Thermorheological and magnetorheological effects on Marangoni-Ferroconvection with internal heat generation

Marangoni convectiveinstability in a ferromagnetic fluid layer in the presence of a spatial heat sourceand viscosity variation is examined by means of the classical linear stability analysis. The higher order Rayleigh-Ritz technique is used to compute the critical Marangoni number. The effective viscosity of the ferromagnetic liquid is taken to be a quadratic function of both the temperature and magnetic field strength. It is shown that the ferromagnetic fluid is significantly influenced by the effect of viscosity variation and is more prone to instability in the presence of heat source compared to that when viscosity is constant. On comparing the corresponding results of heat source and heat sink it is found that heat sink works in tandem with the effect of viscosity variation if magnetic field dependence of viscosity dominates over temperature dependence. If the temperature dependence of viscosity dominates, the effects of viscosity variation and heat sink are mutually antagonistic.


Introduction
The manifestation of cellular convective instability in non-magnetic fluid layers heated from below is generally credited to the buoyancy and surface tension mechanisms. The buoyancy driven convection (usually known as Rayleigh-Bénard Convection (RBC))preponderates the surface tension driven convection (typically referred to as Marangoni Convection (MC)) in the case of not-so-thin fluid layers under usual gravity conditions and it is the other way round for thin fluid layersunder microgravity situations (Pearson [1]). Thermally and magnetically induced gradients of magnetization are also responsiblefor the convective motion transpiring in magnetic fluids besides the buoyancy and surface tension candidates. The idea of regulating the properties of magnetic fluidsthrough a magnetic field has led to numerous fascinating applications (Popplewell[2],Berkovskiiet al [3] andHornget al [4]).
Technological and biomedical applications of magnetic liquids indicate that these liquids depend greatly on their rheological properties. Several studies such as those of Rosensweiget al [25], Shliomis[26], Kamiyamaet al [27], Kobori and Yamaguchi [28] and Chen et al [29] specify that the effective viscosity of a ferromagnetic liquid is enhanced by the application of a magnetic field. This reversible effect, known as magnetorheological effect, is a consequence of the fact that the particles magnetize in the presence of a magnetic field and form chain-like clusters that align with the applied field. These chain-like alignments of the dispersed solid particles impede the motion of the liquid, thereby increasing the viscous characteristics of the suspension. The contemporary applications of the magnetorheological effect include dampers, brakes, pumps, clutches, valves, robotic control systems and the like (Carlson et al [30]). Balauet al [31] have pointed out through their experiments that magnetorheological effect is of importance significantly in water-based and kerosene-based solutions, and in physiological-solution-based magnetic liquids even for moderate strengths of applied magnetic field. This is more so in the extraterrestrial context.Prakash [32] studied the effects of magnetic field dependent viscosity and non-uniform basic temperature profiles onthermomagnetic convection in a horizontal layer of ferrofluid.
Another fact about the viscosity of any carrier liquid decreasing with temperature is also well known (Geetha and Nanjundappa [24], Platten and Legros [33],Severin and Herwig [34], Ramanathan and Muchikel [35] and Nanjundappaet al [36]) and is referred to as thermorheological effect. It is imperative therefore to envisage the importance of the MC problem in ferromagnetic liquids involving both magnetic field and temperature dependent effective viscosity. Apart from the rheological effects discussed earlier, the effect of volumetric internal heat source is also important in ferromagnetic liquids from the viewpoint of magnetocaloric pumping. In this paper we aim at studying the effect of internal heat generation on the threshold of MC in a variable viscosity ferromagnetic liquid with a vertical temperature gradient and a vertical magnetic field. The assumed strength of the magnetic field is such that the liquid does not exhibit any non-Newtonian characteristics. The report on the study culminates with an important exploration of the dissimilarity, for Marangoni convection, between heat source and heat sink problems.

Mathematical Formulation
Consider an infinite horizontal layer of a thin ferromagnetic liquid (with a free upper surface) that maintains a temperature gradient and a magnetic field o H in the vertical direction. The gradient in temperature is by virtue of a prescribed temperature difference T (> 0 for fluid heated from below) across the layer and a uniform distribution of heat source/sink of intensity S in the liquid. The liquid is assumed to have an effective variable viscosity μ that depends on the magnitude of the magnetic field and the temperature.The upper boundary interface has a temperature and magnetic field dependent surface tension k is the thermal conductivity,  is the vector differential operator, It is of interest to note that the well-known viscosity variation with temperature is a non-Boussinesq effect (Selak and Lebon [37]). Further, for a ferromagnetic liquid, we have one more non-Boussinesq influencing factor for the viscosity that arises only when the magnetic field is present. The effective viscosity is known to escalatedue to the magnetic field in the case of these synthetic liquids (Chen et al [38]) owing to the reorientation of magnetic particles. There exist a number of correlations of viscosity-temperature and viscosity-magnetic field strength including linear, quadratic and exponential proportionalities. The quadratic and exponential viscosity variations have been brought in owing to the fact that the linear viscosity variation is inadequate in showing the destabilizing nature of temperature dependence of viscosity and stabilizing nature of magnetic field dependence of viscosity. The problem under consideration also necessitates a nonlinear viscosity variation rather than the linear one. In view of this we assume the effective viscosity ( , ) μ H T to be a quadratic function of H and T in the form In arriving at the above solution it has been assumed that o at where d is the thickness of the liquid layer. The dominance of magnetic dependency over temperature dependency of viscosity is signified by the condition 1 0 V< and 2 0 V< , while 1 0 V> and 2 0 V> signifies dominance of temperature dependency. We next study the stability of the systemby resorting to the method of small perturbation (Finlayson [14]).Introducing the magnetic potential ' Φ , eliminating the pressure p and incorporating the solutionsin equation (2.9), we obtain the following equations pertaining to the perturbed state    T Ma is the ratio of thermorheological factors favouringfluid motion to forces opposing motion. Likewise H Ma is the ratio of magnetorheological factors supporting fluid motion to forces opposing motion (which is assumed negligible in the further analysis).Since the occurrence of oscillatory instability is ruled out for the problem at hand (Lam andBayazitoglu [6], Finlayson [14] and Weilepp and Brand [23]), the stability equations associated with the stationary instability therefore read (2.24)

Method of Solution
The system of equations (

Results and discussion
External regulation of rheological properties and thereby the control of surface tension driven instability in a variable viscosity ferromagnetic liquid in the presence of internal heat generation and vertical uniform magnetic field is studied. The critical values pertaining to stationary convection have been computed using the Rayleigh-Ritz technique. The results arrived at in the problem could be understood better if we observe the profile of the basic state temperature distribution which sheds light on the effect of heat source/sink on the stability of the system.    Figures 2 and 3 is that the thermorheological and magnetorheological effects are more pronounced for a uniform heat sink than for a uniform heat source.   Table 1. It is found that the qualitative effect of the magnetization parameter 3 M and the magnetic susceptibility m χ on the onset of convection is akin to that in a constant viscosity ferromagnetic liquid (Finlayson [14]).   As can be seen from

Conclusions
The influence of temperature and magnetic field dependent effective viscosity ofmagnetic fluid on Marangoniconvection with internal heat generationis studied. The following conclusions are arrived at from the study:  Heat source and heat sink have reverse influence on magnetic fluid Marangoni instability.  Thermorheological and magnetorheological effects are markedly pronouncedwhen there is a uniform heat sink in the fluid layer.  Convection cell size is noticeably sensitive to the effect of variable viscosity, internal heat generation and the fluid magnetization. The problem is important in energy conversion devices and in microgravity application situations involving ferromagnetic liquids as working media.