17-03-2014, 08:54 PM
Abstract
The effect of a feedback control on the onset of oscillatory
B´enard-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.
1. Introduction
B´enard convection, sometimes referred as B´enard-
Marangoni convection was first observed by Henri B´enard
[1]. B´enard Convection involves a horizontal layer of fluid
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 B´enard cells. The instability of B´enard-
Marangoni convection is due to the combined effects of the
thermal buoyancy and surface tension.
The instability of the convection driven by buoyancy is
referred to as the Rayleigh-B´enard instability and studied in
[2] and [3].
The latter effect is due to the local variation of surface
tension. This type of convection instability is referred to as
the Marangoni instability and was first theoretically analysed
by Pearson [4]. On the Marangoni instability problem, the
effect of the surface deflection is 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
B´enard-Marangoni instability. Nield [6] first analyses the
B´enard-Marangoni instability problem. Davis and Homsy
[7] later study the effect of the surface deflection on the
combined B´enard-Marangoni problem. P´erez- Garcia and
Cameiro [8] have carried out a systematic study of the
linear stability of the B´enard-Marangoni convection with a
deformable free surface.
In the above B´enard-Marangoti instability analysis, the
convective instability is induced by the the temperature gradient
which is decreasing linearly with liquid layer height.
Sparrow et al. [9] and Roberts [10] analyse the thermal
instability in a horizontal fluid layer with the nonlinear
temperature distribution which is created by an internal
heat generation. Gasser and Kazimi [11] and Kaviany [12]
investigate the effect of the internal heat generation on the
onset of convection in a porous medium.
For delaying the onset of convection by the use of linear
and nonlinear control strategies as described in [13], [14],
[15], [16]. Bau [15] extended the studies of Pearson [4]
and Takashima [17], [18] by including a feedback control
strategy effecting small perturbations in boundary data to
suppress the onset of Marangoni convection. Arifin et al.
[16] applied feedback control strategy to Marangoni instability
in a fluid layer with a free-slip bottom. Then, Siri and
Hashim[19] extend the study of Arifin et al. [16] for the
rotation case. Recently, Hashim and Siri [20] and Siri and
Hashim [21] applied feedback control strategy to Marangoni
instability in a rotating fluid layer. Bau’s [15] control strategy
has recently been applied by Hashim and Awang Kechil
[22] to delay the onset of Marangoni convection in variable
viscosity fluids.
In this work, we use the 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 concerned
with the effect of the feedback control on the B´enard-
Marangoni instability of a horizontal liquid layer with a
deformable upper free surface. In this study, the effects of
the thermal buoyancy, the surface tension of a deformable
upper free surface and its thermal conductivity on the onset
of the B´enard- Marangoni instability are considered.
The effect of a feedback control on the onset of oscillatory
B´enard-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.
1. Introduction
B´enard convection, sometimes referred as B´enard-
Marangoni convection was first observed by Henri B´enard
[1]. B´enard Convection involves a horizontal layer of fluid
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 B´enard cells. The instability of B´enard-
Marangoni convection is due to the combined effects of the
thermal buoyancy and surface tension.
The instability of the convection driven by buoyancy is
referred to as the Rayleigh-B´enard instability and studied in
[2] and [3].
The latter effect is due to the local variation of surface
tension. This type of convection instability is referred to as
the Marangoni instability and was first theoretically analysed
by Pearson [4]. On the Marangoni instability problem, the
effect of the surface deflection is 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
B´enard-Marangoni instability. Nield [6] first analyses the
B´enard-Marangoni instability problem. Davis and Homsy
[7] later study the effect of the surface deflection on the
combined B´enard-Marangoni problem. P´erez- Garcia and
Cameiro [8] have carried out a systematic study of the
linear stability of the B´enard-Marangoni convection with a
deformable free surface.
In the above B´enard-Marangoti instability analysis, the
convective instability is induced by the the temperature gradient
which is decreasing linearly with liquid layer height.
Sparrow et al. [9] and Roberts [10] analyse the thermal
instability in a horizontal fluid layer with the nonlinear
temperature distribution which is created by an internal
heat generation. Gasser and Kazimi [11] and Kaviany [12]
investigate the effect of the internal heat generation on the
onset of convection in a porous medium.
For delaying the onset of convection by the use of linear
and nonlinear control strategies as described in [13], [14],
[15], [16]. Bau [15] extended the studies of Pearson [4]
and Takashima [17], [18] by including a feedback control
strategy effecting small perturbations in boundary data to
suppress the onset of Marangoni convection. Arifin et al.
[16] applied feedback control strategy to Marangoni instability
in a fluid layer with a free-slip bottom. Then, Siri and
Hashim[19] extend the study of Arifin et al. [16] for the
rotation case. Recently, Hashim and Siri [20] and Siri and
Hashim [21] applied feedback control strategy to Marangoni
instability in a rotating fluid layer. Bau’s [15] control strategy
has recently been applied by Hashim and Awang Kechil
[22] to delay the onset of Marangoni convection in variable
viscosity fluids.
In this work, we use the 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 concerned
with the effect of the feedback control on the B´enard-
Marangoni instability of a horizontal liquid layer with a
deformable upper free surface. In this study, the effects of
the thermal buoyancy, the surface tension of a deformable
upper free surface and its thermal conductivity on the onset
of the B´enard- Marangoni instability are considered.