Effect of synchronized and unsynchronized boundary temperature modulation on the regulation of heat transfer in a ferrofluid with Fe3O4 nanoparticles

Linear and non-linear analysis was carried out for a temperature modulated Rayleigh-Bénard ferroconvection (RBF) problem using Lorenz and Ginzburg-Landau models. The parallel and horizontal plates of infinite extension enclosing the ferrofluid (with nanosized Fe3O4 - magnetite), is cooled from the top and heated from the bottom and is exposed to an exterior static magnetic field which manipulates the flow of a ferrofluid. The Lorenz model in its linear form manifests the stationary Rayleigh number expression, whereas the nonlinear form of the model leads to Ginzburg-Landau equation determining the amplitude, which aids to quantify the amount of heat transfer in ferrofluids with the effect of temperature modulation. The influence of various parameters like Lewis number, concentration Rayleigh number, ferromagnetic parameters on the onset of ferroconvection has been discussed in detail using marginal stability curves. On the other hand, the effect of different parameters like ferro-nanoparticle volume fraction, modulation frequency, phase angle, temperature modulation on heat transfer in ferrofluids has been analyzed and represented graphically.


Introduction
In the past few decades, rapid development of nanotechnology lead to a new research domain viz.nanofluids which has found a wide range of applications all over the world in many fields.Nanofluids are the fluids, where the suspension of nano-sized particles takes place in the base fluids which include water, mineral oils, polymeric solutions, ethylene glycol and other common liquids whereas nanoparticles consists of metals, metal oxides, nitrides, carbides and so on.Nanofluids possess unique features that are different from millimeter and micrometer sized particles suspended in the base fluids where the particles accumulate, corrode and clog the flow.Many works on nanofluids are reported (Sheikholeslami et. al.[34], Ali et.al. [35], Siddheshwar and Meenakshi [39][45], Siddheshwar and Kanchana [40], Kanchana et.al. [48], Haddad et.al. [25] [26], Kakac and Pramuanjaroenkij [19], Yimin and Qiang [11], Oztop and Abu-Nada [17], Buongiorno [15], Khanafer et.al. [10], Siddheshwar and Lakshmi [46]) due to their excellent characteristics and extensive applications, specially in the enhancement of heat transfer.
Survey on nanofluids, ferrofluids and external modulation motivated the current work to present the rate of heat transferred by the ferrofluids with boundary temperature modulation, which aids in thermal management systems to greater extent, due to its ability to react towards external magnetic field(either uniform or non-uniform).Also one should know that, ferrofluids in the absence of an exterior static magnetic field acts more or less similar to that of regular nanofluids.

Equations governing the problem
We now will examine a Newtonian ferrofluid layer of depth , parallel to the xy-plane of infinite extent, exposed to a vertical temperature gradient and exterior magnetic field, along with the gravitational acceleration,  ⃗ = − ̂.One of the most fundamental characteristic of the ferrofluid placed in external magnetic field, is the initial magnetic susceptibility which determines the magnetic retort of the ferrofluid.Also, one can notice that, if there is no external magnetic field, then ferromagnetic nanoparticles acts more or less similar to that of, regular nanoparticles suspended in the carrier fluid.As a consequence of which, the ferrofluid layer cooled from the top and heated from the bottom, is subject to external static magnetic field,  ⃗ ⃗⃗ =  0  ̂.We limit the current problem to the study of an electrically non-conducting superparamagnet[see [28][6]].Under the presumption of small scale convective motions(and obeying Boussinesq approximation), the equations governing the considered problem are: (  ⃗⃗⃗⃗⃗⃗  + ( ⃗. ∇)q ⃗⃗) = −∇ +   ∇ 2 q ⃗⃗ + ρ(T, C)g ⃗⃗ + μ 0 (M ⃗⃗⃗⃗ .∇)H ⃗⃗⃗ , where  ⃗ is the velocity vector,  the density,  time,  pressure,  ⃗⃗⃗ is the magnetization,  the temperature,  nanoparticle concentration,  0 is the temperature at upper plate,  0 is the concentration at upper plate,  ⃗⃗ is the magnetic induction,  the scalar magnetic potential,  0 is the magnetic permeability of vacuum,   and   are magentic coefficients,   ,   ,   ,   and   are the coefficients of Brownian diffusion, thermophoretic diffusion, magnetophoretic diffusion, thermal expansion and concentration expansion respectively.where   (= 423 kA/m) is the magnetic saturation of magnetite [38],  1 (= 10 ) is the diameter of magnetite,   (= 1.38 * 10 −23 /) the Boltzmann constant,  1 is the nanoparticle volume fraction,  3 =  1  1 3 ( 1 + 2 ) 3 is the gross volume fraction of the nanoparticles including the nonmagnetic coating at the particle's surface,  2 (= 2) is the non-magnetic layer of the nanoparticle.
The externally imposed boundary criteria for temperature and nanoparticle concentration are given by [23], and where  is the amplitude modulation,  is the smallness in order of magnitude of modulation,  frequency of modulation, Φ phase angle, Δ and Δ are small temperature and small concentration of nanoparticles that is modulated upon across the ferrofluid layer.

