Stability of a Horizontal Porous Layer with Timewise Periodic Boundary Conditions

1979 ◽  
Vol 101 (2) ◽  
pp. 244-248 ◽  
Author(s):  
B. Chhuon ◽  
J. P. Caltagirone
Keyword(s):  
Porous Layer ◽  
Periodic Boundary ◽  
Lower Boundary ◽  
Periodic Boundary Conditions ◽  
Temperature Oscillation ◽  
Linear Stability Theory ◽  
Onset Of Convection ◽  
Thermal Signal ◽  
Periodic Temperature ◽  
The Stability

The stability of a horizontal porous layer bounded by two impermeable planes is investigated. A time dependent periodic temperature profile is imposed on the lower boundary while the upper plane is kept at constant temperature. Starting from the preconvective temperature distribution, and using the linear stability theory, a criterion for the onset of convection is defined as a function of the perturbation wavenumber and of the amplitude and frequency of the temperature oscillation. Experimental work with a setup allowing both the amplitude and the frequency of the thermal signal to vary is done. Finally, the equations are also solved numerically and the results are compared to the previous ones. A synthesis of all results is included.

Experimental investigation of convective stability in a superposed fluid and porous layer when heated from below

1989 ◽  
Vol 207 ◽  
pp. 311-321 ◽  
Author(s):  
Falin Chen ◽  
C. F. Chen
Keyword(s):  
Porous Layer ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Convective Stability ◽  
Onset Of Convection ◽  
Temperature Distributions ◽  
Critical Wavelength ◽  
Crystal Film ◽  
Flux Curve

Experiments have been carried out in a horizontal superposed fluid and porous layer contained in a test box 24 cm × 12 cm × 4 cm high. The porous layer consisted of 3 mm diameter glass beads, and the fluids used were water, 60% and 90% glycerin-water solutions, and 100% glycerin. The depth ratio ď, which is the ratio of the thickness of the fluid layer to that of the porous layer, varied from 0 to 1.0. Fluids of increasingly higher viscosity were used for cases with larger ď in order to keep the temperature difference across the tank within reasonable limits. The top and bottom walls were kept at different constant temperatures. Onset of convection was detected by a change of slope in the heat flux curve. The size of the convection cells was inferred from temperature measurements made with embedded thermocouples and from temperature distributions at the top of the layer by use of liquid crystal film. The experimental results showed (i) a precipitous decrease in the critical Rayleigh number as the depth of the fluid layer was increased from zero, and (ii) an eightfold decrease in the critical wavelength between ď = 0.1 and 0.2. Both of these results were predicted by the linear stability theory reported earlier (Chen & Chen 1988).

A solution of the heat conduction equation in the finite cylinder exposed to periodic boundary conditions: the case of steady oscillation and constant thermal properties

1992 ◽  
Vol 438 (1903) ◽  
pp. 319-329 ◽  
Keyword(s):  
Thermal Properties ◽  
Boundary Conditions ◽  
Heat Conduction Equation ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Finite Cylinder ◽  
Steady Oscillation ◽  
Diffusivity Measurements ◽  
Periodic Temperature ◽  
Reduced Coordinates

The problem of the temperature distribution in the finite cylinder with periodic boundary conditions over all the surfaces is treated analytically and numerically. The solution in the case of steady oscillation and constant thermal properties is presented in reduced coordinates and can be used to correct thermal diffusivity measurements by the Angstrom method or to study, for example, the influence of a destabilized controller which induces a periodic temperature of the sample during thermal treatment.

Effect of modulation on the onset of thermal convection in a viscoelastic fluid-saturated sparsely packed porous layer

1990 ◽  
Vol 68 (2) ◽  
pp. 214-221 ◽  
Author(s):  
N. Rudraiah ◽  
P. V. Radhadevi ◽  
P. N. Kaloni
Keyword(s):  
Porous Layer ◽  
Viscoelastic Fluid ◽  
Phase Modulation ◽  
Applied Field ◽  
Temperature Modulation ◽  
Elastic Parameters ◽  
Oldroyd Model ◽  
Maximum Range ◽  
Periodic Temperature ◽  
The Stability

The stability of a Boussinesq viscoelastic fluid-saturated horizontal porous layer, when the boundaries of the layer are subjected to periodic temperature modulation, is analyzed. The Darcy–Forchheimer–Brinkman–Oldroyd model is employed and only infinitesimal disturbances are considered. Three cases of the oscillating temperature field were examined: (a) symmetric, so that the wall temperatures are modulated in phase, (b) asymmetric, corresponding to out-of-phase modulation, and (c) only the bottom wall is modulated. Perturbation solution in powers of the amplitude of the applied field is obtained. The effect of the frequency of modulation on the stability is clearly shown. Possibilities of the occurrence of subcritical instabilities are also discussed. It is shown that an increase in the elastic parameters A1 and A2 has a stabilizing influence. An increase in the Prandtl number destabilizes the system for small values of the frequency but stabilizes the systems for large values of the frequency. It is shown that the system is most stable when the boundary temperatures are modulated out of phase. The maximum range of ε when subcritical instabilities exist is also determined.

