Effects of rotation and feedback control on Bénard–Marangoni convection

Abstract

The effect of a feedback control on the onset of steady and oscillatory Bénard–Marangoni instability in a rotating horizontal fluid layer is considered theoretically using linear stability theory. It is demonstrated that generally the critical Marangoni number for transition from the no-motion (conduction) to the motion state can be drastically increased by the combined effects of feedback control and rotation. The role of the controller gain parameter on the $\mathit{Pr}-\mathit{Ta}$ and $\mathit{Pr}-R/R_c$ parameter spaces, dividing stability domains into which either steady or oscillatory convection is preferred, is determined.

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.