Basic state
Assuming the basic state's solution in the below form: and employing the above in governing equations, we have basic state solution in the subsequent form: where,

Perturbed state
The main objective here is to analyse the stability theory so, now will disturb the basic state employing the below perturbations: For simplicity, we take only two-dimesional disturbances into consideration and therefore set forth stream function as shown below: which satisfies the continuity equation 1.

Non-dimensionalization
Using the above in governing equations, we arrive at the set of perturbed governing equations.To transform the resultant equations to dimensionless form, we use the below characteristic scales: Eliminating the pressure in momentum equation, the resultant equation is non-dimensionalized along with other equation using the above scales 17 and dropping down all asterisks(*), we have: where , ′ is the derivative of η wrt z. , are ferromagnetic parameters.

Boundary conditions
The aforementioned dimensionless equations 18-21 are solved, subject to the below mentioned freefree isothermal boundary criteria:

Linear stability analysis for the onset of ferromagnetic convection
The linear stability analysis can be undergone by ignoring the nonlinear terms in dimensionless governing equations and by incorporating the normal mode technique method as shown below: The thermal Rayleigh number,   , for stationary mode of convection is given by: where  1 2 =  2 + We now employ the following asymptotic expansion in equations ( 18)-( 21): where  0 is the critical Rayleigh number,   , for stationary mode of convection.Using equation 26 in equation 25 and evaluating for the like powers of  on either side, we have: , ].
The system of equation 27, corresponds to the linear stability equations for the emergence of ferroconvection and the condition for non-triviality of the above system yields R0=Rtc.The solution to the first order system 27 and the solution to the second order system 28 are given by: where  1 and  2 are too lengthy and omitted due to compactness.

Heat transport
The horizontally averaged Nusselt number, (), which measures the quantity of heat transfer is given by: Now substituting equation 31 in equation 32 and solving the integration, we have the final equation for () as We now extract A from the Lorenz model which comes in the upcoming section in order to quantify the amount of heat transport.

