# The Helmholtz equation for convection in two-dimensional porous cavities with conducting boundaries

D Andres Rees, Peder A Tyvand

Research output: Contribution to journalArticle

22 Citations (Scopus)

### Abstract

It is well-known that every two-dimensional porous cavity with a conducting and impermeable boundary is degenerate, as it has two different eigensolutions at the onset of convection. In this paper it is demonstrated that the eigenvalue problem obtained from a linear stability analysis may be reduced to a second-order problem governed by the Helmholtz equation, after separating out a Fourier component. This separated Fourier component implies a constant wavelength of disturbance at the onset of convection, although the phase remains arbitrary. The Helmholtz equation governs the critical Rayleigh number, and makes it independent of the orientation of the porous cavity. Finite-difference solutions of the eigenvalue problem for the onset of convection are presented for various geometries. Comparisons are made with the known solutions for a rectangle and a circle, and analytical solutions of the Helmholtz equation are given for many different domains. © 2004 Kluwer Academic Publishers.
Original language English 181-193 13 Journal of Engineering Mathematics 49 2 https://doi.org/10.1023/B:ENGI.0000017494.18537.df Published - 2004

### Fingerprint

Helmholtz equation
Helmholtz Equation
Convection
Cavity
Eigenvalue Problem
Linear stability analysis
Linear Stability Analysis
Rayleigh number
Rectangle
Finite Difference
Analytical Solution
Circle
Disturbance
Wavelength
Imply
Geometry
Arbitrary

### Keywords

• Heat conduction
• Heat convection
• Cavitation
• Porosity
• Problem solving
• Boundary conditions
• Mechanical permeability

### Cite this

In: Journal of Engineering Mathematics, Vol. 49, No. 2, 2004, p. 181-193.

Research output: Contribution to journalArticle

@article{c8a8cca9f14c4a78a648acd561412f50,
title = "The Helmholtz equation for convection in two-dimensional porous cavities with conducting boundaries",
abstract = "It is well-known that every two-dimensional porous cavity with a conducting and impermeable boundary is degenerate, as it has two different eigensolutions at the onset of convection. In this paper it is demonstrated that the eigenvalue problem obtained from a linear stability analysis may be reduced to a second-order problem governed by the Helmholtz equation, after separating out a Fourier component. This separated Fourier component implies a constant wavelength of disturbance at the onset of convection, although the phase remains arbitrary. The Helmholtz equation governs the critical Rayleigh number, and makes it independent of the orientation of the porous cavity. Finite-difference solutions of the eigenvalue problem for the onset of convection are presented for various geometries. Comparisons are made with the known solutions for a rectangle and a circle, and analytical solutions of the Helmholtz equation are given for many different domains. {\circledC} 2004 Kluwer Academic Publishers.",
keywords = "Heat conduction, Heat convection, Cavitation, Porosity, Problem solving, Boundary conditions, Mechanical permeability",
author = "Rees, {D Andres} and Tyvand, {Peder A}",
year = "2004",
doi = "10.1023/B:ENGI.0000017494.18537.df",
language = "English",
volume = "49",
pages = "181--193",
journal = "Journal of Engineering Mathematics",
issn = "0022-0833",
publisher = "Springer Netherlands",
number = "2",

}

TY - JOUR

T1 - The Helmholtz equation for convection in two-dimensional porous cavities with conducting boundaries

AU - Rees, D Andres

AU - Tyvand, Peder A

PY - 2004

Y1 - 2004

N2 - It is well-known that every two-dimensional porous cavity with a conducting and impermeable boundary is degenerate, as it has two different eigensolutions at the onset of convection. In this paper it is demonstrated that the eigenvalue problem obtained from a linear stability analysis may be reduced to a second-order problem governed by the Helmholtz equation, after separating out a Fourier component. This separated Fourier component implies a constant wavelength of disturbance at the onset of convection, although the phase remains arbitrary. The Helmholtz equation governs the critical Rayleigh number, and makes it independent of the orientation of the porous cavity. Finite-difference solutions of the eigenvalue problem for the onset of convection are presented for various geometries. Comparisons are made with the known solutions for a rectangle and a circle, and analytical solutions of the Helmholtz equation are given for many different domains. © 2004 Kluwer Academic Publishers.

AB - It is well-known that every two-dimensional porous cavity with a conducting and impermeable boundary is degenerate, as it has two different eigensolutions at the onset of convection. In this paper it is demonstrated that the eigenvalue problem obtained from a linear stability analysis may be reduced to a second-order problem governed by the Helmholtz equation, after separating out a Fourier component. This separated Fourier component implies a constant wavelength of disturbance at the onset of convection, although the phase remains arbitrary. The Helmholtz equation governs the critical Rayleigh number, and makes it independent of the orientation of the porous cavity. Finite-difference solutions of the eigenvalue problem for the onset of convection are presented for various geometries. Comparisons are made with the known solutions for a rectangle and a circle, and analytical solutions of the Helmholtz equation are given for many different domains. © 2004 Kluwer Academic Publishers.

KW - Heat conduction

KW - Heat convection

KW - Cavitation

KW - Porosity

KW - Problem solving

KW - Boundary conditions

KW - Mechanical permeability

U2 - 10.1023/B:ENGI.0000017494.18537.df

DO - 10.1023/B:ENGI.0000017494.18537.df

M3 - Article

VL - 49

SP - 181

EP - 193

JO - Journal of Engineering Mathematics

JF - Journal of Engineering Mathematics

SN - 0022-0833

IS - 2

ER -