Role of Heat Generation/Absorption on Mixed Convection Flow in a Vertical Tube Filled with Porous Material Having Time-Periodic Boundary Condition: Steady-Periodic Regime

Abstract

This paper discusses the hydrodynamic and thermal behaviours of a fully developed mixed convective flow of a viscous incompressible heat generating/absorbing fluid in a vertical tube filled with isotropic porous material having time-periodic boundary condition. The dimensionless governing equations of motion and energy subjected to the Boussinesq approximation and asymmetric heat conditions are solved analytically. Closed-form solutions are expressed in terms of modified Bessel function of first kind. The solutions obtained are graphically represented, and the effects of the dimensionless heat generation/absorption H, the dimensionless frequency \(\Omega \) and the Darcy number Da on the fluid flow are 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. The presence of the heat generation parameter is seen to enhance the temperature distribution and this is reflected as increase in the magnitude of the oscillation dimensionless velocity, whereas in the presence of heat absorption a reversed trend occurs.

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Abbreviations

A(t) :

Function of time

Da :

Darcy number

f :

Fanning friction factor

\(f_{a}\) :

Steady dimensionless fanning friction factor

\(f_{b}\) :

Oscillatory fanning friction factor

\(\mathbf g \) :

Gravitational acceleration

Gr :

Grashof number

H :

Dimensionless heat generation parameter

i :

Imaginary unit

\(I_0\) :

Modified Bessel function of first kind and order zero

\(I_1\) :

Modified Bessel function of first kind and order one

k :

Thermal conductivity

K :

Permeability of the porous medium

p :

Pressure

P :

Difference between the pressure and the hydrostatic pressure

Pr :

Prandtl number

q :

Heat flux per unit area

\( Q_0\) :

Dimensional heat generation coefficient

r :

Dimensionless radial coordinate

R :

Radial coordinate

\(R_{0}\) :

Radius of the tube

Re :

Reynolds number

\(\mathfrak {R}e\) :

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

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

Dimensionless complex-valued function in Eq. (27)

\(\mathbf U \) :

Fluid velocity

\(U_{0}\) :

Reference velocity

X :

Longitudinal coordinate

\(\alpha \) :

Thermal diffusivity

\(\beta \) :

Volumetric coefficient of thermal expansion

\(\Delta T\) :

Amplitude of the wall temperature oscillations

\(\gamma \) :

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

\(\lambda \) :

Dimensionless parameter

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

Dimensionless complex-valued function in Eq. (27)

\(\eta \) :

Dimensionless parameter

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

Dimensionless complex-valued function in Eq. (27)

\(\mu \) :

Dynamic viscosity

\(\nu \) :

Kinematic viscosity

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

Effective kinematic viscosity

\(\Phi \) :

Dimensionless heat flux

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

Dimensionless complex value function in Eq. (48)

\(\xi \) :

Dimensionless parameter

\(\varrho \) :

Density

\(\varrho _0\) :

Density at \(T=T_0\)

\(\tau _w\) :

Average wall shear stress

\(\omega \) :

Frequency of the wall temperature oscillation

\(\Omega \) :

Dimensionless frequency

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Correspondence to Deborah Daramola.

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Jha, B.K., Daramola, D. & Ajibade, A.O. Role of Heat Generation/Absorption on Mixed Convection Flow in a Vertical Tube Filled with Porous Material Having Time-Periodic Boundary Condition: Steady-Periodic Regime. Transp Porous Med 111, 681–699 (2016). https://doi.org/10.1007/s11242-015-0620-8

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Keywords

  • Tube
  • Mixed convection
  • Darcy
  • Heat generation/absorption
  • Porous material
  • Steady-periodic