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Fully Developed Mixed Convection Flow in a Vertical Channel Containing Porous and Fluid Layer with Isothermal or Isoflux Boundaries

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Abstract

An analysis of fully developed combined free and forced convective flow in a fluid saturated porous medium channel bounded by two vertical parallel plates is presented. The flow is modeled using Brinkman equation model. The viscous and Darcy dissipation terms are also included in the energy equation. Three types of thermal boundary conditions such as isothermal–isothermal, isoflux–isothermal, and isothermal–isoflux for the left–right walls of the channel are considered. Analytical solutions for the governing ordinary differential equations are obtained by perturbation series method. In addition, closed form expressions for the Nusselt number at both the left and right channel walls are derived. Results have been presented for a wide range of governing parameters such as porous parameter, ratio of Grashof number and Reynolds number, viscosity ratio, width ratio, and conductivity ratio on velocity, and temperature fields. It is found that the presence of porous matrix in one of the region reduces the velocity and temperature.

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Abbreviations

A :

Constant defined in Eq. (8)

b :

Thermal expansion coefficient ratio, β 2/β 1

Br :

Brinkman number, \({\mu _1 U_0^{(1)^{2}} /k_1 \Delta T}\)

C p :

Specific heat at constant pressure

Da :

Darcy number based on h 1, \({K/h_1^2}\)

g :

Acceleration due to gravity

Gr :

Grashoff number, \({g \beta _1 h_1^3 \Delta T/\upsilon_1^2}\)

GR :

Dimensionless parameter, Gr/Re

h :

Width ratio, h 2/h 1

h 1 :

Height of the region-I

h 2 :

Height of the region-II

K :

Permeability of the porous media

k 1 :

Thermal conductivity of the fluid in region-I

k 2 :

Thermal conductivity of the fluid in region-II

m :

Viscosities ratio, μ 1/μ 2

n :

Densities ratio,ρ 2/ρ 1

p :

Dimensional pressure

Re :

Reynolds number, \({U_0^{(1)} h_1/\upsilon_1}\)

T :

Temperature

\({U_0^{\left(i\right)}}\) :

Reference velocity, \({- (dP/dX)(h_i^2 /48 \mu _i )}\)

u :

Velocity

T1 ,T2:

Temperature of the boundaries

X, Y:

Space co-ordinates

α :

Thermal diffusivity

β :

Coefficient of thermal expansion

\({\varepsilon}\) :

Dimensionless parameter, (Gr/Re) Br

κ :

Thermal conductivities ratio, k 1 /k 2

μ :

Viscosity

ν :

Kinematic viscosity

θ i :

Nondimensional temperature, (T i − T 0)/ΔT

ρ :

Density

σ :

Permeability parameter, \({h_1/\sqrt{K}}\)

ΔT :

Difference in temperature, (T 2 − T 1)

1 & 2:

Reference quantities for region-I and II respectively

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Authors and Affiliations

  1. Department of Mathematics, Gulbarga University, Gulbarga, 585106, Karnataka, India

    J. Prathap Kumar, J. C. Umavathi & Basavaraj M. Biradar

  2. Department of Civil Engineering, National Chi Nan University, University Rd, Puli Nantou, 545, Taiwan, ROC

    J. C. Umavathi

  3. Faculty of Mathematics, University of Cluj, 3400, Cluj, CP 253, Romania

    I. Pop

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  1. J. Prathap Kumar
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  2. J. C. Umavathi
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  3. I. Pop
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  4. Basavaraj M. Biradar
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Correspondence to J. C. Umavathi.

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Kumar, J.P., Umavathi, J.C., Pop, I. et al. Fully Developed Mixed Convection Flow in a Vertical Channel Containing Porous and Fluid Layer with Isothermal or Isoflux Boundaries. Transp Porous Med 80, 117–135 (2009). https://doi.org/10.1007/s11242-009-9347-8

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