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Mixed convection flow in a vertical tube filled with porous material with time-periodic boundary condition: steady-periodic regime

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Abstract

An analytical study is reported on the hydrodynamic and thermal behaviour of a fully developed mixed convective flow in a vertical tube filled with isotropic porous material having time-periodic boundary condition. The analysis is performed for fully developed parallel flow and steady-periodic regime. The momentum and energy equations presented in dimensionless form along with the constraint equations for the present physical situation are solved exactly. Closed form solution are expressed in terms of modified Bessel function of first kind. The solution obtained is graphically represented and the effect of the Prandtl number \(Pr\), the dimensionless frequency \(\Omega \), and the Darcy number \(Da\) on the flow is investigated. It is discovered that velocity is maximum at two different locations in the flow domain, one near the surface of the tube and another at the axis of the tube.

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Abbreviations

\(A(t)\) :

Function of time

\(Da\) :

Darcy number

\(f\) :

Fanning friction factor

\(g\) :

Gravitational acceleration

\(Gr\) :

Grashof number

\(i\) :

Imaginary unit

\(I_n\) :

Modified Bessel function of first kind and order n.

\(k\) :

Thermal conductivity

\(K\) :

Permeability of the medium

\(n\) :

Integer number

\(p\) :

Pressure

\(P\) :

Difference between the pressure and the hydrostatic pressure

\(Pr\) :

Prandtl number

\(R\) :

Radial coordinate

\(Re\) :

Reynolds number

\(\mathfrak {R}\) :

Real part of a complex number

\(t\) :

Time

\(T\) :

Temperature

\(T_0\) :

Mean temperature in a duct section

\(T_1\) :

Mean wall temperature

\(u\) :

Dimensionless velocity

\(u^*\) :

Dimensionless complex-valued function

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

Dimensionless complex-valued function

\(U\) :

Fluid velocity

\(X \) :

Longitudinal coordinate

\(\propto \) :

Thermal diffusivity

\(\beta \) :

Volumetric coefficient of thermal expansion

\(\Delta T\) :

Amplitude of the wall temperature oscillations

\(\Gamma \) :

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

\(\lambda \) :

Dimensionless parameter

\(\lambda ^*\) :

Dimensionless complex-valued function

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

Dimensionless complex-valued function

\(\eta \) :

Dimensionless parameter

\(\theta \) :

Dimensionless temperature

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

Dimensionless complex-valued function

\(\mu \) :

Dynamic viscosity

\(\nu \) :

Kinematic viscosity

\(\nu _{eff}\) :

Effective kinematic viscosity

\(\Phi \) :

Dimensionless heat flux

\(\Phi _a^*,\phi _b^*\) :

Dimensionless complex value function

\(\varrho \) :

Mass density

\(\varrho _0\) :

Mass density for \(T=T_0\)

\(\tau _w\) :

Average wall shear stress

\(\omega \) :

Frequency of the wall temperature oscillation

\(\Omega \) :

Dimensionless frequency

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Authors and Affiliations

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

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

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

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Jha, B.K., Ajibade, A.O. & Daramola, D. Mixed convection flow in a vertical tube filled with porous material with time-periodic boundary condition: steady-periodic regime. Afr. Mat. 26, 529–543 (2015). https://doi.org/10.1007/s13370-013-0222-y

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