Afr. Mat. DOI 10.1007/s13370-013-0222-y

Mixed convection flow in a vertical tube filled with porous material with time-periodic boundary condition: steady-periodic regime Basant K. Jha · Abiodun O. Ajibade · Deborah Daramola

Received: 18 June 2013 / Accepted: 23 December 2013 © African Mathematical Union and Springer-Verlag Berlin Heidelberg 2014

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 , 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. Keywords

Tube · Mixed convection · Darcy number

Mathematics Subject Classification

80A20

List of symbols A(t) Da f g Gr

Function of time Darcy number Fanning friction factor Gravitational acceleration Grashof number

B. K. Jha · A. O. Ajibade · D. Daramola (B) Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria e-mail: [email protected] B. K. Jha e-mail: [email protected] A. O. Ajibade e-mail: [email protected]

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i In k K n p P Pr R Re  t T T0 T1 u u∗ u a∗ , u ∗b U X

Imaginary unit Modified Bessel function of first kind and order n. Thermal conductivity Permeability of the medium Integer number Pressure Difference between the pressure and the hydrostatic pressure Prandtl number Radial coordinate Reynolds number Real part of a complex number Time Temperature Mean temperature in a duct section Mean wall temperature Dimensionless velocity Dimensionless complex-valued function Dimensionless complex-valued function Fluid velocity Longitudinal coordinate

Greek symbols ∝ β T  λ λ∗ λa∗ , λ∗b η θ θa∗ , θb∗ μ ν νe f f

a∗ , φb∗ 0 τw ω 

Thermal diffusivity Volumetric coefficient of thermal expansion Amplitude of the wall temperature oscillations νe f f = ν Dimensionless parameter Dimensionless complex-valued function Dimensionless complex-valued function Dimensionless parameter Dimensionless temperature Dimensionless complex-valued function Dynamic viscosity Kinematic viscosity Effective kinematic viscosity Dimensionless heat flux Dimensionless complex value function Mass density Mass density for T = T0 Average wall shear stress Frequency of the wall temperature oscillation Dimensionless frequency

1 Introduction Heat transfer in channels with oscillatory surface temperature occurs in many industrial, engineering processes and natural phenomenon. Most of interest in this subject is due to its

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practical applications such as the automatic control systems, electrical and electronic component frequently subjected to periodic heating. Among the foundational works on effect of periodic heating on flows was the work of [1] that considered a case in which the surface temperature varies slightly about a mean level, which is higher than the ambient temperature. Perturbation analysis in term of amplitude of surface temperature variation was performed. The same problem was solved by [2] using a slightly different perturbation expansion their results were restricted to small amplitudes. Nanda and Sharma [3] studied the effect of sinusoidal variation in temperature surface. They circumvented the limitation of the result for only small amplitude by separating the temperature and velocity into steady and oscillatory components. The steady-periodic natural convection in a square enclosure was solved numerically with both the upper and lower walls insulated and the left vertical wall kept at a constant temperature. Bar-Cohen and Rohsenow [4] studied the fully developed convection between two periodically heated parallel plates. Wang [5] investigated the effect of Strouhal number on the development of boundary layers in a vertical channel whose boundary are subjected to periodic heating and periodic heat flux. Kazmierczak and Chinoda [6] and Kwak et al. [7] considered a boundary conditions for the right vertical wall having a uniform temperature which varies in time with a sinusoidal law while a uniform heat flux which varies periodically in time with square wave pulses for the right vertical wall were considered by [8,9]. Kwak et al. [7], Lage and Bejan [9] and Antohe and Lage [8] predicted a resonance phenomenon; the heat flux through a vertical surface which fluctuates with an amplitude that for fixed values of the other parameters reach a maximum for a given value of the amplitude frequency called resonance frequency. Result obtained in [7] shows that a large amplitude wall temperature oscillation causes an increase in the time of average heat transfer rate and that increase is maximum at a resonance frequency. On the other hand, [10] studied analytically the time-periodic laminar mixed convection in an inclined channel with the temperature of one wall constant and the other wall a sinusoidal function of time and ignoring viscous dissipation. It is found that for every Prandtl number greater than 0.277, there exist a resonance frequency that maximizes the amplitude of the friction factor oscillations at the unsteady temperature wall. In another article, [11] studied mixed convection flow in a vertical circular duct with time-periodic boundary conditions: steady-periodic regime, it was also found that there exists resonance frequency for which the velocity oscillation reaches a maximum and this depends on the value assumed by the Prandtl number and the radial position. The resonance frequency which maximizes the amplitude of the friction factor is a decreasing function of Pr. Nawaf [12] investigated the free convection flow in a square porous cavity with an oscillating wall temperature, it is found that when the hot wall temperature oscillates with high amplitude of frequency, the Nusselt number becomes negative over part of period for Rayleigh number 103 . Arash et al. [13] investigated the periodic mixed convection of a nanofluid in a cavity with top lid sinusoidal motion, it is found that higher value of Richardson number, Ri corresponds to a lower value of the amplitude of Nusselt number, N u in the steady-periodic state. Convective heat transfer in fluid-saturated porous media has received great attention in recent years. This is due to its various applications, for example, in packed sphere bed, in high performance insulation for buildings, in chemical catalytic reactors and in grain storage. Much of these can be found in recent works by [14–17]. There are previous work that investigated convective flow in the annular geometry filled with porous material. Kou and Lu [18] studied mixed convection in a vertical channel embedded in a porous media with asymmetric wall heat flux, it was found that reverse flow depends on the value of Gr/Re. Jha [19] analytically studied free-convection flow through an annular porous medium, it is found that skin friction is Darcy number dependent. Rossi di Schio et al. [24] have analytically studied buoyant flow

