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Mixed Convection in an Inclined Channel Filled with Porous Material Having Time-Periodic Boundary Conditions: Steady-Periodic Regime

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Abstract

This paper reports an investigation of the hydrodynamic and thermal behaviour in a steady-periodic regime of a fully developed laminar mixed convection flow in an inclined channel filled with porous material. One of the channel walls is kept at a constant temperature, while the other is heated sinusoidally. The flow formation inside the porous media is modelled using Darcy–Brinkman model. The resulting governing dimensionless momentum and energy equations are separated into steady and periodic parts and solved analytically by the method of undetermined coefficients. In order to see the effect of the governing parameters on the thermal and hydrodynamic behaviour of the fluid flow, the results are depicted pictorially. These are seen to depend strongly on the dimensionless frequency of the periodic heating, Darcy number and the Prandtl number of the working fluid. The result is also seen to be in strong agreement with the existing analytical work, when the Darcy number is large. For sufficiently high Prandtl number, it is observed that the amplitude of the friction factor oscillation is maximised at a resonance frequency at the wall where there is periodic heating; also increasing Darcy number (Da) increases the amplitude of the friction factor oscillation, while it reduces the resonance frequency.

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Abbreviations

A, B :

Functions defined by Eq. (7)

D :

Hydraulic diameter, 4L

Da :

Darcy number

\(f_{1}\) :

Fanning friction factor at the wall \(Y=-L\)

\(f_{2}\) :

Fanning friction factor at the wall \(Y=L\)

\(f_{1a}^*,\,f_{1b}^*\) :

Complex fanning friction factor defined in Eq. (51)

\(f_{2a}^*,\,f_{2b}^*\) :

Complex fanning friction factor defined in Eq. (52)

\(\mathbf{g}\) :

Gravitational acceleration

g :

Magnitude of the gravitational acceleration

Gr :

Grashof number

i :

Imaginary unit

k :

Thermal conductivity

K :

Permeability of the medium

L :

Channel half width

p :

Pressure

P :

Difference between the pressure and the hydrostatic pressure

Pr :

Prandtl number

q :

Heat flux per unit area

Re :

Reynolds number

\(\mathfrak {R}e\) :

Real part of a complex number

t :

Time

T :

Temperature

\(T_0\) :

Average temperature in a channel section

\(T_1\) :

Temperature of the wall \(Y=-L\)

\(T_2\) :

Time average temperature of the wall \(Y=L\)

u :

Dimensionless velocity

\(u^*,\,u_a^*,\,u_b^*\) :

Dimensionless complex-valued function

U :

Fluid velocity

\(U_{0}\) :

Average velocity

X, Y :

Rectangular Cartesian coordinates

y :

Dimensionless coordinate

\(\alpha \) :

Thermal diffusivity

\(\beta \) :

Volumetric coefficient of thermal expansion

\(\Delta \) :

Amplitude of the wall temperature oscillations

\(\gamma \) :

\(=\dfrac{\nu _\mathrm{eff}}{\nu }\)

\(\lambda \) :

Dimensionless parameter

\(\lambda ^*,\,\lambda _a^*,\,\lambda _b^*\) :

Dimensionless complex-valued function

\(\eta \) :

Dimensionless parameter

\(\theta ^*,\,\theta _a^*,\,\theta _b^*\) :

Dimensionless complex-valued function

\(\mu \) :

Dynamic viscosity

\(\nu \) :

Kinematic viscosity

\(\nu _\mathrm{eff}\) :

Effective kinematic viscosity

\(\varPhi \) :

Dimensionless heat flux

\(\varPhi _a^*,\,\varPhi _b^*\) :

Dimensionless complex value function

\(\xi \) :

Dimensionless parameter defined in Eq. (11)

\(\chi \) :

Dimensionless parameter defined in Eq. (11)

\(\varphi \) :

Tilt angle

\(\varrho \) :

Mass density

\(\varrho _0\) :

Mass density for \(T=T_0\)

\(\tau _w\) :

Average wall shear stress

\(\omega \) :

Frequency of the wall temperature oscillation

\(\varOmega \) :

Dimensionless frequency

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Authors and Affiliations

  1. Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria

    Basant K. Jha, Deborah Daramola & Abiodun O. Ajibade

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  1. Basant K. Jha
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  2. Deborah Daramola
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  3. Abiodun O. Ajibade
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Correspondence to Deborah Daramola.

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Jha, B.K., Daramola, D. & Ajibade, A.O. Mixed Convection in an Inclined Channel Filled with Porous Material Having Time-Periodic Boundary Conditions: Steady-Periodic Regime. Transp Porous Med 109, 495–512 (2015). https://doi.org/10.1007/s11242-015-0533-6

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