Wärme - und Stoffübertragung

, Volume 29, Issue 3, pp 185–193 | Cite as

Natural convection of non-Newtonian fluids in a horizontal porous layer

  • B. Amari
  • P. Vasseur
  • E. Bilgen


The buoyancy-induced flows of non-Newtonian fluids in a horizontal fluid saturated porous layer is studied analytically and numerically using the power-law model to characterize the non-Newtonian fluid behavior. A constant heat flux is applied for heating and cooling the two opposing walls of the layer while the other two walls are insulated. On the basis of a modified Darcy equation the problem is solved analytically, in the limit of a thin layer, using a parallel flow approximation and an integral form of the energy equation. Solutions for the flow and temperature fields, and Nusselt numbers are obtained explicitly in terms of the modified Rayleigh numberR and the power-law indexn. A numerical study of the same phenomenon, obtained by solving the complete system of governing equations, is also conducted. A good agreement is found between the analytical prediction and the numerical simulation.


Heat Flux Nusselt Number Natural Convection Porous Layer Parallel Flow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



aspect ratio of the cavity,L′/H′

a, b

bottom heating:a=1,b=0; side wall heating:a=0,b=1


dimensionless temperature gradient along thex direction


acceleration due to gravity


the consistency index


thickness of the cavity


thermal conductivity of fluid-saturated porous medium


permeability of the porous medium


length of the cavity


power-law index


Nusselt number, Eq. (27)




constant heat flux


modified Rayleigh number,KϱgβΔT′(H′/α) n /ε


Rayleigh number for a Newtonian-fluid,KϱgβΔT′H′/αμ




reference temperature at the geometric center of the cavity

\(\vec V'\)

superficial velocity


superficial velocity in thex′ direction


superficial velocity in they′ direction


horizontal coordinate


vertical coordinate

Greek symbols


effective thermal diffusivity


coefficient of thermal expansion


dimensionless temperature, Eq. (15)


dynamic viscosity for a Newtonian fluid


apparent viscosity for a power-law fluid, Eq. (2)




stream function


porosity of the porous medium


parameter in power-law model, Eq. (3)


dimensional quantities



maximum value


condition at the origin of the coordinate system

Natürliche Konvektion nicht-Newtonscher Fluide in einer horizontalen porösen Schicht


Die durch Auftriebskräfte induzierten Strömmungen nicht-Newtonscher Fluide in einer horizontalen, flüssigkeitsgetränkten porösen Schicht werden analytisch und numerisch unter Verwendung des Potenzgesetz-Modells für das nicht-Newtonsche Fluidverhalten untersucht. Von einer beheizten Wand fließt ein konstanter Wärmestrom an die gekühlte gegenüberliegende, während die beiden übrigen Wände adiabat sind. Das Problem wird auf der Basis einer modifizierten Darcy-Gleichung analytisch gelöst, und zwar für den Grenzfall geringer Schichtdicke, wobei näherungsweise Parallelströmung unterstellt wird und die Energiegleichung in Integralform Verwendung findet. Lösungen für das Strömungs- und Temperaturfeld sowie Nusselt-Zahlen werden in expliziter Form als Funktionen der modifizierten Rayleigh-ZahlR und des Potenzgesetz-Indexn erhalten. Ferner wird eine numerische Untersuchung des gleichen Problems unter Lösung des vollständigen Systems der bestimmenden Gleichungen durchgeführt. Analytische Vorausberechnung und numerische Simulation stimmen gut überein.


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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • B. Amari
    • 1
  • P. Vasseur
    • 1
  • E. Bilgen
    • 1
  1. 1.Ecole PolytechniqueMontrealCanada

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