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in a vertical fluid saturated porous annulus using the Brinkman model, it is found that reversal flow may occur at both the inner and the outer boundary of the duct. The purpose of the present work is to extend the work of [11] to study the impact of porous material on mixed convection flow in a vertical tube having time periodic boundary condition on the surface of the tube. Analytical solutions of the momentum and energy are derived in terms of modified Bessel’s function of first kind. These solution generally deserve great attention, since it allows one to gain a deeper knowledge of the underlying physical situation. Moreover, it provides the possibility to get a bench mark for numerical solvers with reference to basic flow configurations.

2 Governing equations Considering a fully developed mixed convection flow caused by pressure gradient and a buoyancy force in vertical tube, filled with porous material having a periodic variation of temperature with time. The flow is assumed to be fully developed laminar such that fluid − → velocity vector U has the only non-vanishing component U along the X -axis. The X -axis is the axial coordinate which is parallel to the gravitational acceleration g but with opposite direction while the R-axis is the axis in the radial direction (see Fig.1 below). The porous medium is assumed to be isotropic and homogenous.The thermophysical properties of the solid matrix and the fluid are also assumed to be constant except the density variation in the momentum equations. The effect of viscous dissipation in the fluid is neglected, (see Refs. [20–23]). Since only the axial component of U is non vanishing, the mass balance equation ensures that ∂U ∂ X = 0, i.e U = U (R, t). Assuming the wall at R = R0 is kept at an oscillating temperature with time namely T (X, R0 , t) = T1 + T cos(ωt).

Fig. 1 Schematic diagram of the flow

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(1)

Mixed convection flow in a vertical tube

Moreover, since the thermal boundary condition (1) does not yield any net fluid heating or cooling, heat transfer occurs only in the radial direction, so that ∂T = 0. (2) ∂X i.e T = T (R, t). The prescribed mass flow rate is assumed to be stationary, therefore average velocity in a tube cross section, defined as R0

2 U0 = 2 R0

RU (R, t)d R.

(3)

0

is time independent. Assuming the equation of state, = (T ) is considered as linear, = 0 [1 − β(T − T0 )].

(4)

where T0 is the reference temperature with respect to both the tube cross section and to a period of time namely 2π

ω T0 = π R02

ω

R0 RT (R, t)d R

dt 0

(5)

0

∂T ∂X

= 0, T0 is a constant. By invoking the Boussinesq approximation, the momentum Since balance equation along the X -component is   νe f f ∂ ∂U ∂P ∂U νU 0 = + 0 gβ(T − T0 ) + R − . (6) ∂t ∂X R ∂R ∂R K where P = p + 0 gβ(T − T0 ) is the difference between the pressure and the hydrostatic d2 P pressure. By differentiating both sides of Eq. (6) with respect to X , one obtains = 0. d X2 The result implies an existence of a function A(t) such that dP = A(t). dX

(7)

