## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2014, Special Issue (2013), Article ID 695425, 10 pages.

### Eddy Heat Conduction and Nonlinear Stability of a Darcy Lapwood System Analysed by the Finite Spectral Method

Jónas Elíasson

#### Abstract

A finite Fourier transform is used to perform both linear and nonlinear stability analyses of a Darcy-Lapwood system of convective rolls. The method shows how many modes are unstable, the wave number instability band within each mode, the maximum growth rate (most critical) wave numbers on each mode, and the nonlinear growth rates for each amplitude as a function of the porous Rayleigh number. Single amplitude controls the nonlinear growth rates and thereby the physical flow rate and fluid velocity, on each mode. They are called the flak amplitudes. A discrete Fourier transform is used for numerical simulations and here frequency combinations appear that the traditional cut-off infinite transforms do not have. The discrete show a stationary solution in the weak instability phase, but when carried past 2 unstable modes they show fluctuating motion where all amplitudes except the flak may be zero on the average. This leads to a flak amplitude scaling process of the heat conduction, producing an eddy heat conduction coefficient where a Nu-RaL relationship is found. It fits better to experiments than previously found solutions but is lower than experiments.

#### Article information

Source
J. Appl. Math., Volume 2014, Special Issue (2013), Article ID 695425, 10 pages.

Dates
First available in Project Euclid: 1 October 2014

https://projecteuclid.org/euclid.jam/1412183191

Digital Object Identifier
doi:10.1155/2014/695425

Mathematical Reviews number (MathSciNet)
MR3166780

#### Citation

Elíasson, Jónas. Eddy Heat Conduction and Nonlinear Stability of a Darcy Lapwood System Analysed by the Finite Spectral Method. J. Appl. Math. 2014, Special Issue (2013), Article ID 695425, 10 pages. doi:10.1155/2014/695425. https://projecteuclid.org/euclid.jam/1412183191

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