Summary
In this paper a problem of fluid flow and heat transfer in Couette flow through a rigid saturated porous medium is investigated. The fluid flow occurs due to a moving wall and it is described by the Brinkman-Forchheimer-extended Darcy equation. Two heat transfer situations are considered: (A) isoflux fixed wall and insulated moving wall and (B) insulated fixed wall and isoflux moving wall. Analytical solutions for the flow velocity, temperature distribution, and for the Nusselt number are obtained.
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Abbreviations
- A :
-
parameter defined in Eq. (22),\({{\operatorname{Re} _H F} \mathord{\left/ {\vphantom {{\operatorname{Re} _H F} {\sqrt {Da_H } }}} \right. \kern-\nulldelimiterspace} {\sqrt {Da_H } }}\)
- B :
-
parameter defined in Eq. (22),\(\frac{1}{{Da_H }}\)
- D :
-
parameter defined in Eq. (27),\({{\left\{ {1 + \sqrt {1 + \frac{2}{3}\frac{A}{B}} } \right\}} \mathord{\left/ {\vphantom {{\left\{ {1 + \sqrt {1 + \frac{2}{3}\frac{A}{B}} } \right\}} {\left\{ {1 - \sqrt {1 + \frac{2}{3}\frac{A}{B}} } \right\}}}} \right. \kern-\nulldelimiterspace} {\left\{ {1 - \sqrt {1 + \frac{2}{3}\frac{A}{B}} } \right\}}}\)
- Da H :
-
Darcy number,K/H 2
- F :
-
Forchheimer coefficient
- H :
-
width of the channel, m
- K :
-
permeability of the porous medium, m2
- Nu:
-
Nusselt number at the isoflux wall,\({{Hq''} \mathord{\left/ {\vphantom {{Hq''} {\left( {k_{eff} \left( {\tilde T_W - \tilde T_W } \right)} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {k_{eff} \left( {\tilde T_W - \tilde T_W } \right)} \right)}}\)
- Re H :
-
Reynolds number,\(\left\langle {\rho _f } \right\rangle ^f {{\tilde u_W H} \mathord{\left/ {\vphantom {{\tilde u_W H} {\mu _f }}} \right. \kern-\nulldelimiterspace} {\mu _f }}\)
- T :
-
dimensionless temperature,\({{\left( {\left\langle {\tilde T} \right\rangle - \tilde T_W } \right)} \mathord{\left/ {\vphantom {{\left( {\left\langle {\tilde T} \right\rangle - \tilde T_W } \right)} {\left( {\tilde T_m - \tilde T_W } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\tilde T_m - \tilde T_W } \right)}}\)
- \(\left\langle {\tilde T} \right\rangle \) :
-
intrinsic average temperature, K
- \(\tilde T_m \) :
-
mean temperature,\(\frac{1}{{H\tilde U}}\int\limits_0^H {\left\langle {\tilde u_f } \right\rangle \left\langle {\tilde T} \right\rangle d\tilde y,K} \)
- \(\tilde T_W \) :
-
temperature at the isoflux wall, K
- \(\left\langle {\tilde u_f } \right\rangle \) :
-
superficial average velocity, ms−1
- u :
-
dimensionless velocity,\({{\left\langle {\tilde u_f } \right\rangle } \mathord{\left/ {\vphantom {{\left\langle {\tilde u_f } \right\rangle } {\tilde u_W }}} \right. \kern-\nulldelimiterspace} {\tilde u_W }}\)
- \(\tilde u_W \) :
-
velocity of the lower plate, ms−1
- \(\tilde U\) :
-
mean flow velocity,\(\frac{1}{H}\int\limits_0^H {\left\langle {\tilde u_f } \right\rangle d\tilde y,ms^{ - 1} } \)
- \(\tilde x\) :
-
streamwise coordinate, m
- \(\tilde y\) :
-
transverse coordinate, m
- y :
-
dimensionless transverse coordinate,\({{\tilde y} \mathord{\left/ {\vphantom {{\tilde y} H}} \right. \kern-\nulldelimiterspace} H}\)
- δ:
-
minus gradient of the dimensionless velocity at the fixed wall,\(\left. { - \frac{{du}}{{dy}}} \right|_{y = 1} \)
- ε:
-
porosity
- γ:
-
constant,\(\sqrt {\frac{{\mu _{eff} }}{{\mu _f }}} \)
- μf :
-
fluid viscosity, kg m−1s−1
- μeff :
-
effective viscosity in the Brinkman term, kg m−1s−1
- ϱ:
-
density, kg m−3
- f :
-
fluid
- s :
-
solid
- 〈...〉:
-
volume average
- 〈...〉f :
-
intrinsic volume average
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Kuznetsov, A.V. Analytical investigation of heat transfer in Couette flow through a porous medium utilizing the Brinkman-Forchheimer-extended Darcy model. Acta Mechanica 129, 13–24 (1998). https://doi.org/10.1007/BF01379647
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DOI: https://doi.org/10.1007/BF01379647