  νe f f ∂ ∂U ∂U νU 0 = A(t) + 0 gβ(T − T0 ) + R − . ∂t R ∂R ∂R K

(8)

Then Eq. (6) can be rewritten as

The energy balance equation is given by

  ∂T ∂T ∝ ∂ R . = ∂t R ∂R ∂R

(9)

The non-dimensional quantities in the above equations are defined as: 4R02 ω 4R0∗ A(t) T − T0 U R , u= , η = ωt,  = , r= λ= T 2R0 U0 ν μU0 8gβT R03 νe f f ν 2R0 U0 T1 − T0 , Gr = , Pr = ,  = , Re = ξ= 2 ν ν T ∝ ν

θ=

Da =

(10)

K . 4R02

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Pr is the Prandtl number which is inversely proportional to the thermal diffusivity of the working fluid, Da is the permeability of the porous media, λ is the pressure gradient in the fluid and  is the frequency of the temperature oscillations. Substituting Eq. (10) into Eqs. (8) and (9) we have:   ∂u Gr 1 ∂ ∂u u  =λ+ θ + r − . (11) ∂η Re r ∂r ∂r Da   ∂θ 1 ∂ ∂θ Pr = r . (12) ∂η r ∂r ∂r The boundary conditions for the dimensionless velocity distribution u(r, η) and for the dimensionless temperature distribution θ (r, η) are as follows ⏐ ⏐     1 1 ∂u ⏐ ∂θ ⏐ ⏐ ⏐ = 0, θ = 0, (13) u , η = 0, , η = ξ + cos(η), 2 ∂r ⏐r =0 2 ∂r ⏐r =0 From Eqs. (3) and (5), the following two constraint equations for the dimensionless velocity and temperature are respectively 1

2

1 r u(r, η)dr = , 8

0

1

2π

2

θ (r, η)r dr = 0.

dη 0

(14)

0

The Fanning friction factor is defined as f =

⏐ 2 ∂u ⏐ 2τW ⏐ = − . Re ∂r ⏐r = 1 0 U02 2

(15)

Differentiating with respect to η both sides of the integral constraint on u(r, η) expressed in Eq. (14), we have 1

2

dr 0

∂u(r, η) r =0 ∂η

(16)

Multiplying both sides of Eq. (11) by r and integrating with respect to r in the interval [0, 21 ], one obtains 1

1

2

r 0

∂u dr = λ ∂η

1

2

r dr + 0

Gr Re

1

2

2 r θ (r, η)dr + 

0

0

1

  2 ∂u 1 ∂ r dr − r u(r, η)dr. ∂r ∂r Da 0

(17) which gives, 1

⏐ 2 λ Gr 4 1 1 ∂u ⏐ + + = − r θ (r, η)dr f Re = −2 ⏐ ∂r ⏐r = 1 Da 2 2 Re  2

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0

(18)

Mixed convection flow in a vertical tube

3 Analytical solution: velocity and temperature distributions In the steady periodic regime, the momentum and energy balance equations (11) and (12), together with the boundary conditions (13) and the constraints (14) can be solved analytically by considering the functions u(r, η, ) θ (r, η) and λ(η) as the real parts of three complex valued functions, namely: u(r, η, ) = Re[u ∗ (r, η)]. θ (r, η) = Re[θ ∗ (r, η)]. λ(η) = Re[λ∗ (η)].

(19)

On the account of Eqs. (11),(12),(13),(14), the complex valued functions u ∗ (r, η), θ ∗ (r, η) and λ∗ (η) must be solution of the boundary value problem:   ∂u ∗ u∗ Gr ∗ 1 ∂ ∂u ∗ r −  = λ∗ + θ + . ∂η Re r ∂r ∂r Da   1 ∂ ∂θ ∗ ∂θ ∗ = r . Pr ∂η r ∂r ∂r   1 ∂u ∗ u∗ , η = 0, |r =0 = 0, 2 ∂r ⏐   ∂θ ∗ ⏐ 1 ⏐ = 0, (20) θ ∗ , η = ξ + cos(η), 2 ∂r ⏐r =0 Therefore, one has Gr ∗ u (r )ex p(iη). Re b θ ∗ (r, η), = θa∗ (r ) + θb∗ (r )ex p(iη).

u ∗ (r, η), = u a∗ (r ) +

λ∗ (r, η), = λa∗ (r ) + λ∗b (r )ex p(iη).

