International Journal of Heat and Mass Transfer
Effects of rotation and feedback control on Bénard–Marangoni convection
Introduction
Bénard convection, sometimes referred to as Bénard–Marangoni convection, was first observed by Henri Bénard [1]. Bénard convection occurs when a horizontal layer of fluid is heated uniformly from below, which causes the heated fluid to rise because of local density differences. The warm fluid near the bottom is replaced by cooler fluid near the top. If the thickness of the fluid is small in comparison to the expanse of its surface, the fluid will tend to circulate in a series of cells known as Bénard cells. The instability of Bénard–Marangoni convection is due to the combined effects of thermal buoyancy and surface tension.
The instability of the convection driven by buoyancy is referred to as Rayleigh-Bénard instability. It was studied by Chandrasekhar [2] and Drazin and Reid [3]. Another effect is due to local variation in surface tension. This type of convective instability is referred to as Marangoni instability and was first theoretically analysed by Pearson [4]. The effect of the surface deflection on Marangoni instability was later considered by Scriven and Sternling [5]. As these two kinds of instability take place at the same time, the instability mechanism is known as the Bénard–Marangoni instability. Nield [6] first analysed the Bénard–Marangoni instability problem. Davis and Homsy [7] later studied the effect of surface deflection on the combined Bénard–Marangoni effect. Stability analysis of Bénard–Marangoni convection of a low Prandtl number fluid in an open vertical cylinder was studied by Xu et al. [8]. Medale and Cerisier [9] investigate numerical simulation of Bénard–Marangoni convection of fluid layer heated from below in small aspect ratio container. A numerical study of the relative importance of Marangoni effects under microgravity conditions is presented by Giangi et al. [10]. A top free liquid layer is heated from the bottom in a two-dimensional rectangular container with insulated side walls was studied by [11].
Pérez-Garcia and Carneiro [12] have carried out a systematic study of the linear stability of Bénard–Marangoni convection with a deformable free surface. In their study, convective instability was induced by the temperature gradient which decreased linearly with liquid layer height. Sparrow et al. [13] and Roberts [14] analysed thermal instability in a horizontal fluid layer with the nonlinear temperature distribution which is created by internal heat generation. Boeck and Thess [15] studied Bénard–Marangoni convection at low Prandtl number with periodic boundary condition in both horizontal directions and either a free-slip or no-slip bottom wall of the two-dimensional case. Gasser and Kazimi [16] and Kaviany [17] investigated the effect of the internal heat generation on the onset of convection in a porous medium. Very recently, Idris et al. [18] studied the effect of a cubic temperature profile on the onset of steady Bénard–Marangoni convection in a micropolar fluid.
The ability to control complex convective flow patterns is important in both technology and fundamental science. In many technological processes, the naturally occurring flow patterns may not be the optimal ones. By controlling the flow, one may be able to optimize the process, improve product quality, and achieve significant savings. The ability to stabilise otherwise non-stable states may also assist one in gaining deeper insights into the dynamics of flows. Delaying the onset of convection by the use of linear and nonlinear control strategies was described by Tang and Bau [19], [20] and Bau [21]. Bau [21] extended the studies of Pearson [4] and Takashima [22], [23] by including a feedback control strategy effecting small perturbations in boundary data to suppress the onset of Marangoni convection. Or et al. [24] employed a nonlinear feedback control strategy to delay the onset and eliminate the subcritical long-wavelength instability of Marangoni–Bénard convection. Or and Kelly [25] showed that the weakly nonlinear flow properties in the Rayleigh–Bénard–Marangoni problem can be altered by linear and nonlinear proportional feedback control processes and the stabilization of the basic state can be achieved. Remillieux et al. [26] delineated the mechanism that lead to oscillatory Rayleigh-Bénard convection in the presence of large controller gains and the application of derivative controller to suppress oscillatory convection. Recently, Hashim and Siri [27] and Siri and Hashim [28], [29] applied Bau’s [21] feedback control strategy to Marangoni instability in a rotating fluid layer. Bau’s [21] control strategy has also been applied by Hashim and Awang-Kechil [30] to delay the onset of Marangoni convection in variable viscosity fluids.
