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Natural Convection on a Horizontal Cone in a Porous Medium with Non-Uniform Wall Temperature/Concentration or Heat/Mass Flux and Suction/Injection

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Abstract

The steady natural convection flow on a horizontal cone embedded in a saturated porous medium with non-uniform wall temperature/concentration or heat/mass flux and suction/injection has been investigated. Non-similar solutions have been obtained. The nonlinear coupled differential equations under boundary layer approximations governing the flow have been numerically solved. The Nusselt and Sherwood numbers are found to depend on the buoyancy forces, suction/injection rates, variation of wall temperature/concentration or heat/mass flux, Lewis number and the non-Darcy parameter.

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Abbreviations

C :

Species mass fraction or concentration

C 0 :

Species wall mass fraction at x = L

D :

Binary diffusion coefficient

Er x :

Ergun number

f :

Dimensionless stream function

f′:

Dimensionless velocity

g :

Gravitational acceleration

k :

Effective thermal conductivity of the saturated porous medium

K :

Permeability of the porous medium

K * :

Inertial coefficient of the Ergun equation

L :

Characteristic length

Le :

Lewis number

m :

Mass flux of diffusing species

m 0 :

Mass flux of diffusing species at the wall when x = L

M :

Index in the power-law variation of wall temperature/concentration or wall heat/mass flux with the distance x

N :

Index in the power-law variation of surface mass transfer (suction/injection) with the distance x

Nu x :

Local Nusselt number for the PWT/PWC case

\({Nu^{*}_{x}}\) :

Local Nusselt number for the PHF/PMF case

Pr :

Prandtl number

q w :

Local surface heat transfer

q 0 :

Local surface heat transfer at x = L

r :

Local radius of the circular cone

Ra x :

Local Rayleigh number for the PWT/PWC case

\({Ra^{*}_{x}}\) :

Local Rayleigh number for the PHF/PMF case

S, S*:

Ratio of the buoyancy force due to the mass diffusion to the buoyancy force due to the thermal diffusion for the PWT/PWC and PHF/PMF cases, respectively

Sc :

Schmidt number

Sh x :

Local Sherwood number for the PWT/PWC case

\({Sh^{*}_{x}}\) :

Local Sherwood number for the PHF/PMF case

T :

Temperature of the fluid

T 0 :

Wall temperature at x = L

u, v:

Velocity components in x and y directions, respectively

x :

Coordinate along cone ray

y :

Coordinate normal to the surface of the cone

α :

Effective thermal diffusivity of saturated porous medium

β :

Volumetric coefficient of thermal expansion

β * :

Volumetric coefficient of expansion with mass fraction

η, ξ :

Transformed coordinates

η ∞ :

Dimensionless boundary layer thickness

θ :

Dimensionless temperature

λ, λ * :

Non-Darcy parameters for PWT/PWC and PHF/PMF cases, respectively

μ :

Dynamic viscosity of the fluid

ν :

Kinematic viscosity of the fluid

ρ :

Fluid density

\({\phi}\) :

Dimensionless mass fraction

Ω:

Semi-vertical angle of the cone

w :

Condition at the wall

∞:

Ambient condition

′:

Prime denotes derivative with respect to η

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India

    M. Kumari

  2. C/o Dr. S.K. Sinha, KNIT Campus, Type IV/17, KNIT, Sultanpur, 228118, India

    G. Nath

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  1. M. Kumari
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  2. G. Nath
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Correspondence to M. Kumari.

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Kumari, M., Nath, G. Natural Convection on a Horizontal Cone in a Porous Medium with Non-Uniform Wall Temperature/Concentration or Heat/Mass Flux and Suction/Injection. Transp Porous Med 84, 275–284 (2010). https://doi.org/10.1007/s11242-009-9499-6

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