(21)

By substituting Eq. (21) into (20), one obtains two independent boundary value problems. The first boundary value problem is expressed as    d Gr ∗ du ∗ u∗ r a − a = −λa∗ − θ . r dr dr Da Re a   1 d dθ ∗ r a = 0, r dr dr ⏐   1 ∂u a∗ ⏐ ∗ ⏐ = 0, = 0, ua 2 ∂r ⏐r =0 ⏐   1 ∂θa∗ ⏐ ⏐ = 0, θa∗ = ξ, 2 ∂r ⏐r =0 1

2

u a∗ (r )dr =

1 , 8

0 1

2

θa∗ (r )r dr = 0.

(22)

0

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while the second is given by   du ∗ u∗  d r b − b = −λ∗b − θb∗ − iu ∗b r dr dr Da   dθb∗ 1 d r − iPr θb∗ = 0, r dr dr ⏐   ∂u ∗b ⏐ ∗ 1 ⏐ = 0, ub = 0, 2 ∂r ⏐r =0 ⏐   ∂θb∗ ⏐ 1 ⏐ = 1, = 0, θb∗ 2 ∂r ⏐r =0 1

2

1

u ∗b (r )dr = 0,

0

2

θb∗ (r )r dr = 0.

(23)

0

By employing the constraint on θa∗ (r ) yields ξ = 0 i.e T = T0 The solution of Eq. (22) is θa∗ (r ) = 0,

 I0 (0.5D1 ) − I0 (r D1 ) , D1 I0 (0.5D1 ) − 4I1 (0.5D1 )   D1 I0 (0.5D1 ) 1 λa∗ (r ) = . Da D1 I0 (0.5D1 ) − 4I1 (0.5D1 )

u a∗ (r ) = D1



(24)

while the solution of Eq. (23) is I0 (r D2 ) , I0 (0.5D2 )     λ∗b I0 (r D2 ) I0 (r D3 ) 1 I0 (r D3 ) ∗ u b (r ) = 2 1− + − , I0 (0.5D3 ) I0 (0.5D3 ) D3  [D32 − D22 ] I0 (0.5D2 ) θb∗ (r ) =

λ∗b =

4D32 [D32

−

D22 ]

[D3 I0 (0.5D3 )I1 (0.5D2 ) − D2 I0 (0.5D2 )I1 (0.5D3 )] . (D2 I0 (0.5D2 ))[4I1 (0.5D3 ) − D3 I0 (0.5D3 )]

(25)

where, D1 = √

1 Da

√ D2 = iPr ,

 D3 =

 1 1 + i . Da 

and In is the modified Bessel function of the first kind of order n The Fanning friction factor can be written as f Re = Re( f a∗ Re + f b∗ ∗ Gr eiη ).

(26)

where f a∗ and f b∗ are respectively given by

⏐ du a∗ ⏐ ⏐ dr ⏐r = 1 2 ∗⏐ ⏐ du f b∗ Re = −2 b ⏐ dr ⏐r = 1 f a∗ Re = −2

2

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(27)

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Fig. 2 a Radial distribution of |u ∗b | for Pr = 7.0 and Da = 0.1. b Radial distribution of |u ∗b | for Pr = 7.0 and Da = 0.01

The dimensionless heat flux is defined as

=

 ∗ ∂θ ∂θ = Re = Re( ∗ ), ∂r ∂r

∗ = a∗ + ∗b eiη , wher e, ∗b =

∂θb∗ D2 I1 (D2 r )eiη = . ∂r I0 (0.5D2 )

(28)

4 Discussion of result A presentation of the analytical solution above is displayed from Figs. 2, 3, 4, 5, 6, 7, 8, 9. By representing the radial distribution of the oscillating amplitude of the dimensionless velocity for fixed Pr , Gr Re , , Da. Figures 2a, b, 3, 4, 5a, b represent the radial distribution of |u ∗b | for fixed values of the Prandtl number Pr and for different . In Figs 2a, b and 3a, b, it is seen that the oscillation amplitude of the dimensionless velocity has two local maxima, one close to the tube surface and the other one at the axis of the tube. There also exist local minimum close to r ≈ 0.3. A comparison made between the present work and that of [11] shows a slight decrease in the

