temperatures at h /2 and −h/2, respectively. The study is restricted to the case when magnetization induced by the temperature variation is much smaller than that induced by the external magnetic field. The density varies linearly with temperature as(8)
3 Weakly nonlinear stability
In this section we use weakly nonlinear stability analysis to study the evolution of disturbances in a ferromagnetic fluid with two free and two rigid boundary conditions and temperature modulation. Using Eqs. (4) and (5) and assuming that the magnetic field H is collinear with the magnetic induction B, Eq. (2) reduces to(9)(10)
We superimpose small perturbations on the basic state. The perturbed quantities are defined as(13)(14)(15)(16)(17)(18)(19)
where ρoC = ρ0CV,H + μ0KH0 and is the Laplace operator in two dimension. For the clarity we drop the prime from the perturbed quantities and introduce the following dimensionless variables(20)(21)(22)(23)(24)
where is the Taylor number, is the Rayleigh number, is the ratio of the magnetic force to the buoyancy force, is a nondimensional parameter, is the Prandtl number and is a measure of nonlinearity of the magnetization. The magnetic Rayleigh number can be obtained from the formula N = RaM1. Hereafter the asterisk will be dropped from Eqs. (21)-(24).
– Free boundary conditions(25)
– Rigid boundary conditions(26)
3.1 The solution for stress free boundaries
The solution for stress free boundaries has been discussed in [1, 3, 9]. Here we only emphasize the solution aspects which have not been discussed before. We solve the eigenvalue problem with two stress free boundaries to study the onset of instability in the ferromagnetic fluid. We consider a small variation in time scale τ = ϵ2t such that stationary convection occurs at lower orders of ϵ and introduce the following asymptotic expansions(27)(28)(29)(30)(31)(32)
This equation corresponds(33)(34)(35)(36)
A(τ) sin ax sin πz,
where a is a dimensionless wave number. Thus, the stationary Rayleigh number is given as(37)
To find the critical wave number and the corresponding critical Rayleigh number we set a2 = π2x. Then the stationary Rayleigh number can be written as(38)
The critical wave number and the corresponding critical Rayleigh number are obtained from
then we have(39)
For the case Ta = 0, M1 = 0 and M3 = 0, the classical critical wave number is
with corresponding classical critical Rayleigh number
The magnetic Rayleigh number is also of an interest and can be expressed as(40)
For large values of M1 the magnetic Rayleigh number in the absence of buoyancy effects is obtained as(41)
The critical wave number and corresponding critical magnetic Rayleigh number are obtained from solving the equation(42)
At second order O(ϵ2) we obtain the following equations:(43)
The solution at the second order is(45)(46)(47)
A2(τ) sin2 ax cos 2πz.
At the third order, we obtain(48)
where ŵ1, ζ̂1 and θ̂1 are the solutions of the adjoint system of the first order. This gives(54)
The above equation reduces to(55)
In this study we are also interested in heat transfer in ferromagnetic fluids. The Nusselt number for ferromagnetic fluids is defined as 2 1(56)
3.2 The general Lorentz type equations
We restrict the analysis to the case of two-dimensional disturbances so that all physical quantities are independent of y. Using the stream function defined by(57)(58)(59)(60)
where J is the Jacobian matrix. The solution of Eqs. (57)–(60) represented as a minimal double Fourier series of modes (1,1) for the stream function and magnetic potential and modes (0,2) and (1,1) for temperature and vorticity of the finite amplitude convection of the ferromagnetic fluid flows as(61)(62)(63)(64)
where A11, B11, B02, C11, C02 and D11 are time t dependent amplitudes. This is equivalent to a truncated Galerkin method. Substituting and integrating over the domain, we obtain a set of four ordinary differential equations for the time evolution of the amplitudes of convection of a ferromagnetic fluid in the form(65)(66)(67)(68)
To simplify the equations we introduce new variables(69)(70)(71)(72)(73)(74)
3.3 Stability of Lorentz equations
In this section we discuss the stability of the nonlinear systems of differential equations that describe the evolution of the convection amplitudes for a ferromagnetic fluid flow. Firstly, we note that the nonlinear Eqs. (71)–(74) are invariant under the transformation(75)
These equations are also uniformly bounded and dissipative in the phase space(76)
Thus the volume of the phase space moving with the flow for time τ > 0 is given by(77)
– The motionless conduction solutions (0, 0, 0, 0).
– The steady solution represented by the point
The stability of the stationary point associated with the motionless solution X * = (0, 0, 0, 0) is determined by roots of the following characteristic polynomial equation(78)
for the eigenvalues ξi, (i = 2, 3, 4) and It is clear that ξ1 is always negative as Pr > 0. The remaining eigenvalues are obtained from Eq. (78), and using the Routh-Hurwitz criteria , the polynomial Eq. (78) has negative real roots if and only if
and Pr > 1. This implies that the stationary solution is a stable node. Hence the critical value of R where the stationary solution of ferromagnetic fluid flow loses stability and steady convective flow takes over is
The stability of the stationary point corresponding to the steady convective flow is determined by the roots of the characteristic equation(79)
Applying the Routh-Hurwitz criteria to Eq. (79), it is clear that c1 > 0, and c3 > 0 if and only if
Also, c4 > 0 if and only if
Ta + 1 < PrKR.
