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The Onset of Convection in a Heterogeneous Porous Medium with Vertical Throughflow

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Abstract

The effect of vertical heterogeneity of permeability, on the onset of convection in a horizontal layer of a saturated porous medium, uniformly heated from below but with a non-uniform basic temperature gradient resulting from vertical throughflow, is studied analytically using linear stability theory. It is found that, to first order, a linear variation of the reciprocal of permeability with depth has no effect on the critical value of the Rayleigh number Ra c based on the harmonic mean of the permeability, but a quadratic variation increasing in the upwards direction leads to a reduction in Ra c.

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Abbreviations

a :

Dimensionless horizontal wavenumber

c a :

Acceleration coefficient

f(z):

Function characterizing the basic temperature gradient, defined by Eq. 25

g(z):

Function characterizing the reciprocal of the permeability, defined by Eq. 25

g :

Gravitational acceleration

H :

Dimensional layer depth

k m :

Effective thermal conductivity of the porous medium

K :

Permeability of the porous medium

P*:

Pressure

P :

Dimensionless pressure, P* K/μα m

Q :

Péclet number defined by Eq. 14

Ra:

Rayleigh–Darcy number

t*:

Time

t :

Dimensionless time, t*α m /σ H 2

T*:

Temperature

T :

Dimensionless temperature, (T* − T 0)/(T 1 − T 0)

\({T^{\ast}_{0}}\) :

Temperature at the upper wall

\({T^{\ast}_{1}}\) :

Temperature at the lower wall

(u, v, w):

Dimensionless Darcy velocity components, (u*, v*, w*)H/α m

V 0 :

Throughflow velocity

v :

Dimensionless Darcy velocity, \({\frac{(\rho{c})_{\rm f} H}{k_{\rm m}}}\)

v*:

Dimensional Darcy velocity, (u*, v*, w*)

(x, y, z):

Dimensionless Cartesian coordinates, (x*, y*, z*)/H; z is the vertically upward coordinate

(x*, y*, z*):

Cartesian coordinates

α m :

Thermal diffusivity of the porous medium, \({\frac{k_{\rm m}}{(\rho{c}_P )_{\rm f}}}\)

γ a :

Acceleration coefficient defined by Eq. 10

γ :

Permeability linear heterogeneity parameter defined by Eq. 36

δ :

Permeability quadratic heterogeneity parameter defined by Eq. 40

μ :

Viscosity of the fluid

ρ :

Fluid density

(ρc)f :

Heat capacity of the fluid

(ρc)m :

Effective heat capacity of the porous medium

σ :

Thermal capacity ratio defined by Eq. 6

*:

Dimensional variable

′:

Perturbation variable

b:

Basic solution

References

  • Barletta A., di Schio E.R., Storesletten L.: Convective roll instabilities of vertical throughflow with viscous dissipation in a horizontal porous layer. Transp. Porous Med. 81, 461–477 (2010)

    Article  Google Scholar 

  • Braester C., Vadasz P.: The effect of weak heterogeneity of a porous medium on natural convection. J. Fluid Mech. 254, 345–362 (1993)

    Article  Google Scholar 

  • Brevdo L.: Three-dimensional absolute and convective instabilities at the onset of convection in a porous medium with inclined temperature gradient and vertical throughflow. J. Fluid Mech. 641, 475–487 (2009)

    Article  Google Scholar 

  • Brevdo L., Ruderman S.: On the convection in a porous medium with inclined temperature gradient and vertical throughflow. Part I. Normal modes. Transp. Porous Med. 80, 137–151 (2009a)

    Article  Google Scholar 

  • Brevdo L., Ruderman S.: On the convection in a porous medium with inclined temperature gradient and vertical throughflow. Part II. Absolute and convective instabilities, and spatially amplifying waves. Transp. Porous Med. 80, 153–172 (2009b)

    Article  Google Scholar 

  • Hill A.A.: Unconditional nonlinear stability for convection in a porous medium with vertical throughflow. Acta Mech. 193, 197–206 (2007)

    Article  Google Scholar 

  • Hill A.A., Rionero S., Straughan B.: Global stability for penetrative convection with throughflow in a porous material. IMA J. Appl. Math. 72, 635–643 (2007)

    Article  Google Scholar 

  • Kuznetsov A.V., Nield D.A., Simmons C.T.: The effect of strong heterogeneity on the onset of convection in a porous medium: periodic and localized variation. Transp. Porous Med. 81, 123–139 (2010)

    Article  Google Scholar 

  • Kuznetsov A.V., Nield D.A., Simmons C.T.: The onset of convection in a strongly heterogeneous porous medium with transient temperature profile. Transp. Porous Med. 86, 851–865 (2011)

    Article  Google Scholar 

  • Nield D.A.: Convective instability in porous media with throughflow. AIChE J. 33, 1222–1224 (1987)

    Article  Google Scholar 

  • Nield D.A.: General heterogeneity effects on the onset of convection in a porous medium. In: Vadasz, P. (ed.) Emerging Topics in Heat and Mass Transfer in Porous Media—from Bioengineering and Microelectronics to Nanotechnology, pp. 63–84. Springer, New York (2008)

    Chapter  Google Scholar 

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

    Google Scholar 

  • Nield, D.A., Kuznetsov, A.V.: The effects of combined horizontal and vertical heterogeneity on the onset of convection in a porous medium. Int. J. Heat Mass Transf. 50, 2361–2367; erratum 4512–4512 (2007)

    Google Scholar 

  • Nield D.A., Kuznetsov A.V.: The effects of combined horizontal and vertical heterogeneity on the onset of convection in a porous medium: moderate heterogeneity. Int. J. Heat Mass Transfer 51, 2361–2367 (2008)

    Article  Google Scholar 

  • Nield D.A., Simmons C.T.: A discussion on the effect of heterogeneity on the onset of convection in a porous medium. Transp. Porous Med. 68, 413–421 (2007)

    Article  Google Scholar 

  • Nield D.A., Kuznetsov A.V., Simmons C.T.: The effect of strong heterogeneity on the onset of convection in a porous medium. Transp. Porous Med. 77, 169–186 (2009)

    Article  Google Scholar 

  • Nield D.A., Kuznetsov A.V., Simmons C.T.: The effect of strong heterogeneity on the onset of convection in a porous medium: 2D/3D localization and spatially correlated random permeability fields. Transp. Porous Med. 83, 465–477 (2010)

    Article  Google Scholar 

  • Shivakumara I.S., Nanjundappa C.E.: Effects of quadratic drag and throughflow on double diffusive convection in a porous layer. Int. Commun. Heat Mass Transf. 33, 357–363 (2006)

    Article  Google Scholar 

  • Simmons C.T., Kuznetsov A.V., Nield D.A.: The effect of strong heterogeneity on the onset of convection in a porous medium: Importance of spatial dimensionality and geologic controls. Water Resour. Res. 46, W09539 (2010). doi:10.1029/2009WR008606

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland, New Zealand

    D. A. Nield

  2. Department of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, Raleigh, NC, 27695-7910, USA

    A. V. Kuznetsov

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  1. D. A. Nield
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  2. A. V. Kuznetsov
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Correspondence to A. V. Kuznetsov.

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Nield, D.A., Kuznetsov, A.V. The Onset of Convection in a Heterogeneous Porous Medium with Vertical Throughflow. Transp Porous Med 88, 347–355 (2011). https://doi.org/10.1007/s11242-011-9742-9

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