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Fig. 3 a Radial distribution of |u ∗b | for Pr = 0.71 and Da = 0.1. b Radial distribution of |u ∗b | for Pr = 0.71 and Da = 0.01

magnitude of the oscillation amplitude of the fluid velocity in the tube. This is attributed to the presence of porous materials as considered in the present work. However, when the value of Da becomes large (Da > 2), the effect of the porous material becomes negligible and the flow returns to that of clear fluid which is considered by [11]. It is observed that for small Da value, the oscillatory velocity gradient decreases on the duct wall as well on the tube axis. However, for smaller values of Pr , the thermal boundary layer is thickened, thereby resulting in a higher oscillation amplitude of the dimensionless velocity. In Figs. 4a, b and 5a, b, that is for smaller  (0.5 <  < 2.0) and for Pr = 0.71, the magnitude of oscillation amplitude of the dimensionless velocity is much lower than that of Pr = 7.0. Also a minima is also noticed in the duct close to r ≈ 0.3., this is further enhanced by the presence of Da number , a critical look at the fluid section shows that at this point the influences of  as well as that of Pr are negligible. Also, there is flow stagnation at this location of the flow domain. In Figs. 6 and 7, in which the dimensionless velocity distribution is reported for different values of dimensionless frequency. These figures reveal that at different radial positions, there exist a resonance frequency that gives a maximum oscillation amplitude of the dimensionless velocity. However, it is observed that the rate of increase of the oscillation amplitude of the dimensionless velocity is clearly dependent on the Prandtl number of the working fluid. For instance, it requires a larger frequency to drive velocity amplitude of air to maximum in comparison to the frequency required for water. In addition, porous medium barrier acts as an opposition to fluid flow and its effect is revealed in the retardation of fluid velocity as well as the amplitude. These are evident in Figs. 6 and 7, where increasing Darcy number

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Fig. 4 a Radial distribution of |u ∗b | for Pr = 7.0 and Da = 0.1. b Radial distribution of |u ∗b | for Pr = 7.0 and Da = 0.01

leads to an increase in the velocity in the tube. Figure 8 show the oscillation amplitude of the dimensionless pressure drop variation for different dimensionless frequency. It is seen that for increasing  as well as Prandtl number, Pr , the oscillatory amplitude of the dimensionless pressure drop decreases monotonically within the tube. However, it is also evident that as Da increases the dimensionless pressure drop decreases In Fig. 9, the oscillation amplitude of the friction factor is reported for different dimensionless frequency , it is noted that there exist a resonance frequency for every assumed Pr .

5 Conclusion An investigation of a laminar mixed convection flow of viscous incompressible fluid in an infinite vertical tube with circular cross section is carried out in the steady-periodic regime. A fully developed laminar mixed convection flow with an assumed sinusoidal temperature change boundary is considered. The problem was solved analytically. The governing equations of the energy and momentum equations and the constraint equations are written in dimensionless form as the real part of a complex valued function. The solution of the dimensionless velocity, temperature, pressure drop and heat flux are obtained in terms of Prandtl number Pr , the dimensionless frequency  the ratio of the Grashof number Gr to the Reynolds number Re and the Darcy number Da. It is observed that that there exist two local maxima, one near the surface of the tube and the other on the

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Fig. 5 a Radial distribution of |u ∗b | for Pr = 0.71 and Da = 0.1. b Radial distribution of |u ∗b | for Pr = 0.71 and Da = 0.01

Fig. 6 Radial distribution of |u ∗b | for different  at r = 0

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Fig. 7 Radial distribution of |u ∗b | for different  at r = 0.4

Fig. 8 Distribution of |λ∗b | for different 

tube axis. There exists also a minima, close to r ≈ 0.3. The resonance frequencies that give rise to the oscillation amplitudes of the dimensionless velocity and dimensionless fanning friction factor is a function of the assumed Prandtl number and the radial position. It is worthy to note that porous medium barrier acts as an opposition to fluid flow and its effect is revealed in the retardation of fluid velocity as well as the amplitude.

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Fig. 9 Distribution of | f b∗ Re| for different 

It is seen that there exist a point in the tube section where the fluid flow is stagnated. At this point, effect of Darcy number,  and Pr is nullified in the fluid flow. Also, it is possible to control pressure drop inside the tube by using appropriate value of Darcy number.

