Dynamic bifurcations and pattern formation in melting-boundary convection

2011 ◽  
Vol 686 ◽  
pp. 77-108 ◽  
Author(s):  
G. M. Vasil ◽  
M. R. E. Proctor
Keyword(s):  
Pattern Formation ◽  
General Feature ◽  
Fluid Layer ◽  
Landau Equation ◽  
Ginzburg Landau ◽  
Nonlinear Convection ◽  
Weakly Nonlinear ◽  
Dynamic Bifurcation ◽  
Dynamic Bifurcations ◽  
New System

AbstractWe consider weakly nonlinear convection in a fluid layer with a melting top boundary. This leads us to derive a new set of non-autonomous envelope equations as a dynamic generalization to the well-known Ginzburg–Landau equation. However, this new system possesses a number of interesting properties not found in systems close to a traditional dynamic bifurcation, because it involves the interaction of two destabilizing mechanisms. We investigate the system both analytically and numerically; specifically, we find the robust ‘locking in’ of spatially complex patterns, and show this is a general feature of systems of this nature.

Weakly Nonlinear Oscillatory Convection in a Rotating Fluid Layer Under Temperature Modulation

2016 ◽  
Vol 138 (5) ◽  
Author(s):  
Palle Kiran ◽  
B. S. Bhadauria
Keyword(s):  
Heat Transport ◽  
Fluid Layer ◽  
Landau Equation ◽  
Complex Form ◽  
Oscillatory Mode ◽  
Temperature Modulation ◽  
Rotating Fluid ◽  
Nonlinear Stability Analysis ◽  
Ginzburg Landau ◽  
Weakly Nonlinear

A study of thermal instability driven by buoyancy force is carried out in an initially quiescent infinitely extended horizontal rotating fluid layer. The temperature at the boundaries has been taken to be time-periodic, governed by the sinusoidal function. A weakly nonlinear stability analysis has been performed for the oscillatory mode of convection, and heat transport in terms of the Nusselt number, which is governed by the complex form of Ginzburg–Landau equation (CGLE), is calculated. The influence of external controlling parameters such as amplitude and frequency of modulation on heat transfer has been investigated. The dual effect of rotation on the system for the oscillatory mode of convection is found either to stabilize or destabilize the system. The study establishes that heat transport can be controlled effectively by a mechanism that is external to the system. Further, the bifurcation analysis also presented and established that CGLE possesses the supercritical bifurcation.

scholarly journals The Complex Ginzburg Landau Model for an Oscillatory Convection in a Rotating Fluid Layer

2020 ◽  
Vol 25 (1) ◽  
pp. 75-91
Author(s):  
S.H. Manjula ◽  
P. Kiran ◽  
P. Raj Reddy ◽  
B.S. Bhadauria
Keyword(s):  
Heat Transfer ◽  
Thermal Instability ◽  
Fluid Layer ◽  
Landau Equation ◽  
Finite Amplitude ◽  
Oscillatory Mode ◽  
Stationary Mode ◽  
Ginzburg Landau ◽  
Weakly Nonlinear ◽  
Speed Modulation

AbstractA weakly nonlinear thermal instability is investigated under rotation speed modulation. Using the perturbation analysis, a nonlinear physical model is simplified to determine the convective amplitude for oscillatory mode. A non-autonomous complex Ginzburg-Landau equation for the finite amplitude of convection is derived based on a small perturbed parameter. The effect of rotation is found either to stabilize or destabilize the system. The Nusselt number is obtained numerically to present the results of heat transfer. It is found that modulation has a significant effect on heat transport for lower values of ωf while no effect for higher values. It is also found that modulation can be used alternately to control the heat transfer in the system. Further, oscillatory mode enhances heat transfer rather than stationary mode.

scholarly journals Study of Weakly nonlinear Mass transport in Newtonian Fluid with Applied Magnetic Field under Concentration/Gravity modulation

2019 ◽  
Vol 8 (1) ◽  
pp. 513-522 ◽  
Author(s):  
Om Prakash Keshri ◽  
Vinod K. Gupta ◽  
Anand Kumar
Keyword(s):  
Mass Transport ◽  
Fluid Layer ◽  
Landau Equation ◽  
Periodic Part ◽  
Gravity Modulation ◽  
Nonlinear Stability Analysis ◽  
Ginzburg Landau ◽  
Weakly Nonlinear ◽  
Mass Transfer Effect ◽  
Time Periodic

