Darcy–Brinkman Flow in Narrow Crevices
- 132 Downloads
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.
Keywords
Darcy–Brinkman Small aspect ratio Sine duct Thermal convection RitzList of Symbols
- \(a_i ,a_{ci} \)
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
References
- Dejana, E., Orsenigo, F.: Endothelial adherens junction at a glance. J. Cell Sci. 126(12), 2545–2559 (2013)CrossRefGoogle Scholar
- Ding, J., Manglik, R.M.: Analytical solutions for laminar fully developed flows in double sine shaped ducts. Heat Mass Trans. 31, 269–277 (1996)CrossRefGoogle Scholar
- Haji-Sheikh, A.: Fully developed heat transfer to fluid flow in rectangular passages filled with porous materials. J. Heat Trans. 128, 550–556 (2006)CrossRefGoogle Scholar
- Haji-Sheikh, A., Vafai, K.: Analysis of flow and heat transfer in porous media imbedded inside various shaped ducts. Int. J. Heat Mass Trans. 47, 1889–1905 (2004)CrossRefGoogle Scholar
- Haji-Sheikh, A., Sparrow, E.M., Minkowycz, W.J.: Heat transfer to flow through porous passages using extended weighted residuals method—a Green’s function solution. Int. J. Heat Mass Trans. 48, 1330–1349 (2005)CrossRefGoogle Scholar
- Hooman, K., Merrikh, A.A.: Analytical solution of forced convection in a duct of rectangular cross section saturated by a porous medium. J. Heat Trans. 128, 596–600 (2006)CrossRefGoogle Scholar
- Kaviany, M.: Laminar flow through a porous channel bounded by isothermal parallel plates. Int. J. Heat Mass Trans. 28, 851–858 (1985)CrossRefGoogle Scholar
- Kaviany, M.: Principles of Heat Transfer in Porous Media. Springer, New York (1991)CrossRefGoogle Scholar
- Kays, W.M., Crawford, M.E., Weigand, B.: Convective Heat and Mass Transfer. McGraw-Hill, Boston (2005)Google Scholar
- Manglik, R.M., Ding, J.: Laminar flow heat transfer to viscous power-law fluids in double-sine ducts. Int. J. Heat Mass Trans. 40, 1378–1390 (1997)CrossRefGoogle Scholar
- Mortazavi, S.N., Hassanipour, F.: Effect of apex angle, porosity, and permeability on flow and heat transfer in triangular porous ducts. J. Heat Trans. 136, #112602 (2014)CrossRefGoogle Scholar
- Nakayama, A., Koyama, H., Kuwabara, F.: An analysis on forced convection in a channel filled with a Brink–Darcy porous medium: exact and approximate solutions. Warme Stoffubertrag 23, 291–295 (1988)CrossRefGoogle Scholar
- Nield, D.A., Bejan, A.: Convection in Porous Media, 5th edn. Springer, New York (2017)CrossRefGoogle Scholar
- Niu, J.L., Zhang, L.Z.: Heat transfer and friction coefficients in corrugated ducts confined by sinusoidal and arc curves. Int. J. Heat Mass Trans. 45, 571–578 (2002)CrossRefGoogle Scholar
- Parang, M., Keyhani, M.: Boundary effects in laminar mixed convection flow through an annular porous medium. J. Heat Trans. 115, 506–510 (1987)Google Scholar
- Radeva, M.Y, Waschke, J.: Mind the gap: mechanisms regulating the endothelial barrier. Acta Physiol. (2017). doi: 10.1111/alpha.12860
- Shah, R.K.: Laminar flow friction and forced convection heat transfer in ducts of arbitrary geometry. Int. J. Heat Mass Trans. 18, 849–862 (1975)CrossRefGoogle Scholar
- Shah, R.K., London, A.L.: Laminar Flow Forced Convection in Ducts. Academic, New York (1978)Google Scholar
- Sherony, D.F., Solberg, C.W.: Analytical investigation of heat or mass transfer and friction factors in a corrugated duct heat or mass exchanger. Int. J. Heat Mass Trans. 13, 145–159 (1970)CrossRefGoogle Scholar
- Speight, J.G.: Handbook of Hydraulic Fracturing. Wiley, Hoboken (2016)CrossRefGoogle Scholar
- Wang, C.Y.: Analytical solution for forced convection in a semi-circular channel filled with a porous medium. Trans. Porous Med. 73, 369–378 (2008)CrossRefGoogle Scholar
- Wang, C.Y.: Analytical solution for forced convection in a sector duct filled with a porous medium. J. Heat Trans. 132, #084502 (2010a)Google Scholar
- Wang, C.Y.: Flow through super-elliptic ducts filled with a Darcy–Brinkman medium. Trans. Porous Med. 81, 207–217 (2010b)CrossRefGoogle Scholar
- Wang, C.Y.: Flow and heat transfer through a polygonal duct filled with a porous medium. Trans. Porous Med. 90, 321–332 (2011a)CrossRefGoogle Scholar
- Wang, C.Y.: Forced convection in a lens shaped duct filled with a porous medium. J. Porous Med. 14, 743–749 (2011b)CrossRefGoogle Scholar