## 1. Introduction

Convection in a plane horizontal fluid layer heated from below, initially at rest and subject to an adverse temperature gradient, may be produced either by buoyancy forces or surface tension forces. These convective instability problems are known as the Rayleigh-Benard convection and Marangoni convection, respectively. The determination of the criterion for the onset of convection and the mechanism to control has been a subject of interest because of its applications in the heat and momentum transfer research. RayleighRayleigh 1916 was the first to solve the problem of the onset of thermal convection in a horizontal layer of fluid heated from below. His linear analysis showed that Benard convection occurs when the Rayleigh number exceeds a critical value. Theoretical analysis of Marangoni convection was started with the linear analysis by Pearson (1958) who assumed an infinite fluid layer, a nondeformable case and zero gravity in the case of no-slip boundary conditions at the bottom. He showed that thermocapillary forces can cause convection when the Marangoni number exceeds a critical value in the absence of buoyancy forces.

The determination of the criterion for the onset of convection and the mechanism to control convective flow patterns is important in both technology and fundamental Science. The problem of suppressing cellular convection in the Marangoni convection problem has attracted some interest in the literature. The effects of a body force due to an externally-imposed magnetic field on the onset of convection has been studied theoretically and numerically. The effect of magnetic field on the onset of steady buoyancy-driven convection was treated by ChandrasekharChandrasekhar 1961 who showed that the effect of magnetic field is to increase the critical value of Rayleigh number and hence to have a stabilising effect on the layer. The effect of a magnetic field on the onset of steady buoyancy and thermocapillary-driven (Benard-Marangoni) convection in a fluid layer with a nondeformable free surface was first analyzed by Nield Nield 1966. He found that the critical Marangoni number monotonically increased as the strength of vertical magnetic field increased. This indicates that Lorentz force suppressed Marangoni convection. Later, the effect of a magnetic field on the onset of steady Marangoni convection in a horizontal layer of fluid has been discuss in a series of paper by Wilson Wilson 1993, Wilson 1994. The influence of a uniform vertical magnetic field on the onset of oscillatory Marangoni convection was treated by Hashim & WilsonHashim & Wilson 1999 and Hashim & Arifin Hashim & Arifin 2003.

The present work attempts to delay the onset of convection by applying the control. The objective of the control is to delay the onset of convection while maintaining a state of no motion in the fluid layer. Tang and Bau Tang and Bau 1993, Tang and Bau 1994 and HowleHowle 1997)have shown that the critical Rayleigh number for the onset of Rayleigh-Bénard convection can be delayed. Or et al. Or et al. 1999 studied theoretically the use of control strategies to stabilize long wavelength instabilities in the Marangoni-Bénard convection. BauBau 1999 has shown independently how such a feedback control can delay the onset of Marangoni-Bénard convection on a linear basis with no-slip boundary conditions at the bottom. Recently, Arifin et. al.Arifin et. al. 2007 have shown that a control also can delay the onset of Marangoni-Bénard convection with free-slip boundary conditions at the bottom.

Therefore, in this paper, we use a linear controller to delay the onset of Marangoni convection in a fluid layer with magnetic field. The linear stability theory is applied and the resulting eigenvalue problem is solved numerically. The combined effect of the magnetic field and the feedback control on the onset of steady Marangoni convection are studied.

## 2. Problem Formulation

Consider a horizontal fluid layer of depth *d* with a free upper surface heated from below and subject to a uniform vertical temperature gradient. The fluid layer is bounded below by a horizontal solid boundary at a constant temperature

We use Cartesian coordinates with two horizontal x- and y- axis located at the lower solid boundary and a positive z- axis is directed towards the free surface. The surface tension,

where

where

Continuity equation:

Momentum equation:

Energy equation :

Magnetic field equations:

where

To simplify the analysis, it is convenient to write the governing equations and the boundary conditions in a dimensionless form. In the dimensionless formulation, scales for length, velocity, time and temperature gradient are taken to be

Our control strategy basically applies a principle similar to that used by BauBau 1999, which is as follows:

Assumed that the sensors and actuators are continuously distributed and that each sensor directs an actuator installed directly beneath it at the same {x,y} location. The sensor detects the deviation of the free surface temperature from its conductive value. The actuator modifies the heated surface temperature according to the following rule BauBau 1999 :

where K is the scalar controller gain. Equation 8 can be rewritten more conveniently as

where

## 3. Linearized Problem

We study the linear stability of the basic state by seeking perturbed solutions for any quantity

where

subject to

andon the lower rigid boundary z = 0. The operator D = d/dz denotes the differentiation with respect to the vertical coordinate z. The variables w, T and f denote respectively the vertical variation of the z-velocity, temperature and the magnitude of the free surface deflection of the linear perturbation to the basic state with total wave number a in the horizontal x-y plane and complex growth rates.

## 4. Results and disussion

The effect of feedback control on the onset of Marangoni convection in a fluid layer with a magnetic field in the case of a deformable free surface

Figure 2 shows the numerically calculated Marangoni number, M as a function of the wavenumber, a for different values of K in the case

When

## 5. Conclusion

The effect of the feedback control on the onset of steady Marangoni convection instabilities in a fluid layer with magnetic field has been studied. We have shown that the feedback control and magnetic field suppresses Marangoni convection We have also shown numerically that the effect of the controller gain and magnetic field is always to stabilize the layer in the case of a nondeforming surface. However, in the case of a deforming surface, the controller gain is effective depending on the parameter