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Heat transfer in an unevenly heated porous layer

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Abstract

For a uniform saturated porous layer heated from below, the dependence of the quantity of heat transferred on the distribution of the heat source is investigated. It is found, using perturbation methods and numerical techniques, that very small nonuniformities in the heat source having the same wavelength as the preferred convection mode significantly reinforce natural convection.

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Abbreviations

A mn , B mn :

Fourier series coefficients in expansions for ψ, T

c :

fluid specific heat

d :

porous layer thickness

f n , g n :

Fourier series coefficients

\(\hat f_{n, } \hat g_n\) :

‘normalized’ Fourier series coefficients

g :

gravitational acceleration

K :

permeability of porous layer

k :

thermal conductivity of porous layer

l :

system aspect ratio (dimensionless)

m :

wavenumber of flow pattern at onset of convection

Nu:

Nusselt number

p :

pressure

R:

Rayleigh number

R c :

critical value of Rayleigh number

T :

temperature

T mn :

Fourier series coefficient in expansion for T

ΔT :

temperature difference across system

u, w :

nondimensional components of mass flux velocity vector

x, z :

nondimensional horizontal, vertical coordinates.

α :

thermal expansion coefficient for fluid

β m :

power series coefficient

δ :

(small) amplitude of temperature nonuniformity

ε :

(small) amplitude of convection

θ(x) :

nondimensional variation in temperature at base of porous layer

θ n :

Fourier series coefficient in expansion of θ(x)

ν :

kinematic viscosity of fluid

ϱ :

calculation parameter

ϱ a :

reference density of fluid

τ :

calculation parameter

ψ :

stream function

ψ mn :

Fourier series coefficient in expansion for ψ

(n):

refers to coefficient of ε n in series expansions in powers of small amplitude ε

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Authors and Affiliations

  1. Department of Theoretical and Applied Mechanics, University of Auckland, Private Bag, Auckland, New Zealand

    M. J. O'Sullivan & R. McKibbin

Authors
  1. M. J. O'Sullivan
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  2. R. McKibbin
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O'Sullivan, M.J., McKibbin, R. Heat transfer in an unevenly heated porous layer. Transp Porous Med 1, 293–312 (1986). https://doi.org/10.1007/BF00238184

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  • DOI: https://doi.org/10.1007/BF00238184

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