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The Effect of Rotation on the Onset of Double Diffusive Convection in a Horizontal Anisotropic Porous Layer

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Abstract

The effect of rotation and anisotropy on the onset of double diffusive convection in a horizontal porous layer is investigated using a linear theory and a weak nonlinear theory. The linear theory is based on the usual normal mode technique and the nonlinear theory on the truncated Fourier series analysis. Darcy model extended to include time derivative and Coriolis terms with anisotropic permeability is used to describe the flow through porous media. The effect of rotation, mechanical and thermal anisotropy parameters, and the Prandtl number on the stationary and overstable convection is discussed. It is found that the effect of mechanical anisotropy is to allow the onset of oscillatory convection instead of stationary. It is also found that the existence of overstable motions in case of rotating porous medium is not restricted to a particular range of Prandtl number as compared to the pure viscous fluid case. The finite amplitude analysis is performed to find the thermal and solute Nusselt numbers. The effect of various parameters on heat and mass transfer is also investigated.

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Abbreviations

a :

Wsavenumber, \(\sqrt{l^{2}+m^{2}}\)

d :

Height of the porous layer,

Da :

Darcy number, K z /d 2

g :

Gravitational acceleration, (0,0,-g)

H :

Rate of heat transport per unit area

J :

Rate of mass transport per unit area

K :

Permeability tensor, K −1 x (ii + jj ) + K −1 z (kk)

l, m :

Horizontal wavenumbers

Nu :

Thermal Nusselt number

Nu S :

Solute Nusselt number

p :

Pressure

Pr:

Prandtl number, ν /κ Tz

q :

Velocity vector, (u, v, w)

Ra T :

Thermal Rayleigh number, β T gΔTdK z ν /κ Tz

Ra S :

Solute Rayleigh number, β S gΔTdK z ν /κ Tz

R T :

Scaled Rayleigh number, Ra T /π 2

R S :

Scaled solute Rayleigh number, Ra S /π2

t :

Time

t 1 :

Rescaled time, χ t

T :

Temperature

Ta :

Taylor number, (2ΩK z /ν)2

S :

Solute concentration

ΔT :

Temperature difference between the walls

ΔS :

Salinity difference between the walls

x, y, z :

Space coordinates

α :

Scaled wavenumber, a 2/π2

β T :

Thermal expansion coefficient

β S :

Solute expansion coefficient

γ :

Scaled Darcy–Prandtl number, χ /π 2

ε :

Porosity

κ T :

Thermal diffusivity tensor, κ Tx (ii + jj) + κ T_z (kk )

κ S :

Solute diffusivity

χ :

Darcy–Prandtl number, ε Pr/Da

η :

Thermal anisotropy parameter, κ Tx /κ Tz

τ :

Diffusivity ratio κ S /κ Tz

μ :

Dynamic viscosity

ν :

Kinematic viscosity, μ /ρ 0

ρ :

Density

σ :

Growth rate

ω :

Vorticity vector, ∇ × q

Ω :

Angular velocity, (0, 0, Ω )

ξ :

Mechanical anisotropy parameter, K x /K z

ψ :

Stream function

\(\nabla_h^2\) :

\(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial ^{2}}{\partial y^{2}}\)

∇2 :

\(\nabla_h^2 +\frac{\partial ^{2}}{\partial z^{2}} \)

b :

Basic state

c :

Critical

f :

Fluid

h :

Horizontal

0 :

Reference

s :

Solid

*:

Dimensionless quantity

/ :

Perturbed quantity

osc :

Oscillatory state

st :

Stationary

References

  • Amahmid A., Hasnaoui M., Mamou M. and Vasseur P. (1999). Double-diffusive parallel flow induced in a horizontal Brinkman porous layer subjected to constant heat and mass fluxes: analytical and numerical studies. Heat Mass Transfer 35: 409–421

    Article  Google Scholar 

  • Bahloul A., Boutana N. and Vasseur P. (2003). Double-diffusive and soret-induced convection in a shallow horizontal porous layer. J. Fluid Mech. 491: 325–352

    Article  Google Scholar 

  • Bennacer R., Mohamad A.A. and Ganaoui M.El. (2005). Analytical and numerical investigation of double diffusion in thermally anisotropy multilayer porous medium. Heat Mass Transfer 41: 298–305

