Abstract
A linear instability analysis has been performed for the onset of convection in a horizontal layer of a porous medium whose permeability pulsates with time as a result of vertical movement of one of the boundaries. It was found that, to a first-order approximation in the pulsation amplitude, the effect of deformation is destabilizing for disturbances whose period is of the order of the thermal diffusion time scale. The effects of the average porosity, pulsation amplitude, and pulsation frequency were investigated.
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Abbreviations
- \(A(z)\) :
-
Amplitude of a complex number
- \(c_{a}\) :
-
Acceleration coefficient
- \(d_{p}\) :
-
Particle diameter for a porous medium composed of spheres
- \(D\) :
-
d/dz
- \(H\) :
-
Layer height
- \(k\) :
-
\((\omega /2)^{1/2}\)
- \(k_{m}\) :
-
Thermal conductivity of the porous medium
- \(K\) :
-
Permeability of the medium
- \(K_{0}\) :
-
Constant defined in Eq. (8)
- \(K_{1}\) :
-
Constant defined in Eq. (8)
- \(P\) :
-
Excess pressure divided by the reference hydrostatic pressure, \(\frac{K_0 P^*}{\mu \alpha _m}\)
- \(P^*\) :
-
Excess fluid pressure
- \(\text{Pr}_{\mathrm{m}}\) :
-
Overall Prandtl number, \(\frac{\mu}{\rho _0 \alpha _m}\)
- \(q(z)\) :
-
Quantity defined in Eq. (26)
- \(Q\left( \tau \right) \) :
-
Monodromy matrix, defined in Eq. (58)
- Ra:
-
Rayleigh number, \(\frac{\rho _0 g\beta K_0 H\Delta T}{\mu \alpha _m}\)
- \(t\) :
-
Dimensionless time, \(\frac{\alpha _m t^*}{\sigma H^{2}}\)
- \(t^*\) :
-
Time
- \(T_{0}\) :
-
Temperature of the upper wall
- \(T_{0}+\Delta T\) :
-
Temperature of the lower wall
- \(T\) :
-
Dimensionless temperature, \(\frac{T^*-T_0}{\Delta T}\)
- \(T^*\) :
-
Temperature
- v :
-
Dimensionless Darcy velocity, \(\frac{H\mathbf{v}}{\alpha _m}\)
- \(\mathbf{v}^*\) :
-
Darcy velocity, (u*, v*, w*)
- x :
-
Dimensionless position vector, \(\frac{\mathbf{x}^*}{H}\)
- \(\mathbf{x}^*\) :
-
Position vector, (\(x^*, y^*, z^*\))
- \(\upalpha \) :
-
Overall horizontal wavenumber
- \(\upalpha _{m}\) :
-
Thermal diffusivity of the porous medium
- \(\beta \) :
-
Thermal volumetric expansion coefficient
- \(\gamma _{a}\) :
-
Nondimensional acceleration coefficient, \(\frac{{c_a} {K_0}}{\sigma \text{Pr}_\mathrm{m} H^{2}}\)
- \(\varepsilon \) :
-
Oscillation amplitude of the lower boundary
- \(\lambda _j\) :
-
Floquet multiplier (an eigenvalue of \(Q)\)
- \(\mu \) :
-
Dynamic viscosity of the fluid
- \(\mu _j\) :
-
Floquet exponents, defined in Eq. (60)
- \(\rho _{f}\) :
-
Fluid density
- (\(\rho \text{c})_{f}\) :
-
Heat capacity of the fluid
- (\(\rho \text{c})_{m}\) :
-
Heat capacity of the medium
- \(\sigma \) :
-
Heat capacity ratio
- \(\phi (z)\) :
-
Argument of a complex number
- \(\phi \) :
-
Porosity of the medium
- \(\phi _{0}\) :
-
Mean porosity
- \(\varPhi \) :
-
Volume fraction of the fluid at time \(t^*\)
- \(\omega \) :
-
Nondimensional frequency, \(\frac{\sigma H^{2}\Omega}{\alpha _m}\)
- \(\Omega \) :
-
Oscillation frequency of the lower boundary
- 0:
-
Reference value
- b:
-
Basic state
- ’:
-
Perturbation quantity
- *:
-
Dimensional quantity
References
Ateshian, G.A., Maas, S., Weiss, J.A.: Solute transport across a contact interface in deformable porous media. J. Biomech. 45, 1023–1027 (2012)
Caltagirone, J.-P.: Stabilite d’une couche poreuse horizontal soumise a des conditions aux periodiques. Int. J. Heat Mass Transf. 19, 815–820 (1976)
Chhuon, B., Caltagirone, J.P.: Stability of a horizontal porous layer with timewise periodic boundary conditions. ASME J. Heat Transf. 101, 244–248 (1979)
Kolditz, O., Bauer, S., Böttcher, N., Elsworth, D., Goerke, U., McDermott, C.-I., Park, C.-H., Singh, A.K., Taron, J., Wang, W.: Numerical simulation of two-phase flow in deformable porous media: Application to carbon dioxide storage in the subsurface. Math. Comput. Simul. 82, 1919–1935 (2012)
Malevich, A.E., Mityushev, V., Adler, P.: Stokes flow through a curvilinear channel. Acta Mech. 182, 151–182 (2006)
Malevich, A.E., Mityushev, V., Adler, P.: Couette flow in a channel with wavy walls. Acta Mech. 197, 247–283 (2008)
Mendes, M.A., Murad, M.A., Pereira, F.: A new computational strategy for solving two-phase flow in strongly heterogeneous poroelastic media of evolving scales. Int. J. Numer. Anal. Methods Geomech. 36, 1683–1716 (2012)
Mihoubi, D., Bellagi, A.: Modeling of heat and moisture transfers with stress-strain formation during convective air drying of deformable media. Heat Mass Transf. 48, 1697–1705 (2012)
Mikelic, A., Wheeler, M.F.: On the interface law between a deformable porous medium containing a viscous fluid and an elastic body. Math. Model. Methods Appl. Sci. 22, 1250031 (2012)
Nield, D.A., Bejan, A.: Convection in Porous Media, 4th edn. Springer, New York (2013)
Nield, D.A., Kuznetsov, A.V.: Investigation of forced convection in an almost circular microtube with rough walls. Int. J. Fluid Mech. Res. 30, 1–10 (2003)
Stoverud, K.H., Darcis, M., Helmig, R., Hassanizadeh, S.M.: Modeling concentration distribution and deformation during convection-enhanced drug delivery into brain tissue. Transp. Porous Media 92, 119–143 (2012)
Vasco, D.W.: On the propagation of a quasi-static disturbance in a heterogeneous, deformable, and porous medium with pressure-dependent properties. Water Resour. Res. 47, W12523 (2011)
Zhang, H.J., Jeng, D.-S., Seymour, B.R., Barry, D.A., Li, L.: Solute transport in partially-saturated deformable porous media: Application to a landfill clay liner. Adv. Water Resour. 40, 1–10 (2012)
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Nield, D.A., Kuznetsov, A.V. The Effect of Pulsating Deformation on the Onset of Convection in a Porous Medium. Transp Porous Med 98, 713–724 (2013). https://doi.org/10.1007/s11242-013-0168-4
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DOI: https://doi.org/10.1007/s11242-013-0168-4