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Mixed convection in a vertical annulus filled with porous material having time-periodic thermal boundary condition: steady-periodic regime

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Abstract

An analytical solution for the problem of a fully developed laminar mixed convection flow in a vertical annulus filled with porous material is presented, where the outer surface of the inner cylinder is heated sinusoidally and the inner surface of outer cylinder is kept at a constant temperature. The solution is based on the assumption that both the velocity and temperature profile far from the annulus entrance do not change due to infinite length of the cylinders forming the annulus. Perturbation method is employed to solve the momentum and energy equations, the graphical interpretation reveals that the effect of the dimensionless frequency Ω of the sinusoidal wall heating on the absolute value of oscillatory term of the velocity is negligible in the range 0.5 ≤ Ω ≤ 2 for air.

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Abbreviations

a :

Radius of the inner cylinder

A(t):

Function of time

b :

Radius of the outer cylinder

Da :

Darcy number

f :

Fanning friction factor

g :

Gravitational acceleration

Gr :

Grashof number

i :

Imaginary unit, \(\sqrt { - 1}\)

I n :

Modified Bessel function of first kind of order n

K n :

Modified Bessel function of second kind of order n

k :

Thermal conductivity

K :

Permeability of the porous medium

n :

Integer

p :

Pressure

P :

Difference between the pressure and the hydrostatic pressure

Pr :

Prandtl number

q :

Heat flux per unit area

r :

Dimensionless radial coordinate

R :

Dimensional radial coordinate

Re :

Reynolds number

ℜe :

Real part of a complex number

t :

Time

T :

Temperature

T 0 :

Reference temperature in the annular section defined in Eq. (5)

T 1 :

Time average temperature of the outer surface of the inner cylinder

u :

Dimensionless velocity

u *, u * a , u * b :

Dimensionless complex-valued function defined in Eqs. (19) and (23)

\(\overrightarrow {U}\) :

Fluid velocity

U :

X-Component of the fluid velocity

U o :

Mean fluid velocity in an annular section

X :

Vertical axial coordinate

α :

Effective thermal diffusivity

β :

Volumetric coefficient of thermal expansion

ΔT :

Amplitude of the wall temperature oscillations

λ :

Dimensionless parameter defined in Eq. (10)

λ * a , λ * b :

Dimensionless complex-valued function defined in Eq. (23)

η :

Dimensionless complex-valued function defined in Eq. (10)

θ :

Dimensionless temperature

θ * a , θ * b :

Dimensionless complex-valued function

μ :

Dynamic viscosity

ν :

Kinematic viscosity

ν eff :

Effective kinematic viscosity

ξ :

Dimensionless parameter defined in Eq. (10)

ρ :

Density

ρ 0 :

Density at T = T 0

τ w :

Average wall shear stress defined in Eq. (40)

ω :

Frequency of the wall temperature oscillation

Ω:

Dimensionless frequency

γ :

Ratio of kinematics viscosity defined in Eq. (10)

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Authors and Affiliations

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

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

Authors
  1. Basant K. Jha
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  2. Deborah Daramola
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  3. Abiodun O. Ajibade
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Corresponding author

Correspondence to Deborah Daramola.

Appendix

Appendix

$$D_{1} = \frac{1}{{\sqrt {\gamma Da} }}\quad D_{2} = \sqrt {i\Omega \Pr } ,\quad D_{3} = \sqrt {\left( {i\Omega + D_{1}^{2} } \right)}$$
$$Z_{1} = \left( {\lambda_{0} I_{1} \left( {\lambda_{0} D_{1} } \right) - I_{1} \left( {D_{1} } \right)} \right)\left( {K_{0} \left( {\lambda_{0} D_{1} } \right) - K_{0} \left( {D_{1} } \right)} \right) - \left( {I_{0} \left( {D_{1} } \right) - I_{0} \left( {\lambda_{0} D_{1} } \right)} \right)\left( {\lambda_{0} K_{1} \left( {\lambda_{0} D_{1} } \right) - K_{1} \left( {D_{1} } \right)} \right)$$
$$Z_{2} = \frac{{I_{0} \left( {\chi_{0} D_{2} } \right)\left( {\chi_{0} K_{1} \left( {\chi_{0} D_{2} } \right) - K_{1} \left( {D_{2} } \right)} \right) + K_{0} \left( {\chi_{0} D_{2} } \right)\left( {\chi_{0} I_{1} \left( {\chi_{0} D_{2} } \right) - I_{1} \left( {D_{2} } \right)} \right)}}{{\left( {\left( {D_{3}^{2} - D_{2}^{2} } \right)} \right)D_{2} \left( {I_{0} \left( {\chi_{0} D_{2} } \right)K_{0} (D_{2} ) - I_{0} \left( {D_{2} } \right)K_{0} (\chi_{0} D_{2} )} \right)}}$$
$$Z_{3} = \frac{{\left( {I_{0} \left( {\chi_{0} D_{3} } \right)K_{0} (D_{3} ) - I_{0} \left( {D_{3} } \right)K_{0} (\chi_{0} D_{3} )} \right)}}{{D_{3}^{2} \left( {\chi_{0}^{2} - 1} \right)\left( {I_{0} \left( {\chi_{0} D_{3} } \right)K_{0} (D_{3} ) - I_{0} \left( {D_{3} } \right)K_{0} (\chi_{0} D_{3} )} \right) + 2\left( {I_{1} \left( {D_{3} } \right) - \chi_{0} I_{1} (\chi_{0} D_{3} )} \right)\left( {K_{0} \left( {D_{3} } \right) - K_{0} (\chi_{0} D_{3} )} \right)}}$$
$$Z_{4\,} \,\, = \left( {\,D_{2}^{2} - D_{3}^{2} } \right)$$
$$Z_{5} \, = \frac{{I_{0} \left( {\chi_{0} \,D_{3} } \right)\left( {\chi_{0} \,K_{1} \left( {\chi_{0} \,D_{3} } \right)\, - K_{1} \left( {\,D_{3} } \right)\,} \right)\, + K_{0} \left( {\chi_{0} \,D_{3} } \right)\left( {\chi_{0} \,I_{1} \left( {\chi_{0} \,D_{3} } \right)\, - I_{1} \left( {\,D_{3} } \right)\,} \right)\,}}{{Z_{4} \left( {\left( {D_{3}^{2} \left( {\chi_{0}^{2} - 1} \right)\left( {I_{0} \left( {\chi_{0} \,D_{3} } \right)K_{0} (D_{3} )\, - \,I_{0} \left( {D_{3} } \right)K_{0} (\chi_{0} \,D_{3} )} \right)\, + 2\left( {I_{1} \left( {D_{3} } \right) - \chi_{0} I_{1} (\chi_{0} \,D_{3} )} \right)\left( {K_{0} \left( {D_{3} } \right) - K_{0} (\chi_{0} \,D_{3} )} \right)} \right)} \right)}}\,\,\,$$

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Jha, B.K., Daramola, D. & Ajibade, A.O. Mixed convection in a vertical annulus filled with porous material having time-periodic thermal boundary condition: steady-periodic regime. Meccanica 51, 1685–1698 (2016). https://doi.org/10.1007/s11012-015-0328-4

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