^{1,a)}and L. Storesletten

^{2,b)}

### Abstract

The onset of thermal convection in a vertical porous cylinder is studied by considering the heating from below and the cooling from above as caused by external forced convection processes. These processes are parametrised through a finite Biot number, and hence through third-kind, or Robin, temperature conditions imposed on the lower and upper boundaries of the cylinder. Both the horizontal plane boundaries and the cylindrical sidewall are assumed to be impermeable; the sidewall is modelled as a thermally insulated boundary. The linear stability analysis is carried out by studying separable normal modes, and the principle of exchange of stabilities is proved. It is shown that the Biot number does not affect the ordering of the instability modes that, when the radius-to-height aspect ratio increases, are displayed in sequence at the onset of convection. On the other hand, the Biot number plays a central role in determining the transition aspect ratios from one mode to its follower. In the limit of a vanishingly small Biot number, just the first (non-axisymmetric) mode is displayed at the onset of convection, for every value of the aspect ratio.

This work was financially supported by Italian government, MIUR Grant No. PRIN–2009KSSKL3.

I. INTRODUCTION

II. GOVERNING EQUATIONS

III. DIMENSIONLESS EQUATIONS

IV. BASIC SOLUTION

V. LINEAR DISTURBANCES

VI. NORMAL MODES

A. Exchange of stabilities

B. Analytical solution for neutral stability

VII. NEUTRAL STABILITY CURVES

A. Instability patterns

VIII. CONCLUSIONS

### Key Topics

- Normal modes
- 12.0
- Convection
- 11.0
- Boundary value problems
- 10.0
- Porous media
- 6.0
- Heat transfer
- 5.0

## Figures

A sketch of the fluid saturated porous cylinder.

A sketch of the fluid saturated porous cylinder.

Neutral stability: plots of R versus a for the most unstable modes, each frame is for a different value of B. The horizontal dotted line is for R = R c ; the vertical dotted lines bound the regions where each specified (n, m)-mode is preferred.

Neutral stability: plots of R versus a for the most unstable modes, each frame is for a different value of B. The horizontal dotted line is for R = R c ; the vertical dotted lines bound the regions where each specified (n, m)-mode is preferred.

Neutral stability: plots of versus a for the most unstable modes, each frame is for a different value of B. The horizontal dotted line is for ; the vertical dotted lines bound the regions where each specified (n, m)-mode is preferred.

Neutral stability: plots of versus a for the most unstable modes, each frame is for a different value of B. The horizontal dotted line is for ; the vertical dotted lines bound the regions where each specified (n, m)-mode is preferred.

Selected modes, (n, m), at onset of convection in the plane (a, B); the solid lines bound the regions where each specified mode is preferred.

Selected modes, (n, m), at onset of convection in the plane (a, B); the solid lines bound the regions where each specified mode is preferred.

Qualitative plots of the time-independent normal modes Θ(r, θ, z), given by Eq. (13) , as functions of (r, θ) at a fixed plane z = constant, with 0 < z < 1, for the pairs (n, m) ordered according to Table I .

## Tables

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