Advertisement

Applied Mathematics and Mechanics

, Volume 31, Issue 5, pp 565–574 | Cite as

Nonlinear convection in a non-Darcy porous medium

  • M. K. ParthaEmail author
Article

Abstract

In this paper, the natural convection in a non-Darcy porous medium is studied using a temperature-concentration-dependent density relation. The effect of the two parameters responsible for the nonlinear convection is analyzed for different values of the inertial parameter, dispersion parameters, Rayleigh number, Lewis number, Soret number, and Dufour number. In the aiding buoyancy, the tangential velocity increases steeply with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter when the inertial effect is zero. However, when the inertial effect is non-zero, the effect of the nonlinear temperature parameter and the nonlinear concentration parameter on the tangential velocity is marginal. The concentration distribution varies appreciably and spreads in different ranges for different values of the double dispersion parameters, the inertial effect parameter, and also for the parameters which control the nonlinear temperature and the nonlinear concentration. Heat and mass transfer varies extensively with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter depending on Dacry and non-Darcy porous media. The variation in heat and mass transfer when all the effects, i.e., the inertial effect, double dispersion effects, and Soret and Dufour effects, are simultaneously zero and non-zero. The combined effects of the nonlinear temperature parameter, the nonlinear concentration parameter and buoyancy are analyzed. The effect of the nonlinear temperature parameter and the nonlinear concentration parameter and also the cross diffusion effects on heat and mass transfer are observed to be more in Darcy porous media compared with those in non-Darcy porous media. In the opposing buoyancy, the effect of the temperature parameter is to increase the heat and mass transfer rate, whereas that of the concentration parameter is to decrease.

Key words

non-Darcy porous medium natural convection double dispersion 

Nomenclature

C

dimensional concentration

Cs

concentration susceptibility

Cp

specific heat at constant pressure

c

inertial coefficient

d

porediameter

D

constant molecular diffusivity

DX

effective solutal diffusivity inthe X direction

DY

effective solutal diffusivity in the Y direction

g

acceleration due to gravity

h

heat transfer coefficient

K

permeability of the porous medium

k

effective thermal conductivity

ke

thermal conductivity of the porous medium

kT

thermal diffusion ratio

T

dimensional temperature

U

dimensional velocity component along the X direction

V

dimensional velocity component along the Y direction

υ

velocity representation in three dimensions

q

local surface heat flux

\( \frac{{\partial p}} {{\partial X}} \)

pressure gradient in the flow direction

α

constant thermal diffusivity

αd

dynamic viscosity

αX

effective thermal diffusivity in the X direction

αY

effective thermal diffusivity in the Y direction

β0, β1

thermal expansion coefficients

β2, β3

solutal expansion coefficients

γ

thermal dispersion coefficient

ξ

solutal dispersion coefficient

ν

fluid kinematic viscosity

ρ

fluid density

µ

fluid viscosity

η

similarity variable

ψ

dimensional stream function

θ

non-dimensional temperature

ϕ

non-dimensional concentration

θw, θw

Tw - T

ψw, ϕw

Cw - C

Le = \( \frac{\alpha } {D} \)

diffusivity ratio (Lewis number)