Abstract In the present paper, a weakly nonlinear stability analysis is used to analyze the effect of time-periodic concentration/gravity modulation on mass transport. We have considered an infinite horizontal fluid layer with constant appliedmagnetic flux salted from above, subjected to an imposed time-periodic boundary concentration (ITBC) or gravity modulation (ITGM). In the case of ITBC, the concentration gradient between the plates of the fluid layer consists of a steady part and a time-dependent oscillatory part. The concentration of both walls is modulated. In the case of ITGM, the gravity fleld consists of two parts: a constant part and an externally imposed time periodic part, which can be realized by oscillating the fluid layer. We have expanded the infinitesimal disturbances in terms of power series of an amplitude of modulation, which is assumed to be small. Ginzburg-Landau equation is derived for dinding the rate of mass transfer. Effect of various parameters on the mass transport is also discussed. It is found that the mass transport can be controlled by suitably adjusting the frequency and amplitude of modulation.

PATTERNS, DEFECTS, AND EVOLUTION OF BÉNARD–MARANGONI CELLS

1996 ◽  
Vol 06 (09) ◽  
pp. 1665-1671 ◽  
Author(s):  
J. BRAGARD ◽  
J. PONTES ◽  
M.G. VELARDE
Keyword(s):  
Fluid Layer ◽  
Ambient Air ◽  
Finite Geometry ◽  
Amplitude Equations ◽  
Ginzburg Landau ◽  
Weakly Nonlinear ◽  
Numerical Exploration ◽  
Ginzburg Landau Equations ◽  
Rigid Plane ◽  
Selection Of

We consider a thin fluid layer of infinite horizontal extent, confined below by a rigid plane and open above to the ambient air, with surface tension linearly depending on the temperature. The fluid is heated from below. First we obtain the weakly nonlinear amplitude equations in specific spatial directions. The procedure yields a set of generalized Ginzburg–Landau equations. Then we proceed to the numerical exploration of the solutions of these equations in finite geometry, hence to the selection of cells as a result of competition between the possible different modes of convection.

scholarly journals Weakly Nonlinear Magnetic Convection in a Nonuniformly Rotating Electrically Conductive Medium Under the Action of Modulation of External Fields

2020 ◽  
Keyword(s):  
Magnetic Field ◽  
Angular Velocity ◽  
Landau Equation ◽  
Temperature Modulation ◽  
External Fields ◽  
Ginzburg Landau ◽  
Electrically Conductive ◽  
Weakly Nonlinear ◽  
Conductive Fluid ◽  
Ginzburg Landau Equations

In this paper we studied the weakly nonlinear stage of stationary convective instability in a nonuniformly rotating layer of an electrically conductive fluid in an axial uniform magnetic field under the influence of: a) temperature modulation of the layer boundaries; b) gravitational modulation; c) modulation of the magnetic field; d) modulation of the angular velocity of rotation. As a result of applying the method of perturbation theory for the small parameter of supercriticality of the stationary Rayleigh number nonlinear non-autonomous Ginzburg-Landau equations for the above types of modulation were obtaned. By utilizing the solution of the Ginzburg-Landau equation, we determined the dynamics of unsteady heat transfer for various types of modulation of external fields and for different profiles of the angular velocity of the rotation of electrically conductive fluid.

PATTERN FORMATION IN SMALL ARRAYS OF LOCALLY COUPLED OSCILLATORS

2005 ◽  
Vol 15 (07) ◽  
pp. 2283-2293 ◽  
Author(s):  
REBECCA ELLISON ◽  
VIRGINIA GARDNER ◽  
JOEL LEPAK ◽  
MEGHAN O'MALLEY ◽  
JOSEPH PAULLET ◽  
...  
Keyword(s):  
Pattern Formation ◽  
Landau Equation ◽  
Current Model ◽  
Relaxation Oscillation ◽  
Chaotic Regime ◽  
Phase Oscillators ◽  
Ginzburg Landau ◽  
Brusselator Model ◽  
Small Domain ◽  
Oscillatory Media

We investigate small two-dimensional arrays of locally coupled phase oscillators which are shown to exhibit a surprising variety of stable structures which include: single spiral waves, spiral pairs and spirals with secondary periodic core motion. This periodic core motion is not the core meander familiar to many models of active media, but is in fact induced by the boundary of the small domain. Such boundary motion was investigated by Sepulchre and Babloyantz [1993] for the complex Ginzburg–Landau equation and for the Brusselator model in a relaxation oscillation parameter regime. The current model confirms the findings in [Sepulchre & Babloyantz, 1993] and sheds new light on the origin of such motion. The model also exhibits other patterns, as well as a chaotic regime. We discuss the transition between patterns as the form of the coupling is changed as well as implications for pattern formation in general oscillatory media.