Results and discussions
In Rayleigh--Bénard convection, the instability of the system occurs due to buoyancy effects, when the Rayleigh number crosses certain critical value which is also named as critical Rayleigh number(below which there is a transfer of heat due to conduction and above which there is a transfer of heat due to convection),    .In the case of RBF, suppose the buoyancy effects are negligible, then the instability occurs when magnetic number(also called as magnetic Rayleigh number) crosses certain critical value, on the contrary, if the magnetic effects are negligible, then the instability occurs, when Rayleigh number crosses the critical value.In the presence of both buoyancy and magnetic effects, the plot of   versus wave number,  for variant values of magnetic numbers are shown in figures 2a-2d.These figures indicates the point of critical value for the onset of ferroconvection.Increase in  1 ,  2 and  4 magnetic numbers causes decrease in   which tends to early onset of convection, and on the other hand, increase in  3 increases the   thereby delaying the onset of convection.As a result, the system's instability arises due to the magnetic influence of  1 ,  2 and  4 whereas the same is stable due to the magnetic influence of  3 .Figure 2e shows the variation of   with respect to  for variant values of Lewis number and fixed values of other parameters.Lewis number is the ratio of thermal diffusivity to Brownian diffusion coefficient and signifies the fluid flow which comprises of both heat and mass transfer(here is due to the motion of ferromagnetic nanoparticles, also known as Brownian motion).Increase in Lewis number specifies the dominance of thermal diffusivity over Brownian diffusivity.As and when the Lewis number increases, we can observe sharp rise in   and , which postpones the onset of convection.Figure 2f depicts the influence of   on the onset of ferroconvection.As the value of   increases,   also increases suggesting the stabilization of the system.The impact of modulated frequency and amplitude on heat transfer are represented in the figures 3a, 3c, 4a, 4c and 5a, 5c for IPM, OPM and LPMO respectively.The impact of altering both the modulation( and  1 ) on heat transport for IPM are negligibly small as the value of  does not alter as shown in figures 3a, 3c.But, this is not the case for OPM and LPMO.Though the magnitude of  does not alter on increasing the frequency modulation values, there is a change in the wavelength of oscillations.The wavelength of  becomes shorter on increasing the values of  thereby indicating the faster heat transfer as shown in figures 4a and 5a.The magnitude of  increases with increasing values of amplitude modulation,  1 , thereby increasing the rate of heat transport as shown in figures 4c and 5c.From figures 3b-5b one can see the effect of nanoparticle volume fraction on the rate of heat transport.The value of  rises as the value of  1 does, thereby enhancing the amount and speed of heat transfer.The impact of   is to boost the value of  thereby increasing the rate of heat transfer and destabilizing the system in all the types of temperature modulations viz.IPM, OPM and LPMO as shown in figures 3d-5d.The plot of  vs  for variant values of magnetic parameters and different temperature profiles can be seen in figures 3efgh-5efgh.One can notice that the influence of magnetic parameters  1 and  2 is to enhance  whereas that of  3 and  4 is to diminish the value of .The comparison between all types of modulation and their effects on heat transport are depicted graphically in figure 6 which shows that heat transfer is greater in OPM when compared with LPMO and IPM.  .(42) which is in accordance with the earlier work carried out by Finlayson [2]. Plots of different temperature profiles for variant values of amplitude modulation, frequency of modulation and   are in line with the earlier findings carried out by Bhadauria et. al.([23], [29]- [33], [37]).

Conclusions
A weakly nonlinear stability analysis of a ferroconvection with the influence of synchronized and unsynchronized boundary temperature modulation(BTM) has been analysed in the present study and following conlusions are drawn accordingly: The effect of magnetic parameters  1 ,  2 and  4 is to advance the onset of ferroconvection whereas that of  3 is to postpone the emergence of ferroconvection.
 Increase in Lewis number and   results in the early emergence of ferroconvection. The impact of synchronized BTM are almost nil on the heat transport. The impact of asynchronized BTM on the heat transport are oscillatory. Comparison among synchronized and asynchronized BTM in the domain of heat transfer brings in the relation as below:   <   <   . Suspension of nano-sized ferromagnetic particles in the base fluid boosts the rate of heat transfer. One can regulate the quantity of heat transport by setting the values of magnetic parameters, phase angle, frequency and amplitude modulation. The consequence of   is to boost the amount of heat transfer in both synchronized and asynchronized BTMs.

Figure 6 : 2 𝑘 2 [
Figure 6: Plot of  vs  for different temperature profiles.6.Validation In case of omission of ferromagnetic nanoparticles(and therefore exterior static magnetic field), we have ferromagnetic parameters  1 ,  2 ,  3 ,  4 = 0 and in the absence of concentration expansion coefficient and magnetic expansion coefficient,   = 0 (which implies   = 0) and   = 0.Both when applied in the equation 24, reverts back to Rayleigh--Bénard convection for stationary mode[1],  =  1 6  2 .(41)Just in the absence of concentration expansion coefficient and magnetic expansion coefficient,   = 0 (which implies   = 0) and   = 0.Both when applied in the equation 24, the stationary Rayleigh number reduces to,  =