Abstract
The governing equation for DarcyForchheimer flow of nonNewtonian inelastic powerlaw fluid through porous media has been derived from first principles. Using this equation, the problem of DarcyForchheimer natural, forced, and mixed convection within the porous media saturated with a powerlaw fluid has been solved using the approximate integral method. It is observed that a similarity solution exists specifically for only the case of an isothermal vertical flat plate embedded in the porous media. The results based on the approximate method, when compared with existing exact solutions show an agreement of within a maximum error bound of 2.5%.
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Abbreviations
 A :

crosssectional area
 b _{i} :

coefficient in the chosen temperature profile
 B _{1} :

coefficient in the profile for the dimensionless boundary layer thickness
 C :

coefficient in the modified Forchheimer term for powerlaw fluids
 C _{1} :

coefficient in the Oseen approximation which depends essentially on pore geometry
 C _{i} :

coefficient depending essentially on pore geometry
 C _{D} :

drag coefficient
 C _{t} :

coefficient in the expression forK ^{*}
 d :

particle diameter (for irregular shaped particles, it is characteristic length for averagesize particle)
 f _{p} :

resistance or drag on a single particle
 F _{R} :

total resistance to flow offered byN particles in the porous media
 g :

acceleration due to gravity
 g _{x} :

component of the acceleration due to gravity in thexdirection
 \(Gr_{K^* }\) :

Grashof number based on permeability for powerlaw fluids
 K :

intrinsic permeability of the porous media
 K ^{*} :

modified permeability of the porous media for flow of powerlaw fluids
 l _{c} :

characteristic length
 m :

exponent in the gravity field
 n :

powerlaw index of the inelastic nonNewtonian fluid
 N :

total number of particles
 Nux,_{D,F} :

local Nusselt number for Darcy forced convection flow
 Nux,_{DF,F} :

local Nusselt number for DarcyForchheimer forced convection flow
 Nux,_{D,M} :

local Nusselt number for Darcy mixed convection flow
 Nux,_{DF,M} :

local Nusselt number for DarcyForchheimer mixed convection flow
 Nux,_{D,N} :

local Nusselt number for Darcy natural convection flow
 Nux,_{DF,N} :

local Nusselt number for DarcyForchheimer natural convection flow
 \(\bar p\) :

pressure
 p :

exponent in the wall temperature variation
 _{Pe} c :

characteristic Péclet number
 _{Pe} x :

local Péclet number for forced convection flow
 _{Pe′} x :

modified local Péclet number for mixed convection flow
 _{Ra} c :

characteristic Rayleigh number
 _{Ra} x :

local Rayleigh number for Darcy natural convection flow
 _{Ra′} x :

local Rayleigh number for DarcyForchheimer natural convection flow
 Re:

convectional Reynolds number for powerlaw fluids
 \(\operatorname{Re} _{K^* }\) :

Reynolds number based on permeability for powerlaw fluids
 T :

temperature
 T _{e} :

ambient constant temperature
 T w,_{ref} :

constant reference wall surface temperature
 T _{w}(X):

variable wall surface temperature
 δT _{w} :

temperature difference equal toT w,_{ref}−T _{e}
 T _{1} :

term in the DarcyForchheimer natural convection regime for Newtonian fluids
 T _{2} :

term in the DarcyForchheimer natural convection regime for nonNewtonian fluids (first approximation)
 T _{N} :

term in the Darcy/Forchheimer natural convection regime for nonNewtonian fluids (second approximation)
 u :

Darcian or superficial velocity
 u _{1} :

dimensionless velocity profile
 u _{e} :

external forced convection flow velocity
 u _{s} :

seepage velocity (local average velocity of flow around the particle)
 u _{w} :

wall slip velocity
 U c _{M} :

characteristic velocity for mixed convection
 U c _{N} :

characteristic velocity for natural convection
 x, y :

boundarylayer coordinates
 x _{1},y _{1} :

dimensionless boundary layer coordinates
 X :

coefficient which is a function of flow behaviour indexn for powerlaw fluids
 α :

effective thermal diffusivity of the porous medium
 α′:

shape factor which takes a value ofΜ/4 for spheres
 Β′:

shape factor which takes a value ofΜ/6 for spheres
 Β _{0} :

expansion coefficient of the fluid
 δ _{T} :

boundarylayer thickness
 δ T _{1} :

dimensionless boundary layer thickness
 ε :

porosity of the medium
 η :

similarity variable
 θ :

dimensionless temperature difference
 λ′:

coefficient which is a function of the geometry of the porous media (it takes a value of 3Μ for a single sphere in an infinite fluid)
 Μ _{0} :

viscosity of Newtonian fluid
 Μ ^{*} :

fluid consistency of the inelastic nonNewtonian powerlaw fluid
 ξ :

constant equal toX(2ε ^{2−n} λ′)/α′
 ρ :

density of the fluid
 Φ :

dimensionless wall temperature difference
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Shenoy, A.V. DarcyForchheimer natural, forced and mixed convection heat transfer in nonNewtonian powerlaw fluidsaturated porous media. Transp Porous Med 11, 219–241 (1993). https://doi.org/10.1007/BF00614813
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DOI: https://doi.org/10.1007/BF00614813