Stability of Thermal Convection in a Vertical Porous Layer

1987 ◽  
Vol 109 (4) ◽  
pp. 889-893 ◽  
Author(s):  
L. P. Kwok ◽  
C. F. Chen
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Porous Layer ◽  
Thermal Convection ◽  
Variable Viscosity ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Lateral Heating ◽  
The Stability ◽  
Allowable Temperature

Experiments were carried out to study the stability of thermal convection generated in a vertical porous layer by lateral heating in a tall, narrow tank. The porous medium, consisting of glass beads, was saturated with distilled water. It was found that the flow became unstable at a critical ΔT of 29.2°C (critical Rayleigh number of 66.2). Linear stability analysis was applied to study the effects of the Brinkman term and of variable viscosity separately using a quadratic relationship between the density and temperature. It was found that with the Brinkman term, no instability could occur within the allowable temperature difference across the tank. With the effect of variable viscosity included, linear stability theory predicts a critical ΔT of 43.4°C (Rayleigh number of 98.3).

scholarly journals Rayleigh-Bénard convection in an elastico-viscous Walters’ (model B’) nanofluid layer

2015 ◽  
Vol 63 (1) ◽  
pp. 235-244 ◽  
Author(s):  
G.C. Rana ◽  
R. Chand
Keyword(s):  
Fluid Model ◽  
Normal Mode Analysis ◽  
Sufficient Conditions ◽  
Lewis Number ◽  
Horizontal Layer ◽  
Linear Stability Theory ◽  
Mode Analysis ◽  
Physical Parameters ◽  
Onset Of Convection ◽  
The Stability

Abstract In this study, the onset of convection in an elastico-viscous Walters’ (model B’) nanofluid horizontal layer heated from below is considered. The Walters’ (model B’) fluid model is employed to describe the rheological behavior of the nanofluid. By applying the linear stability theory and a normal mode analysis method, the dispersion relation has been derived. For the case of stationary convection, it is observed that the Walters’ (model B’) elastico-viscous nanofluid behaves like an ordinary Newtonian nanofluid. The effects of the various physical parameters of the system, namely, the concentration Rayleigh number, Prandtl number, capacity ratio, Lewis number and kinematics visco-elasticity coefficient on the stability of the system has been numerically investigated. In addition, sufficient conditions for the non-existence of oscillatory convection are also derived.

Stability regions for multi-parameter systems of periodic second order ordinary differential equations

1979 ◽  
Vol 84 (3-4) ◽  
pp. 249-257 ◽  
Author(s):  
Patrick J. Browne ◽  
B. D. Sleeman
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Parameter System ◽  
Stability Regions ◽  
The Stability ◽  
Image Position

SynopsisThis paper studies the stability regions associated with the multi-parameter systemwhere the functions qr(xr), ars(xr) are periodic and the system is subjected to periodic or semi-periodic boundary conditions.

BIFURCATION ANALYSIS OF THE 1D AND 2D GENERALIZED SWIFT–HOHENBERG EQUATION

2010 ◽  
Vol 20 (03) ◽  
pp. 619-643 ◽  
Author(s):  
HONGJUN GAO ◽  
QINGKUN XIAO
Keyword(s):  
Boundary Conditions ◽  
Bifurcation Analysis ◽  
Trivial Solution ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Spatial Dimension ◽  
Nontrivial Solutions ◽  
Asymptotic Expressions ◽  
Spatial Dimensions ◽  
The Stability

In this paper, bifurcation of the generalized Swift–Hohenberg equation is considered. We first study the bifurcation of the generalized Swift–Hohenberg equation in one spatial dimension with three kinds of boundary conditions. With the help of Liapunov–Schmidt reduction, the original equation is transformed to the reduced system, and then the bifurcation analysis is carried out. Secondly, bifurcation of the generalized Swift–Hohenberg equation in two spatial dimensions with periodic boundary conditions is also considered, using the perturbation method, asymptotic expressions of the nontrivial solutions bifurcated from the trivial solution are obtained. Moreover, the stability of the bifurcated solutions is discussed.

Stability of Liquid Films

2010 ◽  
Author(s):  
Aneet D. Narendranath ◽  
Jeramy Kimball ◽  
James C. Hermanson ◽  
Robert W. Kolkka ◽  
Jeffrey S. Allen
Keyword(s):  
Evolution Equation ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Liquid Films ◽  
Published Data ◽  
Structural Factors ◽  
Linear Evolution ◽  
Linear Evolution Equation ◽  
The Stability

Macroscopic thin liquid films are entities that are important in biophysics, physics, and engineering, as well as in natural settings. They can be composed of common liquids such as water or oil, rheologically complex materials such as polymers solutions or melts, or complex mixtures of phases or components. When the films are subjected to the action of various mechanical, thermal, or structural factors, they display interesting dynamic phenomena such as wave propagation, wave steepening, and development of chaotic responses. Such films can display rupture phenomena creating holes, spreading of fronts, and the development of fingers. The present work examines, through the solution of a onesided evolution equation as an initial value problem with periodic boundary conditions, the various mechanisms that affect the stability of liquid films. The numerical program employed to solve the non-linear evolution equation is validated by comparing the results produced with previously published data. The wavenumber associated with various destabilizing mechanisms is extracted. The effect of pinned boundary conditions versus periodic boundary conditions will be discussed.

On the Effect of Mechanical and Thermal Anisotropy on the Stability of Gravity Driven Convection in Rotating Porous Media

2005 ◽  
Author(s):  
Saneshan Govender ◽  
Peter Vadasz
Keyword(s):  
Rayleigh Number ◽  
Critical Rayleigh Number ◽  
Equation Model ◽  
Linear Stability Theory ◽  
Mechanical Anisotropy ◽  
Onset Of Convection ◽  
Thermal Anisotropy ◽  
Gravity Driven ◽  
The Stability ◽  
Rayleigh Benard

We investigate Rayleigh-Benard convection in a porous layer subjected to gravitational and Coriolis body forces, when the fluid and solid phases are not in local thermodynamic equilibrium. The Darcy model (extended to include Coriolis effects and anisotropic permeability) is used to describe the flow whilst the two-equation model is used for the energy equation (for the solid and fluid phases separately). The linear stability theory is used to evaluate the critical Rayleigh number for the onset of convection and the effect of both thermal and mechanical anisotropy on the critical Rayleigh number is discussed.

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