Abstract
The linear stability of a viscoelastic liquid saturated horizontal anisotropic porous layer heated from below and cooled from above is investigated by considering the Oldroyd type liquid. A generalized Darcy model, which takes into account the viscoelastic properties, the mechanical and thermal anisotropy is employed as momentum equation. The critical Rayleigh number, wavenumber, for stationary and oscillatory states and frequency of oscillation are determined analytically. It is shown that oscillatory instabilities can set in before stationary modes are exhibited. The effect of the viscoelastic parameter, the mechanical and thermal anisotropy parameters and specific heat ratio on the linear stability of the system is analyzed and presented graphically.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- a :
-
wavenumber, \(\sqrt{l^2+m^2} \)
- d :
-
height of the fluid layer
- D :
-
thermal diffusivity tensor, D x (ii + jj ) + D z (kk )
- g :
-
gravitational acceleration, (0, 0, − g )
- K :
-
permeability tensor, K −1 x (ii + jj ) + K −1 z (kk )
- p :
-
pressure
- q :
-
velocity vector (u, v, w )
- R D :
-
Darcy–Rayleigh number, β g ΔT d K z / ν D z .
- T :
-
temperature
- ΔT :
-
temperature difference between the walls
- t :
-
time
- x, y, z :
-
space coordinates
- Greek symbols:
-
- β:
-
thermal expansion coefficient
- γ:
-
ratio of specific heats (ρc) m / (ρc) f .
- \(\varepsilon \) :
-
porosity
- η:
-
thermal anisotropy parameter, D x / D z
- \({\overline{\lambda}}_{1} \) :
-
stress relaxation time
- λ 1 :
-
dimensionless stress relaxation time, \(D_{z} {\overline{\lambda}}_{1} \big/ d^{2}\)
- \(\overline{\lambda} _2 \) :
-
strain retardation time
- λ 2 :
-
dimensionless strain retardation time, \(D_{z} \overline{\lambda}_{2} \big/ d^{2}\)
- μ:
-
dynamic viscosity
- ν:
-
kinematic viscosity, μ / ρ 0
- Θ:
-
dimensionless amplitude of temperature perturbation
- ρ:
-
density
- σ:
-
growth rate
- ξ:
-
mechanical anisotropy parameter, K x / K z
- ψ :
-
stream function
- Ψ:
-
dimensionless amplitude of stream function
- ω :
-
frequency
- Subscripts:
-
- b :
-
basic state
- c :
-
critical
- 0:
-
reference value
- Superscripts:
-
- *:
-
dimensionless quantity
- /:
-
perturbed quantity
- Osc:
-
oscillatory
- S :
-
stationary
References
Ingham D.B., Pop I., (1998) Transport Phenomena In Porous Media. Pergamon, Oxford
Kim M.C., Lee S.B., Kim S., Chung B.J. (2003) Thermal instability of viscoelastic fluids in porous media. Int. J. Heat Mass Transf. 46, 5065–5072
Khuzhayorov B., Auriault J.L., Royer P. (2000) Derivation of macroscopic filtration law for transient linear viscoelastic fluid flow in porous media. Int. J. Eng. Sci. 38, 487–504
Kladias N., Prasad V. (1990) Flow transitions in buoyancy-induced non-Darcy convection in a porous medium heated from below. ASME J. Heat Transf. 112, 675–684
Lapwood E.R. (1948) Convection of a fluid in a porous medium. Proc. Camb. Phil. Soc. 44, 508–521
Li Z., Khayat R.E. (2005) Finite-amplitude Rayleigh-Benard convection and pattern selection for viscoelastic fluids. J. Fluid Mech. 529, 221–251
Malashetty, M.S., Shivakumara, I.S., Sridhar, K., Mahantesh, S.: Convective instability of Oldroyd-B fluid saturated porous layer heated from below using a thermal non-equilibrium model. Transp. Porous Media (2006)
McKibbin R. (1985) Thermal convection in layered and anisotropic porous media: a review. In: Wooding R.A., White I. (eds), Convective Flows In Porous Media. Department of Scientific and Industrial Research, Wellington NZ, pp. 113–127
McKibbin R. (1992) Convection and heat transfer in layered and anisotropic porous media. In: Quintard M., Todorovic M. (eds) Heat and Mass Transfer in Porous Media. Elsevier, Amsterdam, pp. 327–336
Nield D.A., Bejan A. (1999) Convection in Porous Media, 2nd edn. Springer-Verlag, New York
Rosenblat S. (1986) Thermal convection of a viscoelastic fluid. J. Non-Newtonian Fluid Mech. 21, 201–223
Rudraiah N., Kaloni P.N., Radhadevi P.V. (1989) Oscillatory convection in a viscoelastic fluid through a porous layer heated from below. Rheol. Acta. 28, 48–53
Storesletten L. (1998) Effects of anisotropy on convective flow through porous media. In: Ingham D.B., Pop I. (eds), Transport Phenomena In Porous Media. Pergamon Press, Oxford, pp. 261–283
Storesletten L. (2004) Effects of anisotropy on convection in horizontal and inclined porous layers. In: Ingham D.B. et al (eds), Emerging Technologies and Techniques in Porous Media. Kluwer Academic Publishers, Netherlands, pp. 285–306
Tan W., Masuoka T. (2005) Stokes’ first problem for an Oldroyd-B fluid in a porous half space. Phys. Fluids 17: 23101–23107
Vafai K. (2000) Handbook of Porous Media. Marcel Dekker, New York
Yoon Do-Young Kim M.C., Choi C.K. (2004) The onset of oscillatory convection in a horizontal porous layer saturated with viscoelastic liquid. Transport Porous Media 55, 264–275
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Malashetty, M.S., Swamy, M. The onset of convection in a viscoelastic liquid saturated anisotropic porous layer. Transp Porous Med 67, 203–218 (2007). https://doi.org/10.1007/s11242-006-9001-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-006-9001-7