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Similarity solutions for buoyancy-induced flows over a non-isothermal curved surface in a thermally stratified porous medium

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Abstract

A generalized similarity transformation procedure was proposed for the analysis of buoyancy-induced flows over a curved heated surface embedded in a thermally stratified porous medium. The analysis considers two-dimensional and axisymmetric non-isothermal smooth bodies of arbitrary geometrical configuration. A generalized similarity variable which adjusts its vertical scaling according to the geometry as well as the surface temperature variation was introduced to show that, for any two-dimensional or axisymmetric smooth body shape, there exists a certain class of the surface and ambient temperature distributions which admit similarity solutions. Subsequently, the set of the governing partial differential equations were transformed into a single ordinary differential equation, which was, then, solved by a standard shooting procedure based on the Runge-Kutta method, for numerous sets of parameters. The results presented here may readily be translated for the problem of free convection over any particular two-dimensional or axisymmetric smooth body within a porous medium. The effects of the surface temperature and thermal stratification on the temperature profile and isotherms were also discussed in connection with the local surface heat flux.

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Abbreviations

f :

dimensionless stream function

g :

acceleration due to gravity

I :

function associated with a deviation from unity

k :

thermal conductivity

K :

permeability

m :

parameter associated with thermal stratification

Nu x :

local Nusselt number

q w :

local surface heat flux

r :

function representing wall geometry

r*:

1 for plane flow andr for axisymmetric flow

Ra x :

local Rayleigh number

T :

temperature

u, v :

Darcian velocity components

x, y :

boundary layer coordinates

α :

thermal diffusivity

β :

coefficient of thermal expansion

γ :

ratio of a horizontal axis to a vertical axis

η :

similarity variable

λ :

parameter associated with the surface temperature increase

ν :

kinematic viscosity

ξ :

transformed variable in the streamwise direction

ψ :

stream function

e :

boundary layer edge

w :

wall

r :

reference

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Authors and Affiliations

  1. Department of Energy and Mechanical Engineering, Shizuoka University, 3-5-1 Johoku, 432, Hamamatsu, Japan

    Akira Nakayama & Hitoshi Koyama

Authors
  1. Akira Nakayama
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  2. Hitoshi Koyama
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Nakayama, A., Koyama, H. Similarity solutions for buoyancy-induced flows over a non-isothermal curved surface in a thermally stratified porous medium. Appl. Sci. Res. 46, 309–322 (1989). https://doi.org/10.1007/BF01998548

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  • DOI: https://doi.org/10.1007/BF01998548

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