References 1. Sparrow, E.M., Gregg, J.L.: Newly quasi-steady free-convection heat transfer in gases. J. Heat Mass Transf. Trans. ASME Ser. 82, 258–260 (1960) 2. Chung, P.M., Anderson, A.D.: Unsteady laminar free convection. ASME J. Heat Mass Transf. 83, 473–478 (1961) 3. Nanda, R.J., Sharma, V.P.: Free convection laminar boundary layer in oscillatory flow. J. Fluid Mech. 15, 419–428 (1963) 4. Bar-Cohen, A., Rohsenow, W.M.: Thermally optimum spacing of vertical natural convection cooled parallel plates. ASME J. Heat Transf. 106, 116–123 (1984) 5. Wang, C.Y.: Free convection between vertical plates with periodic heat input. ASME J. Heat Transf. 110, 508–511 (1988) 6. Kazmierczak, M., Chinoda, Z.: Buoyancy-driven flow in an enclosure with time periodic boundary conditions. Int. J. Heat Mass Transf. 35, 1507–1518 (1992) 7. Kwak, H.S., Kvwahara, J.M., Hyun, J.M.: Resonant enhancement of natural convection heat transfer in a square enclosure. Int. J. Heat Mass Transf. 41, 2837–2846 (1998) 8. Antohe, B.V., Lage, J.L.: Amplitude effect on convection induced by time periodic boundary conditions. Int. J. Heat Mass Transf. 39, 1121–1133 (1996) 9. Lage, J.L., Bejan, A.: The resonance of natural convection in an enclosure heated periodically from the side. Int. J. Heat Mass Transf. 36, 2027–2038 (1993) 10. Barletta, A., Zanchini, E.: Time-periodic laminar mixed convection in an inclined channel. Int. J. Heat Mass Transf. 46, 551–563 (2002) 11. Barletta, A., Rossi di Schio, E.: Mixed convection flow in a vertical circular duct with time-periodic boundary conditions: steady-periodic regime. Int. J. Heat Mass Transf. 47, 3187–3195 (2004)

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Mixed convection flow in a vertical tube 12. Nawaf, S.H.: Natural convection in a square porous cavity with an oscillating wall temperature. Arab. J. Sci. Eng. 31(1b), 35–46 (2006) 13. Arash, K., Afrand, M., Bazofti, M.M.: Periodic mixed convection of a nanofluid in a cavity with toplid sinusoidal motion. World Acad. Sci. Eng. Technol. 47, 11–29 (2010) 14. Ingham, D.B., Pop, I. (eds.): Transport Phenomena in Porous Media. Pergamon, Oxford (1998) 15. Neild, D.A., Bejan, A.: Convection in Porous Media, 2nd edn. Springer, New York (1999) 16. Pop, I., Ingham, D.B.: Convective Heat Transfer, Mathematical and Computational Modelling of Viscous fluids and Porous Media. Pergamon, Oxford (2001) 17. Vafai, K. (ed.): Handbook of Porous Media. Marcel Dekker, New York (2000) 18. Kou, H.S., Lu, K.T.: The analytical solution of mixed convection in a vertical channel embedded in a porous media with asymmetric wall heat fluxes. Int. J. Heat Mass Transf. 20, 737–750 (1993) 19. Jha, B.K.: Free-convection flow through an annular porous medium. Heat Mass Transf. 41, 675–679 (2005) 20. Aung, W., Worku, G.: Theory of fully developed combined convection including flow reversal. ASME J. Heat Transf. 108, 485–488 (1986) 21. Barletta, A., Zanchini, E.: On the choice of the reference temperature for fully-developed mixed convection in a vertical channel. Int. J. Heat Mass Transf. 42, 3169–3181 (1999) 22. Cheng, C.H., Kou, S.H., Huang, W.H.: Flow reversal and heat transfer of fully developed mixed convection in vertical channels. J. Thermophys. Heat Transf. 4, 375–383 (1990) 23. Lavine, A.S.: Analysis of fully developed opposing mixed convection between inclined parallel plates. Warme-und Stoffubertragung 23, 249–257 (1988) 24. Rossi di Schio, E., Celli, M., Pop, I.: Buoyant flow in a vertical fluid saturated porous annulus: Brinkman model. Int. J. Heat Mass Transf. 54(7–8), 1665–1670 (2011)

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