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Impact of heat generation/absorption on transient natural convective flow in an annulus filled with porous material subject to isothermal and adiabatic boundaries

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Abstract

This article deliberated the effects of heat generation/absorption on transient natural convective flow in infinite vertical concentric cylinder filled with a porous material. The natural convective flow is as a result of constant heating at the inner cylinder while the inner surface of the outer cylinder is thermally insulated. A combination of Laplace transform technique and Riemann-sum approximation approach has been used to transform and invert the governing equation from the Laplace domain to the time domain respectively. The numerical values obtained from the Riemann-sum approximation excellently agree with the steady state solution at large time. The obtained results are represented graphically and the effects of the governing parameters on the velocity field, temperature field, mass flow rate, heat transfer as well as the skin-friction on both surfaces of the annulus are studied in detail. In the course of numerical computations, it is found that the fluid velocity and temperature both increases with time, while the fluid velocity and skin friction on both walls are seen to be proportional to Darcy number but inversely proportional to the viscosity ratio. Furthermore, it is found that the mass flow rate can be controlled by increasing/decreasing the heat generation/absorption parameter.

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Abbreviations

t′:

Dimensional time (s)

r′:

Dimensional radial coordinate (m)

u′:

Axial velocity (m/s)

U :

Dimensionless axial velocity

R :

Dimensionless radial coordinate

T 0 :

Ambient temperature (K)

T w :

Temperature of the hot cylinder (K)

θ :

Dimensionless temperature

r 1 :

Radius of the inner cylinder (m)

r 2 :

Radius of the outer cylinder (m)

g :

Gravitational acceleration (m/s2)

V :

Dimensionless mass flow rate

Nu :

Dimensionless nusselt number

H:

Dimensionless heat generating/absorbing parameter

Q 0 :

Dimensional heat generating/absorbing parameter (K)

Pr :

Prandtl number (μcp/k)

C p :

Specific heat at constant pressure (kJ/kgK)

t :

Dimensionless time

K :

Permeability of the Porous medium (m2)

k :

Thermal conductivity of the fluid (W/mK)

Da :

Darcy number

υ :

Fluid kinematic viscosity (m2/s)

υ eff :

Fluid kinematic effective viscosity (m2/s)

τ :

Skin friction

ρ :

Density (Pa s)

η :

Laplace parameter

λ :

Ratio of radii (r2/r1)

γ :

Viscosity ratio

β :

Coefficient of thermal expansion

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Authors and Affiliations

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

    Yusuf S. Taiwo & Gambo Dauda

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  1. Yusuf S. Taiwo
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  2. Gambo Dauda
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Correspondence to Yusuf S. Taiwo.

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Taiwo, Y.S., Dauda, G. Impact of heat generation/absorption on transient natural convective flow in an annulus filled with porous material subject to isothermal and adiabatic boundaries. Int J Geomath 10, 20 (2019). https://doi.org/10.1007/s13137-019-0132-8

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