    Article  Google Scholar 

  • Brand H. and Steinberg V. (1983). Nonlinear effects in the convection instability of a binary mixture in a porous medium near threshold. Phys. Lett. 93A: 333–336

    Google Scholar 

  • Castinel G. and Combarnous M. (1974). Critere d’apparition de la convection naturelle dans une couche poreuse anisotrope horizontal. C.R. Acad. Sci. B 278: 701–704

    Google Scholar 

  • Chakrabarti A. and Gupta A.S. (1981). Nonlinear thermohaline convection in a rotating porous medium. Mech. Res. Commun. 8: 9

    Article  Google Scholar 

  • Chandrashekhar S. (1981). Hydrodynamic and hydromagnetic stability. Dover, New York

    Google Scholar 

  • Chen C.F. and Chen F. (1993). Double-diffusive fingering convection in a porous medium. Int. J. Heat Mass Transfer 36: 793–807

    Article  Google Scholar 

  • Epherre J.F. (1977). Criterion for the appearance of natural convection in an anisotropic porous layer. Int. Chem. Eng. 17: 615–616

    Google Scholar 

  • Govender S. (2006). On the effect of anisotropy on the stability of convection in a rotating porous media. Transp. Porous Media 64: 413–422

    Article  Google Scholar 

  • Govender S. (2007). Coriolis effect on the stability of centrifugally driven convection in a rotating anisotropic porous layer subjected to gravity. Transp. Porous Media 67: 219–227

    Article  Google Scholar 

  • Govender S. and Vadasz P. (2007). The effect of mechanical and thermal anisotropy on the stability of gravity driven convection in rotating porous media in the presence of thermal non-equilibrium. Transp. Porous Media 69: 55–66

    Article  Google Scholar 

  • Guo J. and Kaloni P.N. (1995). Nonlinear stability problem of a rotating doubly diffusive porous layer. J. Math. Anal. Appl. 190: 373–390

    Article  Google Scholar 

  • Hill A.A. (2005). Double-diffusive convection in a porous medium with a concentration based internal heat source. Proc. R. Soc. A 461: 561–574

    Google Scholar 

  • Knobloch E. (1986). Oscillatory convection in binary mixtures. Phys. Rev. A34: 1538–1549

    Google Scholar 

  • Kvernvold O. and Tyvand P.A. (1979). Nonlinear thermal convection in anisotropic porous media. J. Fluid Mech. 90: 609–624

    Article  Google Scholar 

  • Lombardo S. and Mulone G. (2002). Necessary and sufficient conditions of global nonlinear stability for rotating double-diffusive convection in a porous medium. Continuum Mech. Thermodyn. 14(16): 527–540

    Article  Google Scholar 

  • Malashetty M.S. (1993). Anisotropic thermo convective effects on the onset of double diffusive convection in a porous medium. Int. J. Heat Mass Transfer 36: 2397–2401

    Article  Google Scholar 

  • Malashetty M.S. and Swamy M. (2007). The effect of rotation on the onset of convection in a horizontal anisotropic porous layer. Int. J. Thermal Sci. 46: 1023–1032

    Article  Google Scholar 

  • Mamou M. (2002). Stability analysis of double-diffusive convection in porous enclosures. In: Ingham, D.B. and Pop, I. (eds) Transport Phenomena in Porous Media II, pp 113–154. Elsevier, Oxford

    Chapter  Google Scholar 

  • Mamou M. and Vasseur P. (1999). Thermosolutal bifurcation phenomena in porous enclosures subject to vertical temperature and concentration gradients. J. Fluid Mech. 395: 61–87

    Article  Google Scholar 

  • Mansour A., Amahmid A., Hasnaoui M. and Bourich M. (2006). Multiplicity of solutions induced by thermosolutal convection in a square porous cavity heated from below and submitted to horizontal concentration gradient in the presence of Soret effect. Numerical Heat Transfer, Part A 49: 69–94

    Article  Google Scholar 

  • McKibbin R. (1985). Thermal convection in layered and anisotropic porous media: a review. In: Wooding, R.A. and White, I. (eds) Convective Flows in Porous Media, pp 113–127. Department of Scientific and Industrial Research, Wellington

    Google Scholar 

  • (1992). Convection and heat transfer in layered and anisotropic porous media. In: Quintard, M. and Todorovic, M. (eds) Heat and Mass Transfer in Porous Media, pp 327–336. Elsevier, Amsterdam