Hydromagnetic Stability of a Thin Viscoelastic Magnetic Fluid on Coating Flow Using Landau Equation

2016 ◽  
Vol 32 (5) ◽  
pp. 643-651 ◽  
Author(s):  
C.-K. Chen ◽  
M.-C. Lin
Keyword(s):  
Viscoelastic Fluid ◽  
Nonlinear Stability ◽  
Film Flow ◽  
Landau Equation ◽  
Ginzburg Landau ◽  
Viscoelastic Parameter ◽  
Weakly Nonlinear ◽  
Linear Mode ◽  
Coating Flow ◽  
The One

AbstractThis paper investigates the weakly nonlinear stability of a thin axisymmetric viscoelastic fluid with hydromagnetic effects on coating flow. The governing equation is resolved using long-wave perturbation method as part of an initial value problem for spatial periodic surface waves with the Walter's liquid B type fluid. The most unstable linear mode of a film flow is determined by Ginzburg-Landau equation (GLE). The coefficients of the GLE are calculated numerically from the solution of the corresponding stability problem on coating flow. The effect of a viscoelastic fluid under an applied magnetic field on the nonlinear stability mechanism is studied in terms of the rotation number, Ro, viscoelastic parameter, k, and the Hartmann constant, m. Modeling results indicate that the Ro, k and m parameters strongly affect the film flow. Enhancing the magnetic effects is found to stabilize the film flow when the viscoelastic parameter destabilizes the one in a thin viscoelastic fluid.

Throughflow and G-Jitter Effects on Oscillatory Convection in a Rotating Porous Medium

2020 ◽  
Vol 12 (6) ◽  
pp. 781-791
Author(s):  
S. H. Manjula ◽  
Palle Kiran ◽  
B. S. Bhadauria
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Heat Transport ◽  
Landau Equation ◽  
Nonlinear Stability Analysis ◽  
Periodic Mode ◽  
Ginzburg Landau ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Analysis ◽  
The Impact

The impact of vertical throughflow and g-jitter effect on rotating porous medium is investigated. A feeble nonlinear stability analysis associate to complex Ginzburg-Landau equation (CGLE) has been studied. This weakly nonlinear analysis performed for a periodic mode of convection and quantified heat transport in terms of the Nusselt number, which is governed by the non-autonomous advanced CGLE. Each idea, rotation and throughflow is used as an external mechanism to the system either to extend or decrease the heat transfer. The results of amplitude and frequency of modulation on heat transport are analyzed and portrayed graphically. Throughflow has dual impact on heat transfer either to increase or decrease heat transfer in the system. Particularly the outflow enhances and inflow diminishes the heat transfer. High centrifugal rates promote heat transfer and low centrifugal rates diminish heat transfer. The streamlines and isotherms area portrayed graphically, the results of rotation and throughflow on isotherms shows convective development.

scholarly journals ASYMPTOTIC COMPACTNESS OF STOCHASTIC COMPLEX GINZBURG–LANDAU EQUATION ON AN UNBOUNDED DOMAIN

2010 ◽  
Vol 10 (04) ◽  
pp. 613-636 ◽  
Author(s):  
DIRK BLÖMKER ◽  
YONGQIAN HAN
Keyword(s):  
Dynamical System ◽  
Pattern Formation ◽  
White Noise ◽  
Unbounded Domain ◽  
Landau Equation ◽  
Random Dynamical System ◽  
Ginzburg Landau ◽  
Type Complex ◽  
Regularity Properties ◽  
Formation Systems

The Ginzburg–Landau-type complex equations are simplified mathematical models for various pattern formation systems in mechanics, physics and chemistry. In this paper, we consider the complex Ginzburg–Landau (CGL) equations on the whole real line perturbed by an additive spacetime white noise. Our main result shows that it generates an asymptotically compact stochastic or random dynamical system. This is a crucial property for the existence of a stochastic attractor for such CGL equations. We rely on suitable spaces with weights, due to the regularity properties of spacetime white noise, which gives rise to solutions that are unbounded in space.

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