Abstract
Analytic approximate formulas for flow and heat transfer through a porous medium in narrow crevices are derived. The Poiseuille number and the Nusselt number depend on the crevice geometry and the product of the aspect ratio and the porous medium factor, the latter being inversely proportional to the square root of the Darcy number. Exact numerical solutions show the approximate formulas are valid up to an aspect ratio of 0.3. The results are applicable to flow through porous rock fissures and biological clefts.
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Abbreviations
- \(a_i ,a_{ci} \) :
-
Coefficients
- A :
-
Non-dimensional area
- \(A_{ij} ,B_i ,C_{ij} \) :
-
Area integrals defined by Eq. (50)
- b :
-
Aspect ratio
- \(c_\mathrm{p} \) :
-
Effective specific heat
- \(D_\mathrm{h} \) :
-
Hydraulic diameter
- \(f_i \) :
-
Base functions
- G :
-
Pressure gradient
- \(\bar{{h}}\) :
-
Convection coefficient
- i, j :
-
Integers
- \(I_1 ,I_2 \) :
- J :
-
Functional defined by Eq. (47)
- \(k_\mathrm{e} \) :
-
Effective thermal conductivity
- K :
-
Permeability
- L :
-
Width
- N :
-
Integer
- Nu:
-
Nusselt number
- P :
-
Non-dimensional perimeter
- Po:
-
Poiseuille number
- q :
-
Heat flux input
- Q :
-
Non-dimensional flow rate
- s :
-
Porous medium factor \(L\sqrt{\mu /(\mu _\mathrm{e} K)}\)
- S :
-
Energy integral defined by Eq. (12)
- T :
-
Temperature
- \(T_\mathrm{m},T_\mathrm{s} \) :
-
Mean temperature, surface temperature
- w :
-
Non-dimensional longitudinal velocity
- \(w_\mathrm{c} \) :
-
Non-dimensional clear fluid velocity
- x, y, z :
-
Non-dimensional Cartesian coordinates
- \(\lambda \) :
-
Scaled porous medium factor = sb
- \(\mu \) :
-
Viscosity of fluid
- \(\mu _\mathrm{e} \) :
-
Effective viscosity of matrix
- \(\eta \) :
-
Scaled normal coordinate = y/b
- \(\rho \) :
-
Density of fluid
- \(\tau \) :
-
Non-dimensional temperature deviation defined by Eq. (7)
- \(_0 \) :
-
Zeroth-order perturbation
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Wang, C.Y. Darcy–Brinkman Flow in Narrow Crevices. Transp Porous Med 120, 101–113 (2017). https://doi.org/10.1007/s11242-017-0911-3
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DOI: https://doi.org/10.1007/s11242-017-0911-3