    Google Scholar 

  • Mojtabi A. and Charrier-Mojtabi M.C. (2000). Double-diffusive convection in porous media. In: Vafai, K. (eds) Handbook of Porous media, pp 559–603. Marcel Dekker, New York

    Google Scholar 

  • Mojtabi A. and Charrier-Mojtabi M.C. (2005). Double-diffusive convection in porous media. In: Vafai, K. (eds) Handbook of Porous media, 2nd edn., pp 269–320. Taylor and Francis, New York

    Google Scholar 

  • Murray B.T. and Chen C.F. (1989). Double diffusive convection in a porous medium. J. Fluid Mech. 201: 147–166

    Article  Google Scholar 

  • Nield D.A. (1968). Onset of thermohaline convection in a porous medium. Water Resour. Res. 4: 553–560

    Article  Google Scholar 

  • Nield D.A. and Bejan A. (2006). Convection in Porous Media, 3rd edn. Springer-Verlag, New York

    Google Scholar 

  • Nilsen T. and Storesletten L. (1990). An analytical study on natural convection in isotropic and anisotropic porous channels. Trans. ASME J. Heat Transfer 112: 396–401

    Article  Google Scholar 

  • Patil Prabhamani R., Parvathy C.P. and Venkatakrishnan K.S. (1989). Thermohaline instability in a rotating anisotropic porous medium. Appl. Sci. Res. 46: 73–88

    Article  Google Scholar 

  • Poulikakos D. (1986). Double diffusive convection in a horizontal sparsely packed porous layer. Int. Commun. Heat Mass Transfer 13: 5897–5898

    Google Scholar 

  • Rosenberg N.D. and Spera F.J. (1992). Thermohaline convection in a porous medium heated from below. Int. J. Heat Mass Transfer 35: 1261–1273

    Article  Google Scholar 

  • Rudraiah N. and Malashetty M.S. (1986). The influence of coupled molecular diffusion on the double diffusive convection in a porous medium. ASME J. Heat Transfer 108: 872–876

    Article  Google Scholar 

  • Rudraiah N., Shrimani P.K. and Friedrich R. (1982). Finite amplitude convection in a two component fluid saturated porous layer. Int. J. Heat Mass Transfer 25: 715–722

    Article  Google Scholar 

  • Rudraiah N., Shivakumara I.S. and Friedrich R. (1986). The effect of rotation on linear and nonlinear double diffusive convection in a sparsely packed porous medium. Int. J. Heat Mass Transfer 29: 1301–1317

    Article  Google Scholar 

  • Storesletten L. (1998). Effects of anisotropy on convective flow through porous media. In: Ingham, D.B. and Pop, I. (eds) Transport Phenomena in Porous Media, pp 261–283. Pergamon Press, Oxford

    Chapter  Google Scholar 

  • Storesletten L.: 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, pp. 285–306. Kluwer Academic Publishers, Netherlands (2004)

  • Straughan B. and Hutter K. (1999). A priori bounds and structural stability for double diffusive convection incorporating the Soret effect. Proc. R. Soc. Lond. A 455: 767–777

    Google Scholar 

  • Trevisan O.V. and Bejan A. (1999). Combined heat and mass transfer by natural convection in a porous medium. Adv. Heat Transfer 20: 315–352

    Google Scholar 

  • Tyvand A.P. (1980). Thermohaline instability in anisotropic media. Water Resour. Res. 16(2): 325–330

    Article  Google Scholar 

  • Tyvand P.A. and Storesletten L. (1991). Onset of convection in an anisotropic porous medium with oblique principal axes. J. Fluid Mech. 226: 371–382

    Article  Google Scholar 

  • Vadasz P. (1998). Coriolis effect on gravity-driven convection in a rotating porous layer heated from below. J. Fluid Mech. 376: 351–375

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Gulbarga University, Jnana Ganga Campus, Gulbarga, 585 106, India

    M. S. Malashetty & Rajashekhar Heera

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  1. M. S. Malashetty
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  2. Rajashekhar Heera
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Correspondence to M. S. Malashetty.

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Malashetty, M.S., Heera, R. The Effect of Rotation on the Onset of Double Diffusive Convection in a Horizontal Anisotropic Porous Layer. Transp Porous Med 74, 105–127 (2008). https://doi.org/10.1007/s11242-007-9183-7

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