Hence the fixed point is stable if the condition(80)
is satisfied where
3.4 The method of solution
In this section, we describe the multi-domain spectral collocation method [15, 16, 17, 18] used to obtain the solutions to Eqs. (71)–(74). The multi-domain technique assumes that the interval Λ = [0, T] can be decomposed into p non overlapping sub-intervals. The sub-intervals are defined as(81)(82)
with initial conditions(83)
Here αn,k and gn (n, k = 1, 2, 3, 4) are constants while fn is the nonlinear component of each equation. Each subinterval Λi is transformed to [−1, 1] using the transformation(84)
The Chebyshev-Gauss-Lobatto collocation points are used to discretize the unknown functions(85)
The derivative of the unknown function at the collocation point is given by(86)
with δτi = τi−1 − τi and D is Chebyshev differentiation matrix. The vector functions at the collocation points are(87)
with An = D + αn,nI and(88)(89)(90)(91)
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
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  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 directly comparable, of interest is the general trend observed, namely that in both cases, increasing Ta increases the critical wave number and the Rayleigh numbers suggesting that the influence of Taylor number is to stabilize the system.
on the stationary Rayleigh number. It can be seen in Figures 1(a)and 2(a) that as Ta increases from 0 to 50 the values of the stationary Rayleigh and the magnetic Rayleigh numbers both increase. This shows that rotation has a stabilizing effect on the system. This result is similar to that of an ordinary viscous fluid. Rotation has a stabilizing influence on ferromagnetic fluid flow. Figures 1(b) and 1(c) show the relative influence of the size of the magnetic
force to the buoyancy force parameter M1. As M1 increases from 0 to 5 the stationary Rayleigh number is reduced. This suggest the magnetic and the buoyancy force are both destabilizing to the ferromagnetic fluid flow. Further, as M1 increases with Ta = 10 fixed, the stationary Rayleigh number decreases, suggesting M1 has a destabilizing effect for both low and high Taylor numbers. From Figures 1(d) and 2(b) it is observed that increasing M3 from 1 to 20 reduces both the Rayleigh number and the magnetic Rayleigh number, this is destabilization to the system.
The Ginzburg-Landau equation is obtained using the nonlinear stability analysis at the third order of ϵ. The equation was solved using a multi-domain spectral method. The heat transfer coefficient, represented by the Nusselt number, is presented graphically for in-phase and out-phase modulation in Figures 3–4. Figure 3(a)–3(d) show the effect of Pr, Ta, M1 and M3 on the Nusselt number with time τ. It can be observed that on increasing the Pr and M3, the Nusselt number decreases. Hence increasing these parameters reduces the rate of heat transfer. Increasing Ta and M1 increases the Nusselt number, thus the rate of heat transfer increases. Figures 4(a)–4(d) show changes in the Nusselt number with respect to time τ due to the influence of various parameters in the case of out of phase modulation. It can be observed that the Nusselt number for in-phase modulation is less than for out of phase modulation.
A multi-domain spectral collocation method was used to find the nonlinear amplitudes in ferromagnetic fluid convection equations for various values of R. The solution sets were obtained using initial conditions selected in the neighborhood of the stationary points corresponding to the motionless solutions. The simulations were done to a maximum time τmax = 20. For a sense of the accuracy of the method, the solutions were compared with solutions obtained using the Runge-Kutta based ode45 routine. Figures 5(a)–5(d) show the time series solution of X1(τ) for different supercritical values of R. As R increases, periodic solutions are obtained. Here a comparison between the multi-domain spectral collocation
method and the ode45 is given. In Figures 6–11 we present a projection of the trajectories onto the (X1, X2), (X1, X3), (X1, X4), (X2, X3), (X2, X4) and (X3, X4) phase planes, respectively. The initial supercritical convective solution R = 2 is presented in part a in each figure. We observe that the trajectories attracted to equilibrium points that correspond to the motionless solution are stable spirals. The solutions are presented in part c and d of each figure when R = 20 and R = 25, respectively. For these Rayleigh numbers, chaotic solutions are obtained. These changes in solutions are further presented in Figures 6(c)–6(d), 7(c)–7(d) and 9(c)–9(d). The results presented
Figure 12 shows the streamlines patterns for the flow of a ferromagnetic fluid. Two different eddies are observed. The clockwise and anti-clockwise flows are shown via negative and positive stream function values, respectively. With the Rayleigh number increasing from 2 to 200, the magnitude of the stream function values increase. The sense of motion in the subsequent cells is opposite that of an adjoining cell, indicating symmetry in the formation
of ferromagnetic convective cells. Figure 13 shows the isotherm patterns as the Rayleigh number changes from 2 to 200. Three different eddies are observed. The small eddy at the left corner diminishes as R increases from 2 to 200.
Also, increasing R reduces the density of the isotherms implying a delay of the onset of instability.
We investigated the thermoconvective instability in a rotating ferromagnetic fluid layer with time periodic temperature boundary conditions. The influence of flow parameters such as the Rayleigh number on the onset of instability was determined using a weakly nonlinear stability analysis. The results are broadly in line with the earlier findings in [1, 3, 8]. The heat transport has been analyzed for both the in-phase and out of phase temperature modulations. The influence of the parameters such as the Prandtl number, Taylor number and the magnetization parameter on the Nusselt number for the in-phase modulation was found to be less significant compared to the case of out of phase modulation.
The set of nonlinear differential equations for convection amplitude was solved using a multi-domain spectral collocation method. The accuracy of the solutions was determined by comparison with solutions using a different independent method, namely the Runge-Kutta based ode45 Matlab solver. The stability of the equilibrium solutions of the nonlinear differential equations has been analyzed. Transitions from different states have been demonstrated for different parameter values, for example from steady convection to chaotic solutions at high Rayleigh numbers.
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Online erschienen: 31.12.2018
Quellenangabe: Open Physics, Band 16, Heft 1, Seiten 868–888, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0109.
© 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. BY-NC-ND 4.0