Rayleigh-Bénard Chandrasekhar Convection in An Electrically Conducting Fluid using Maxwell- Cattaneo Law with Temperature Modulation of the Boundaries

The effect of imposed time-periodic boundary temperature (ITBT, also called temperature modulation) and magnetic field at the onset of convection is investigated by making a linear analysis. The classical Fourier heat law is replaced by the non-classical Maxwell-Cattaneo law. The classical approach predicts an infinite speed for the propagation of heat. The adopted non-classical theory involves wave type of heat transport and does not suffer from the physically unacceptable drawback of infinite heat propagation speed. The Venezian approach is adopted in arriving at the critical Rayleigh number, correction Rayleigh number and wave number for small amplitude of ITBT. Three cases of oscillating temperature field are examined: (a) symmetric, so that the wall temperatures are modulated in phase, (b) asymmetric, corresponding to out-of phase modulation and (c) only the lower wall is modulated. The temperature modulation is shown to give rise to sub-critical motion. The shift in the critical Rayleigh number is calculated as a function of frequency and it is found that it is possible to advance or delay the onset of convection by time modulation of the wall temperatures. It is shown that the system is most stable when the boundary temperatures are modulated out-of-phase. It is also found that the results are noteworthy at short times and the critical eigenvalues are less than the classical ones.


I. INTRODUCTION
The Classical Fourier law of heat conduction expresses that the heat flux within a medium is proportional to the local temperature gradient in the system. A well known consequence of this law is that heat perturbations propagate with an infinite velocity. This drawback of the classical law motivated Maxwell [1], Cattaneo [2], Lebon and Cloot [3], Dauby et al. [4], Straughan [5], Siddheshwar [6], Pranesh [7], Pranesh and Kiran [8,9,10] and Pranesh and Smita [11] to adopt a non-classical Maxwell-Cattaneo heat flux law in studying Rayleigh-Bénard/Marangoni convection to get rid of this unphysical results. This Maxwell-Cattaneo heat flux law equation contains an extra inertial term with respect to the Fourier law.
is the heat flux,  is a relaxation time and  is the heat conductivity. This heat conductivity equation together with conservation of energy equation introduces the hyperbolic equation, which describes heat propagation with finite speed. Puri and Jordan [12,13] and Puri and Kythe [14,15] have studied other fluid mechanics problems by employing the Maxwell-Cattaneo heat flux law.
Thomson [16] and Chandrasekhar [17] studied theoretically and Nakagawa [18,19] and Jirlow [20] studied experimentally, the thermal stability in a horizontal layer of fluid with magnetic field and found that vertical magnetic field delays the onset of thermal convection. The application of magnetoconvection is primarily found in astrophysics and in particular by the observation of sunspots. Motivated by the above applications several authors investigated the suppression of convection by strong magnetic fields.
One of the effective mechanisms of controlling the convection is by maintaining a non-uniform temperature gradient across the boundaries. Such a temperature gradient may be generated by (i) an appropriate heating or cooling at the boundaries, (ii) injection/suction of fluid at the boundaries, (iii) an appropriate distribution of heat source, and (iv) radiative heat transfer. These methods are mainly concerned with only space-dependent temperature gradients. However in practice, the non-uniform temperature gradients find its origin in transient heating or cooling at the boundaries. Hence, basic temperature profile depends explicitly on position and time. This problem, called the thermal modulation problem, involves solution of the energy equation under suitable time-dependent boundary conditions. This temperature profile which is a function of both space and time can be used as an effective mechanism to control the convective flow by proper adjustment of its parameters, namely, amplitude and frequency of modulation. It can be used to control the convection in material processing applications to achieve higher efficiency and to advance convection in achieving major enhancement of heat, mass and momentum transfer.
The objective of the present problem is to investigate the combined effect of magnetic field and temperature modulation on the onset of Raleigh-Bénard magnetoconvection using Maxwell-Cattaneo law.

II. MATHEMATICAL FORMULATION
Consider an infinite horizontal layer of a Boussinesquian, electrically conducting fluid, of depth'd'. Cartesian co-ordinate system is taken with origin in the lower boundary and z-axis vertically upwards. Let T be the temperature difference between the lower and upper boundaries. (See Fig. 1 The basic governing equations are: Conservation of energy: Maxwell -Cattaneo heat flux law: respectively, where  is the small amplitude,  is the frequency and  is the phase angle.

III. BASIC STATE
The basic state of the fluid being quiescent is described by: The temperature b The solution of " (10)" that satisfies the thermal boundary conditions of "(7)" and "(8)" is The steady temperature field s T and an oscillating field 1 T  are given by where, (15) .
In the "(16)" (17) . e e e e 2 and Re stands for the real part.
We now superpose infinitesimal perturbation on this basic state and study the stability of the system.

