Combined Effect of Magnetic field and Internal Heat Generation on the Onset of Marangoni Convection

Marangoni convection in a horizontal layer with a uniform internal heat source and vertical magnetic field is analyzed. The boundaries are considered to be rigid, however permeable, and insulated to temperature perturbations. The upper surface of a fluid layer is deformably free. The eigen value equations of the perturbed state obtained from the normal mode analysis are solved by using regular perturbation method with a as wave number. The results show that the critical Marangoni number c M become larger as the Chandrasekhar number Q increases, internal heat source and the Crispation number Cr decreases.


Introduction
The analysis of Benard-Marangoni convection in a thin fluid layer induced by thermal buoyancy and surface tension is important for many applications in science and engineering. Examples include energy storage in molten salts, crystal growth from a melt in space, and paints, colloids and detergents in chemical engineering. Nield [1] initiated the study of the Benard-Marangoni convective instability in a planar horizontal fluid layer with a non-deformable free surface. Later, Davis and Homsy [2] extended the work of Nield [1] to take the effect of the surface deflection into account. When the Crispation effect of a deformably free upper surface is considered, oscillatory instabilities may occur in the Benard-Marangoni problem. Takashima, M [3] presented a detailed numerical study of the linear stability analysis of Benard-Marangoni convection, including stationary and oscillatory modes, and focused the influence of the Crispation number on the conditions for a competition between two of these kinds of modes. Char and Chiang [4] examined the boundary effects on the Benard-Marangoni instability problem in the presence of an electric field, and found that the boundary effects of the solid plate have great influences on the stability of the system. Recently, Hashim and Wilson [5] advanced the analyses of [3,4] to the Benard-Marangoni instability of a horizontal liquid layer in the most physically-relevant case when Rayleigh number and Marangoni number are linearly dependent. All these previous investigators are restricted to the convective system in the absence of a magnetic field. Chandrasekhar [6] studied the Rayleigh-Benard convective instability induced by buoyancy in a magnetic field. The onset of steady Marangoni instability in an electrically conducting fluid layer with a nondeformable free surface in a magnetic field was first considered by Nield [1]. He found that the Lorentz force is to contract Marangoni convection. Later, Maekawa and Tanasawa [7] extended the analysis of [1] to investigate the effect of orientation of the magnetic field and the aspect ratio of the liquid layer on the onset of Marangoni convection.
The effect of quadratic basic state temperature gradient caused by uniform internal heat generation plays a decisive role in understanding control of convection. Copious literature is available on coupled Benard-Marangoni convection in a horizontal ordinary viscous fluid layer with uniform distribution of internal heat generation (Char and Chiang [4], Wilson [8], Bachok and Arifin [9] and references therein). The problem of penetrative convection in an ordinary viscous fluid-saturated porous layer has also received considerable attention in the recent past because of its applications in many science and engineering problems (with current highly relevant literature including Carr [10], ), Gangadharaiah and Suma [14], ).
In the present study, we have considered the problem of combined buoyancy and surface tension driven convection in a horizontal fluid layer in the presence of uniform vertical magnetic field including the additional effect of internal heat generation.. The lower rigid and upper free boundary at which the temperature-dependent surface tension forces are accounted for are considered to be perfectly insulated to temperature perturbations. The resulting eigenvalue problem is solved by regular perturbation technique with wave number as a perturbation parameter.

Mathematical Formulation
We consider penetrative convection via internal heating The governing equations for the fluid layer are: In the above equations, is the velocity vector, p is the pressure, T is the temperature, q is the heat source in the fluid layer, κ is the thermal diffusivity and 0 ρ is the reference fluid density. The basic state is quiescent and is of the form The basic steady state is assumed to be quiescent and temperature distributions are found to be ( ) ( ) Where 0 T is the interface temperature. In order to investigate the stability of the basic solution, infinitesimal disturbances are introduced in the form where the primed quantities are the perturbations and assumed to be small. Eq. (9) The linearized boundary conditions are:

Method of Solution
Since the critical wave number is exceedingly small for the assumed temperature boundary conditions (Nield and The boundary conditions (13) The boundary conditions ( ) ( ) 13 16 − become The general solution of (25) is The differential Equation (26) involving 2 1 D Θ provide the solvability requirement which is given by The expressions for 1 W is back substituted into Eq. (31) and integrated to get Marangoni number

Results and Discussion
The effect of internal heat generation on the criterion for the onset of Marangoni instability in the presence of vertical magnetic field with upper surface of a fluid layer is deformably free is investigated theoretically. The resulting eigen value problem is solved using a regular perturbation technique with wave number a as a perturbation parameter.
The presence of internal heating makes the basic temperature and magnetic field distributions to deviate from linear to parabolic in terms of the porous layer height which in turn has significant influence on the control of Marangoni convection. To assess the impact of internal heat source strength on the stability of the system, the dimensionless basic temperature ( ) ,   The effect of Crispation number on the critical Marangoni numbers for different values of Chandrasekhar number Q is plotted in Figure 5. From Figure 5, it can be seen that in the absence of magnetic field (Q = 0) the critical Marangoni number M c decreases as Cr increases. This effect is attributed to the fact that a higher value of Cr, representing a lower rigidity of the free upper surface of the fluid layer, makes the system more unstable. In addition, the dotted lines in the figure indicated the critical Marangoni number corresponding to an imposed magnetic field with Q = 4.0. An inspection of the figure reveals that the critical Marangoni number is higher in the presence of magnetic field when compared with the case of no applied magnetic field. This indicates that the applied magnetic field reduces the intensity of Marangoni convection and thus leads to a more stable system.

Conclusions
The problem of Marangoni convection in an electrically conducting fluid layer permeated by a uniform, vertical magnetic field has been studied theoretically. Of interest are the influences of internal heat source, imposed magnetic field, and the Crispation on the onset of Marangoni instability. The following conclusions may be made from this study.
1. When the layer is heated from below, the critical Marangoni number decrease monotonically with the increase internal hear source strength.
2. The effects of the Chandrasekhar number on the onset of Marangoni convections are more pronounced, especially for the fluid layer.
3. The critical Marangoni number increase as the Chandrasekhar number increases and decreases with the increasing Crispation number.