α1 = \( \frac{{\beta _0 }} {{2\beta _1 (T_w - T_\infty )}} \),

nonlinear temperature parameter

α2 = \( \frac{{\beta _2 }} {{2\beta _3 (C_w - C_\infty )}} \),

nonlinear concentration parameter

F = \( \frac{{2c\alpha \sqrt K }} {{\nu X}} \),

inertial effect parameter

Rad = \( \frac{{KgB_0 \theta _w d}} {{\alpha \nu }} \),

pore dependent Rayleigh number

RaX = \( \frac{{KgB_0 \theta _w X}} {{\alpha \nu }} \),

local Rayleigh number

Raξ = ξRad

solutal dispersion parameter

Raγ = γRad

thermal dispersion parameter

N = \( \frac{{\beta _2 \varphi _w }} {{\beta _0 \theta _w }} \),

buoyancy parameter

Rep

Reynolds number based on a typical pore or particle diameter

Df

Dufour effect parameter

Sr

Soret effect parameter

w

on the wall

at the outer edge of the boundary layer

Chinese Library Classification

O357.1 

2000 Mathematics Subject Classification

76R10 76S05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Lai, F. C. and Kulacki, F. A. Coupled heat and mass transfer by natural convection from vertical surfaces in porous media. Int. J. Heat Mass Trans. 34(4–5), 1189–1194 (1991)CrossRefGoogle Scholar
  2. [2]
    Angirasa, D., Peterson, G. P., and Pop, I. Combined heat and mass transfer by natural convection with opposing buoyancy effects in a fluid saturated porous medium. Int. J. Heat Mass Trans. 40(12), 2755–2773 (1977)CrossRefGoogle Scholar
  3. [3]
    Murthy, P. V. S. N. Effect of double dispersion on mixed convection heat and mass transfer in non-Darcy porous medium. ASME J. Heat Trans. 122(3), 476–484 (2000)CrossRefMathSciNetGoogle Scholar
  4. [4]
    Nield, D. A. and Bejan, A. Convection in Porous Media, 3rd Ed, Springer, New York (2006)Google Scholar
  5. [5]
    Postelnicu, A. Effects of thermophoresis particle deposition in free convection boundary layer from a horizontal flat plate embedded in a porous medium. Int. J. Heat Mass Trans. 50(15–16), 2981–2985 (2007)zbMATHCrossRefGoogle Scholar
  6. [6]
    Anghel, M, Takhar, H. S., and Pop, I. Dufour and Soret Effects on Free Convection Boundary Layer Flow over a Vertical Surface Embedded in a Porous Medium, Studia Universitatics Babes-Bolyai Mathematica, XLV (2000)Google Scholar
  7. [7]
    Cheng, P. and Minkowcyz, W. J. Free convection about a vertical flat plate embedded in a porous media with application to heat transfer from a dike. J. Geophys. Res. 82(B14), 2040–2044 (1977)CrossRefGoogle Scholar
  8. [8]
    Cheng, P. The influence of lateral mass flux on free convection boundary layers in saturated porous medium. Int. J. Heat Mass Trans. 20(3), 201–206 (1977)CrossRefGoogle Scholar
  9. [9]
    Bejan, A. and Khair, K. R. Heat and mass transfer by natural convection in a porous medium. Int. J. Heat Mass Trans. 28(5), 909–918 (1985)zbMATHCrossRefGoogle Scholar
  10. [10]
    Cheng, P. Buoyancy induced boundary layer flows in geothermal reservoirs. Proceedings of the 2nd Workshop Geothermal Reservoir Engineering, Standford University, Standford, California, 236–246 (1976)Google Scholar
  11. [11]
    Cengel, Y. A. Heat Transfer—a Practical Approach, 2nd Ed., McGraw-Hill, New Delhi, India (2003)Google Scholar
  12. [12]
    Verms, G. Thermophoresis-enhanced deposition rates in combustion turbine blade passages. J. Eng. Power 101, 542–548 (1979)Google Scholar
  13. [13]
    Goren, S. L. On free convection in water at 40 °C. Chem. Eng. Sci. 21, 515–518 (1966)CrossRefGoogle Scholar
  14. [14]
    Sinha, P. C. Fully developed laminar free convection flow between vertical parallel plates. Chem. Eng. Sci. 24(1), 33–38 (1969)CrossRefGoogle Scholar
  15. [15]
    Gilpin, R. R. Cooling of a horizontal cylinder of water through its maximum density point at 4 °C. Int. J. Heat Mass Trans. 18(11), 1307–1315 (1975)CrossRefGoogle Scholar
  16. [16]
    Vanier, C. R. and Tien, C. Effect of maximum density and melting on natural convection heat transfer from a vertical plate. Chem. Eng. Prog. Symp. Ser. 82, 64 (1968)Google Scholar
  17. [17]
    Yen, Y. C. Effects of density inversion on free convective heat transfer in porous layer heated from below. Int. J. Heat Mass Trans. 17(11), 1349–1356 (1974)CrossRefGoogle Scholar
  18. [18]
    Partha, M. K., Murthy, P. V. S. N., and Raja Shekhar, G. P. Soret and Dufour effects in a non-Darcy porous medium. ASME J. Heat Trans. 128(6), 605–610 (2006)CrossRefGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Department of MathematicsMalnad College of EngineeringKarnatakaIndia

Personalised recommendations