In this work, we use classical linear stability analysis to obtain the thresholds and codimension-2 points for the onset of steady and oscillatory convection in a rotating fluid layer in the presence of a feedback control strategy. We are concerned with the effect of the feedback control on the Bénard–Marangoni instability of a horizontal liquid layer with a non-deformable upper free surface.
Section snippets
Mathematical formulation
In this study, the stability of a horizontal layer of quiescent fluid of thickness d which is unbounded in the horizontal x- and y-directions (see Fig. 1). The layer is kept rotating uniformly around the vertical z-axis with a constant angular velocity . The surface tension takes the form where and are reference values, and is the rate of change of surface tension with the temperature. The upper free surface is assumed non-deformable. In the reference state, the fluid
The linearised problem
Following the classical lines of linear stability theory as presented in [2], the linearised and dimensionless governing equations can be written asThe equations have been written in dimensionless form using , and as the scales for distance, time and temperature, respectively, where is the kinematic viscosity, is the thermal diffusivity and is the temperature difference between the top and bottom surfaces.
Solution approach
Since the solution method is standard, we only give a brief description of the solution approach in this section. Following a similar procedure as employed in Hashim and Sarma [32], combining Eqs. (13), (14), (15), then gives a single linear eight order ordinary differential equation for . This eight order ordinary differential equation together with the boundary conditions (16), (17), (18), (19), (20), (21) can be turned into the eigenvalue problem of the formfrom
Results and discussion
In this work, we studied the effect of on the onset of steady and oscillatory Bénard–Marangoni convection in a rotating fluid layer. Also, in this study we describe the results for the case of a non-deformable upper surface and an insulated upper surface, . The most relevant parameters of the current problem are M, and .
To verify the accuracy of our numerical results, test computations were performed for steady Bénard–Marangoni convection in the case of no feedback control
Conclusions
Feedback control strategies to alter the flow patterns in Benard–Marangoni convection in a fluid layer heated from below and cooled from above were studied theoretically. It was demonstrated that the transition from the no-motion state to time-dependent convection can be significantly postponed. We have demonstrated that feedback control can be effectively used to stabilize the no-motion state of a fluid layer heated from below. More specifically, with the use of controller, one can increase
Acknowledgements
Authors gratefully acknowledge the financial support, received from the University of Malaya Grant FS328/2008A, the UKM GUPGrant UKM-GUP-BTT-07-25-173and the Ministry of Higher Education, Malaysia.
References (34)
- et al.
On effect of non-uniform basic temperature gradient on Bénard–Marangoni convection in micropolar fluid
Int. Commun. Heat Mass Transfer
(2009) Control of Marangoni–Bénard convection
Int. J. Heat Mass Transfer
(1999)- et al.
Control of oscillatory Marangoni convection in a rotating fluid layer
Int. Commun. Heat Mass Transfer
(2008) - et al.
On oscillatory Marangoni convection in a rotating fluid layer subject to a uniform heat flux from below
Int. Commun. Heat Mass Transfer
(2007) - et al.
On oscillatory Marangoni convection in rotating fluid layer with flat free surface subject to uniform heat flux from below
Int. J. Heat Mass Transfer
(2007) Les tourbillons cellulaires dans une napple liquide
Revue Générale des Sciences Pures et Appliqués
(1900)Hydrodynamic and Hydromagnetic Stability
(1961)- et al.
Hydrodynamic Stability
(1961) On convection cells induced by surface tension
J. Fluid Mech.
(1958)- et al.
On cellular convection driven by surface-tension gradients: effects of mean surface tension and surface viscosity
J. Fluid Mech.
(1964)
Surface tension and buoyancy effect in cellular convection
J. Fluid Mech.
Energy stability theory for free-surface problem: buoyancy-thermocapillary layers
J. Fluid Mech.
Rayleigh–Bénard–Marangoni instabilities of low-Prandtl-number fluid in a vertical cylinder with lateral heating
Numer. Heat Transferm. A: Appl.
Numerical simulation of Bénard–Marangoni convection in small aspect ratio containers
Numer. Heat Transfer A: Appl.
A numerical study of solidification in the presence of a free surface under microgravity conditions
Numer. Heat Transfer A: Appl.
Thermocapillary-driven flow past the Marangoni instability
Numer Heat Transfer A: Appl.
Linear stability analysis of Bénard–Marangoni convection in fluids with a deformable free surface
Phys. Fluids A
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