IV. LINEAR STABILITY ANALYSIS
Let the basic state be disturbed by an infinitesimal thermal perturbation. We now have , The primes indicate that the quantities are infinitesimal perturbations and subscript 'b' indicates basic state value.
where, the asterisks have been dropped for simplicity and the non-dimensional parameters Pm Pr, , Q , R and C are as defined as In "(30)", Equations "(28)" to "(30)" are solved subject to the following conditions: Eliminating T and z H from "(28)" to "(30)", we get a differential equation of order 8 for W in the form In dimensionless form, the velocity boundary conditions for solving "(35)" are obtained from "(28)" to "(30)" and "(34)" in the form: We seek the eigen-function W and eigenvalues R of the "(35)" for the basic temperature distribution "(31)" that departs from the linear profile 1 z T 0     by quantities of order  . Thus, the eigenvalues of the present problem differ from those of the ordinary Bénard convection by quantities of order  . We seek the solution of "(35)" in the form The expansion "(37)" is substituted into "(35)" and the coefficients of various powers of  are equated on either side of the equation. The resulting system of equation is The zeroth order problem is equivalent to the Rayleigh-Bénard problem with magnetic field using Maxwell-Cattaneo law in the absence of temperature modulation. The stability of the system in the absence of thermal modulation is investigated by introducing vertical velocity perturbation 0 W lowest mode of convection as where, l and m are wave numbers in xy-plane with . m l a To solve "(45)", we expand the right-hand side using Fourier series expansion and obtain 1 W by inverting the operator L term by term as   We shall not solve "(48)", but will use this to determine. The solvability condition requires that the time-independent part of the right hand side of "(48)" must be orthogonal to   Now, we can calculate the maximum range of  , by assigning values to the physical parameters involved in the above condition. Thus, the range of the amplitude of modulation, which causes sub critical instabilities in different physical situations, can be explained.

X. RESULTS AND DISCUSSION
In this paper we make an analytical study of the effects of temperature modulation and magnetic field at the onset of convection in Newtonian fluid by replacing the classical Fourier law of heat conduction by non-classical Maxwell-Cattaneo law.
The analysis presented in this paper is based on the assumption that the amplitude of the modulating temperature is small compared with the imposed steady temperature difference. The validity of the results obtained here depends on the value of the modulating frequency  . When 1   , the period of modulation is large and hence the disturbance grows to such an extent as to make finite amplitude effects important. When The results have been presented in Fig. 2 to Fig. 13, from these figures we observe that the value of  R decreases for small values of  and increases for the moderate values of  . Thus small values of  destabilize the system and moderate values of  stabilize the system. This is due to the fact that when the frequency of modulation is low, the effect of modulation on the temperature field is felt throughout the fluid layer. If the plates are modulated in phase, the temperature profile consists of the steady straight line section plus a parabolic profile which oscillates in time. As the amplitude of modulation increases, the parabolic part of the profile becomes more and more significant. It is known that a parabolic profile is subject to finite amplitude instabilities so that convection occurs at lower Rayleigh number than those predicted by the linear theory. We find that c 2 R increases with increase in Q . When the magnetic field strength permeating the medium is considerably strong, it induces viscosity into the fluid, and the magnetic lines are distorted by convection. Then these magnetic lines hinder the growth of disturbances, leading to the delay in the onset of instability. R becomes more and more negative.
We can infer from this is that the effect of increasing the viscosity of the fluid is to destabilize the system. R in both in-phase and out-of-phase and these can be seen in Fig. 2 and Fig. 6. The above results are due to the fact that the temperature field has essentially a linear gradient varying in time, so that the instantaneous Rayleigh number is super critical for half cycle and subcritical during the other half cycle.
The above results on the effect of various parameters on c 2 R for out-of-phase modulation do not qualitatively change in the case of temperature modulation of just the lower boundary. This is illustrated with the help of Fig. 10 to Fig. 13.
From the above result, we can conclude that the system is more stable when boundary temperature is modulated in out-of-phase in compare to only lower wall modulation and in-phase modulation. In-phase temperature modulation leads to sub critical motions. Sub critical motions are ruled out in the case of out-of-phase modulation and lower wall temperature modulation.  increases the Rayleigh number R also increases thus amplitude of modulation stabilizes the system. It can be clearly seen that as  increases R increases for smaller values of  and decreases for moderate values of  which reconforms our earlier discussion on this.
The effect of 1  on R with respect to out-of-phase modulation and only lower wall modulation is illustrated in Fig. 15 and Fig. 16. We observe that as 1  increases R increases thus increase in 1  stabilizes the system. It is also observed from these figures that the frequency of modulation  destabilizes the system.
The results of this study are useful in controlling the convection by thermal modulation with Maxwell-Cattaneo law.

XI. CONCLUSIONS
Following conclusions are drawn from the problem: 1. The system is more stable when boundary temperature is modulated in out-of-phase.
2. In-phase temperature modulation leads to sub critical motions.
3. The results of the study throw light on an external means of controlling Rayleigh-Bénard convection either advancing or delaying convection by thermal modulation.
4. It is observed that for large frequencies, the effect of modulation disappears.
5. The nonclassical Maxwell -Cattaneo heat flux law involves a wave type heat transport and does not suffer from the physically unacceptable drawback of infinite heat propagation speed. The classical Fourier flux law overpredicts the critical Rayleigh number compared to that predicted by the non-classical law. Overstability is the preferred mode of convection.