The effects of boundary imperfections on convection in a saturated porous layer: near-resonant wavelength excitation

1989 ◽  
Vol 199 ◽  
pp. 133-154 ◽  
Author(s):  
D. A. S. Rees ◽  
D. S. Riley
Keyword(s):  
Porous Layer ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Resonant Forcing ◽  
Saturated Porous Layer ◽  
The Stability ◽  
Spatially Varying ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Complementary Study

Weakly nonlinear theory is used to study the porous-medium analogue of the classical Rayleigh-Bénard problem, i.e. Lapwood convection in a saturated porous layer heated from below. Two particular aspects of the problem are focused upon: (i) the effect of thermal imperfections on the stability characteristics of steady rolls near onset; and (ii) the evolution of unstable rolls.For Rayleigh-Bénard convection it is well known (see Busse and co-workers 1974, 1979, 1986) that the stability of steady two-dimensional rolls near onset is limited by the presence of cross-roll, zigzag and sideband disturbances; this is shown to be true also in Lapwood convection. We further determine the modifications to the stability boundaries when small-amplitude imperfections in the boundary temperatures are present. In practice imperfections would usually consist of broadband thermal noise, but it is the Fourier component with wavenumber close to the critical wavenumber for the perfect problem (i.e. in the absence of imperfections) which, when present, has the greatest effect due to resonant forcing. This particular case is the sole concern of the present paper; other resonances are considered in a complementary study (Rees & Riley 1989).For the case when the modulations on the upper and lower boundaries are in phase, asymptotic analysis and a spectral method are used to determine the stability of roll solutions and to calculate the evolution of the unstable flows. It is shown that steady rolls with spatially deformed axes or spatially varying wavenumbers evolve. The evolution of the flow that is unstable to sideband disturbances is also calculated when the modulations are π out of phase. Again rolls with a spatially varying wavenumber result.

The effects of boundary imperfections on convection in a saturated porous layer: non-resonant wavelength excitation

1989 ◽  
Vol 421 (1861) ◽  
pp. 303-339 ◽  
Keyword(s):  
Porous Layer ◽  
Three Dimensional ◽  
Resonant Interaction ◽  
Mean Values ◽  
Weakly Nonlinear ◽  
Special Cases ◽  
Wide Range ◽  
Saturated Porous Layer ◽  
Convection Patterns ◽  
Rayleigh Benard

The onset of Rayleigh-Bénard convection in a horizontally unbounded saturated porous layer is considered when the temperatures of both horizontal boundaries vary periodically in one direction about their respective mean values. Attention is focused on small-amplitude thermal modulations with a wavenumber not close to the critical value for the perfect layer. A stability analysis of weakly nonlinear convection is performed and the effects of different wavenumbers and symmetries of the thermal modulations are deduced systematically. It is shown that there are many special cases to be considered and that the convection patterns depend crucially on the particular configuration. Intuitively it might be expected that one-dimensional thermal modulation would always stimulate a two-dimensional motion. Surprisingly, however, for a wide range of modulation wavenumber a three-dimensional motion with a rectangular planform results from a resonant interaction between a pair of oblique rolls and the boundary forcing.

Rayleigh–Bénard convection and pattern formation in magnetohydrodynamics

1998 ◽  
Vol 60 (3) ◽  
pp. 529-539 ◽  
Author(s):  
RENU BAJAJ ◽  
S. K. MALIK
Keyword(s):  
Magnetic Field ◽  
Steady State ◽  
Thermal Instability ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid ◽  
Steady State Bifurcation ◽  
The Stability ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
The Magnetic Field

A nonlinear thermal instability in a layer of electrically conducting fluid in the presence of a magnetic field is discussed. Steady-state bifurcation results in the formation of patterns: rolls, squares and hexagons. The stability of various patterns is also investigated. It is found that in the absence of a magnetic field only rolls are stable, but when the magnetic field strength exceeds a certain finite value, squares and hexagons also become stable.

scholarly journals Air-Aided Shear on a Thin Film Subjected to a Transverse Magnetic Field of Constant Strength: Stability and Dynamics

2013 ◽  
Vol 2013 ◽  
pp. 1-24 ◽  
Author(s):  
Mohammed Rizwan Sadiq Iqbal
Keyword(s):  
Magnetic Field ◽  
Inertial Force ◽  
Transverse Magnetic ◽  
Nonlinear Analyses ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Constant Strength ◽  
Long Time ◽  
The Stability ◽  
The One

The effect of air shear on the hydromagnetic instability is studied through (i) linear stability, (ii) weakly nonlinear theory, (iii) sideband stability of the filtered wave, and (iv) numerical integration of the nonlinear equation. Additionally, a discussion on the equilibria of a truncated bimodal dynamical system is performed. While the linear and weakly nonlinear analyses demonstrate the stabilizing (destabilizing) tendency of the uphill (downhill) shear, the numerics confirm the stability predictions. They show that (a) the downhill shear destabilizes the flow, (b) the time taken for the amplitudes corresponding to the uphill shear to be dominated by the one corresponding to the zero shear increases with magnetic fields strength, and (c) among the uphill shear-induced flows, it takes a long time for the wave amplitude corresponding to small shear values to become smaller than the one corresponding to large shear values when the magnetic field intensity increases. Simulations show that the streamwise and transverse velocities increase when the downhill shear acts in favor of inertial force to destabilize the flow mechanism. However, the uphill shear acts oppositely. It supports the hydrostatic pressure and magnetic field in enhancing films stability. Consequently, reduced constant flow rates and uniform velocities are observed.

Feedback control of weakly nonlinear Rayleigh–Bénard–Marangoni convection

2001 ◽  
Vol 440 ◽  
pp. 27-47 ◽  
Author(s):  
A. C. OR ◽  
R. E. KELLY
Keyword(s):  
Feedback Control ◽  
Nonlinear Flow ◽  
Control Action ◽  
Control Analysis ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Wide Range ◽  
Nonlinear Control Processes ◽  
Gas Layer ◽  
Rayleigh Benard

We study the effect of proportional feedback control on the onset and development of finite-wavelength Rayleigh–Bénard–Marangoni (RBM) convection using weakly nonlinear theory as applied to Nield's model, which includes both thermocapillarity and buoyancy but ignores deformation of the free surface. A two-layer model configuration is used, which has a purely conducting gas layer on top of the liquid. In the feedback control analysis, a control action in the form of temperature or heat flux is considered. Both measurement and control action are assumed to be continuous in space and time. Besides demonstrating that stabilization of the basic state can be achieved on a linear basis, the results also indicate that a wide range of weakly nonlinear flow properties can also be altered by the linear and nonlinear control processes used here. These include changing the nature of hexagonal convection and the amount of subcritical hysteresis associated with subcritical bifurcation.

Sign in / Sign up

Export Citation Format

Share Document