Abstract
The paper presents the results of an investigation of the effect of imperfectly insulated sidewalls on natural convection in porous media at slightly supercritical Rayleigh numbers. An analytical solution for a rectangular domain with imperfectly insulated sidewalls and heated from below, was obtained through the weak nonlinear theory. The solution enables the determination of the amplitude of the convection and the direction of the flow. The amplitude results from an ordinary nonhomogeneous differential equation, with a forcing term representing the heat leakage through the lateral walls. The steady state amplitude solution shows that the transition through the critical Rayleigh number is smooth, differing from the case of perfectly insulated sidewalls where a bifurcation usually appears at the critical Rayleigh number. As a result, within a certain range of slightly supercritical Rayleigh number values, the amplitude and the direction of the convection currents are uniquely determined by the heat leakage through the lateral walls and they are independent of the initial conditions. A subcritical convection occurs as a result of the imperfectly insulated sidewalls, enabling the smooth transition through the critical Rayleigh value. A three-branch bifurcation develops at a higher Rayleigh number. A stability analysis of the solutions, corresponding to these branches, shows that the amplitudes which correspond to the two highest values are stable, while the third is unstable.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 215-233 |
| Number of pages | 19 |
| Journal | Acta Mechanica |
| Volume | 91 |
| Issue number | 3-4 |
| DOIs | |
| State | Published - Sep 1992 |
| Externally published | Yes |
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ASJC Scopus subject areas
- Mechanics of Materials
- Computational Mechanics
Cite this
The effect of imperfectly insulated sidewalls on natural convection in porous media. / Vadasz, Peter; Braester, C.
In: Acta Mechanica, Vol. 91, No. 3-4, 09.1992, p. 215-233.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - The effect of imperfectly insulated sidewalls on natural convection in porous media
AU - Vadasz, Peter
AU - Braester, C.
PY - 1992/9
Y1 - 1992/9
N2 - The paper presents the results of an investigation of the effect of imperfectly insulated sidewalls on natural convection in porous media at slightly supercritical Rayleigh numbers. An analytical solution for a rectangular domain with imperfectly insulated sidewalls and heated from below, was obtained through the weak nonlinear theory. The solution enables the determination of the amplitude of the convection and the direction of the flow. The amplitude results from an ordinary nonhomogeneous differential equation, with a forcing term representing the heat leakage through the lateral walls. The steady state amplitude solution shows that the transition through the critical Rayleigh number is smooth, differing from the case of perfectly insulated sidewalls where a bifurcation usually appears at the critical Rayleigh number. As a result, within a certain range of slightly supercritical Rayleigh number values, the amplitude and the direction of the convection currents are uniquely determined by the heat leakage through the lateral walls and they are independent of the initial conditions. A subcritical convection occurs as a result of the imperfectly insulated sidewalls, enabling the smooth transition through the critical Rayleigh value. A three-branch bifurcation develops at a higher Rayleigh number. A stability analysis of the solutions, corresponding to these branches, shows that the amplitudes which correspond to the two highest values are stable, while the third is unstable.
AB - The paper presents the results of an investigation of the effect of imperfectly insulated sidewalls on natural convection in porous media at slightly supercritical Rayleigh numbers. An analytical solution for a rectangular domain with imperfectly insulated sidewalls and heated from below, was obtained through the weak nonlinear theory. The solution enables the determination of the amplitude of the convection and the direction of the flow. The amplitude results from an ordinary nonhomogeneous differential equation, with a forcing term representing the heat leakage through the lateral walls. The steady state amplitude solution shows that the transition through the critical Rayleigh number is smooth, differing from the case of perfectly insulated sidewalls where a bifurcation usually appears at the critical Rayleigh number. As a result, within a certain range of slightly supercritical Rayleigh number values, the amplitude and the direction of the convection currents are uniquely determined by the heat leakage through the lateral walls and they are independent of the initial conditions. A subcritical convection occurs as a result of the imperfectly insulated sidewalls, enabling the smooth transition through the critical Rayleigh value. A three-branch bifurcation develops at a higher Rayleigh number. A stability analysis of the solutions, corresponding to these branches, shows that the amplitudes which correspond to the two highest values are stable, while the third is unstable.
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U2 - 10.1007/BF01194111
DO - 10.1007/BF01194111
M3 - Article
VL - 91
SP - 215
EP - 233
JO - Acta Mechanica
JF - Acta Mechanica
SN - 0001-5970
IS - 3-4
ER -