Abstract
Transient non-Darcy free convection between two parallel vertical plates in a fluid saturated porous medium is investigated using the generalized momentum equation proposed by Vafai and Tien. The effects of porous inertia and solid boundary are considered in addition to the Darcy flow resistance. Exact solutions are found for the asymptotic states at small and large times. The large time solutions reveal that the velocity profiles are rather sensitive to the Darcy number Da when Da<1. It has also been found that boundary friction alters the velocity distribution near the wall, considerably. Finite difference calculations have also been carried out to investigate the transient behaviour at the intermediate times in which no similarity solutions are possible. This analytical and numerical study reveals that the transient free convection between the parallel plates may well be described by matching the two distinct asymptotic solutions obtained at small and large times.
Similar content being viewed by others
Abbreviations
- C :
-
empirical constant for the Forchheimer term
- f :
-
velocity function for the small time solution
- F :
-
velocity function for the large time solution
- g :
-
acceleration due to gravity
- Gr*:
-
micro-scale Grashof number
- H :
-
a half distance between two infinite plates
- K :
-
permeability
- Nu:
-
Nusselt number
- Pr:
-
Prandtl number
- t :
-
time
- T :
-
temperature
- u, v :
-
Darcian velocity components
- x, y :
-
Cartesian coordinates
- α:
-
effective thermal diffusivity
- β:
-
coefficient of thermal expansion
- ε:
-
porosity
- ζ:
-
dimensionless time
- η:
-
similarity variable
- θ:
-
dimensionless temperature
- μ:
-
viscosity
- ν:
-
kinematic viscosity
- ρ:
-
density
- σ:
-
the ratio of heat capacities
References
Siegel, R., Transient free convection from a vertical flat plate.Trans. ASME 80 (1958) 347–359.
Goldstein, R. J. and Briggs, D. G., Transient free convection about vertical plates and cylinders.J. Heat Transfer 86C (1984) 490–500.
Nanbu, B. K., Unsteady Falkner-Skan flow.Z. Angew. Math. Phys. 22 (1971) 1167–1172.
Watkins, C. B., Heat transfer in the boundary layer over an impulsively started flat plate.J. Heat Transfer 97 (1975) 482–484.
Sparrow, E. M. and Siegel, R., Transient film condensation.J. Applied Mechanics 81(1) (1959) 120–121.
Wilson, S. D. R., Unsteady and two-dimensional flow of condensate film.J. Heat Transfer 98 (1976) 313–315.
Johnson, C. H. and Cheng, P., Possible similarity solution for free convection boundary layers adjacent to flat plates in porous media.Int. J. Heat Mass Transfer 21 (1978) 709–718.
Ingham, D. B., Merkin, J. H. and Pop, I., Flow past a suddenly cooled vertical flat surface in a saturated porous medium.Int. J. Heat Mass Transfer 25 (1982) 1916–1919.
Ingham, D. B. and Brown, S. N., Flow past a suddenly heated vertical plate in a porous medium.Proc. R. Soc. Lond. A403 (1986) 51–80.
Jang, J.-Y. and Ni, J.-R., Transient free convection with mass transfer from an isothermal vertical flat plate embedded in a porous medium.Int. J. Heat Fluid Flow 10 (1989) 59–65.
Cheng, P. and Pop, I., Transient free convection about a vertical flat plate embedded in a porous medium.Int. J. Engng Sci. 22 (1984) 253–264.
Cheng, P. and Chui, D. K., Transient film condensation on a vertical surface in a porous medium.Int. J. Heat Mass Transfer 27 (1984) 795–798.
Chen, C.-K., Hung, C.-I. and Horng, H.-C., Transient natural convection on a vertical flat plate embedded in a high-porosity medium.J. Heat Transfer 109 (1989) 112–118.
Nakayama, A., Kokudai, T. and Koyama, H., Non-Darcy boundary layer flow and forced convective heat transfer over a flat plate in a fluid-saturated porous medium.J. Heat Transfer 112 (1990) 157–162.
Vafai, K. and Tien, C. L., Boundary and inertia effects on flow and heat transfer in porous media.Int. J. Heat Mass Transfer 30 (1981) 1391–1405.
Katsuki, M. and Nakayama, A.,Numerical Simulation of Heat and Fluid Flow. Tokyo: Morikita Shuppan (1990) pp. 35–117 (in Japanese).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nakayama, A., Kuwahara, F. & Koyama, H. Transient non-Darcy free convection between parallel vertical plates in a fluid saturated porous medium. Appl. Sci. Res. 50, 29–42 (1993). https://doi.org/10.1007/BF01086451
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01086451