Mixed Convection in an Inclined Channel Filled with ...

Mixed Convection in an Inclined Channel Filled with Porous Material Having TimePeriodic Boundary Conditions: SteadyPeriodic Regime Basant K. Jha, Deborah Daramola & Abiodun O. Ajibade

Transport in Porous Media ISSN 0169-3913 Transp Porous Med DOI 10.1007/s11242-015-0533-6

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Author's personal copy Transp Porous Med DOI 10.1007/s11242-015-0533-6

Mixed Convection in an Inclined Channel Filled with Porous Material Having Time-Periodic Boundary Conditions: Steady-Periodic Regime Basant K. Jha1 · Deborah Daramola1 · Abiodun O. Ajibade1

Received: 3 February 2015 / Accepted: 10 June 2015 © Springer Science+Business Media Dordrecht 2015

Abstract This paper reports an investigation of the hydrodynamic and thermal behaviour in a steady-periodic regime of a fully developed laminar mixed convection flow in an inclined channel filled with porous material. One of the channel walls is kept at a constant temperature, while the other is heated sinusoidally. The flow formation inside the porous media is modelled using Darcy–Brinkman model. The resulting governing dimensionless momentum and energy equations are separated into steady and periodic parts and solved analytically by the method of undetermined coefficients. In order to see the effect of the governing parameters on the thermal and hydrodynamic behaviour of the fluid flow, the results are depicted pictorially. These are seen to depend strongly on the dimensionless frequency of the periodic heating, Darcy number and the Prandtl number of the working fluid. The result is also seen to be in strong agreement with the existing analytical work, when the Darcy number is large. For sufficiently high Prandtl number, it is observed that the amplitude of the friction factor oscillation is maximised at a resonance frequency at the wall where there is periodic heating; also increasing Darcy number (Da) increases the amplitude of the friction factor oscillation, while it reduces the resonance frequency. Keywords

B

Mixed Convection · Darcy Number · Periodic heating and inclined channel

Deborah Daramola [email protected] Basant K. Jha [email protected] Abiodun O. Ajibade [email protected]

1

Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria

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Author's personal copy B. K. Jha et al.

List of symbols A, B D Da f1 f2 ∗ , f∗ f 1a 1b ∗ , f∗ f 2a 2b g g Gr i k K L p P Pr q Re e t T T0 T1 T2 u u ∗ , u a∗ , u ∗b U U0 X, Y y

Functions defined by Eq. (7) Hydraulic diameter, 4L Darcy number Fanning friction factor at the wall Y = −L Fanning friction factor at the wall Y = L Complex fanning friction factor defined in Eq. (51) Complex fanning friction factor defined in Eq. (52) Gravitational acceleration Magnitude of the gravitational acceleration Grashof number Imaginary unit Thermal conductivity Permeability of the medium Channel half width Pressure Difference between the pressure and the hydrostatic pressure Prandtl number Heat flux per unit area Reynolds number Real part of a complex number Time Temperature Average temperature in a channel section Temperature of the wall Y = −L Time average temperature of the wall Y = L Dimensionless velocity Dimensionless complex-valued function Fluid velocity Average velocity Rectangular Cartesian coordinates Dimensionless coordinate

Greek letters α β γ λ λ∗ , λa∗ , λ∗b η θ ∗ , θa∗ , θb∗ μ ν νeff Φ

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Thermal diffusivity Volumetric coefficient of thermal expansion Amplitude of the wall temperature oscillations νeff = ν Dimensionless parameter Dimensionless complex-valued function Dimensionless parameter Dimensionless complex-valued function Dynamic viscosity Kinematic viscosity Effective kinematic viscosity Dimensionless heat flux

Author's personal copy Mixed Convection in an Inclined Channel Filled with Porous. . .

Φa∗ , Φb∗ ξ χ ϕ 0 τw ω Ω

Dimensionless complex value function Dimensionless parameter defined in Eq. (11) Dimensionless parameter defined in Eq. (11) Tilt angle Mass density Mass density for T = T0 Average wall shear stress Frequency of the wall temperature oscillation Dimensionless frequency

1 Introduction Convective flow in a vertical channels filled with or without fluid-saturated porous materials has been studied extensively because of its importance in many practical applications such as packed-bed catalytic reactor, drying of porous solid, waste disposal, storage of grain coal, petroleum industry, electronic cooling, high performance insulation for building and cold storage, aerodynamic heating, electrostatic precipitation, transport of heated and cooled fluid and polymer technology to mention a few. Several studies have been conducted on convective motion in porous medium. Amongst them are the works of Vafai and Tien (1982), in which they investigated the boundary and inertia effects on convective mass transfer in porous media. Kaviany (1985) studied laminar flow through porous channel bounded by isothermal parallel plates and concluded that the Nusselt number for the fully developed fields increase with an increase in the porous media shape parameter. Beckermann and Viskanta (1987) studied forced convection boundary layer flow and heat transfer along a flat plate embedded in a porous medium. Tien and Hunt (1987) studied transport phenomenon for boundary layer flow and heat transfer in packed beds. Vafai and Kim (1989) used the Brinkman–Forcheimer extended Darcy model to obtain a closedform analytical solution for a fully developed flow in a porous channel subject to constant heat flux boundary condition. Although much attention has been directed to the study of free as well as forced convection in a porous channel, studies involving the combination of free and forced (mixed) convection in vertical porous channel are being researched as a result of its significant importance in biomedical and industrial processes as well as in natural and different fields of engineering. Kou and Lu (1993) studied mixed convection in a vertical channel embedded in a porous medium with asymmetric wall heat flux and found that reverse flow depends on the value of Gr/Re. Malashetty et al. (2001) studied convective flow and heat transfer in an inclined composite porous medium. Mixed convection in a vertical porous channel was investigated by Umavathi et al. (2005). Comprehensive fundamental works are documented in the works of Kaviany (1995), Pop and Ingham (2001), Vadasz (2008), Vafai (2010), Nield and Bejan (2013) and Bagchi and Kulacki (2014). Cimpean et al. (2009) studied fully developed mixed convection between inclined plates filled with a porous medium in which an analytical solution based on the Darcy law was obtained in terms of the mixed convection parameter, the Peclet number and the inclination angle of the system. The study reveals that for large values of mixed convection parameter, the formation of boundary layer structures on the plate was observed to occur. Guerroudj and Kahalerras (2012) investigated mixed convection in an inclined channel with heated porous blocks, and their results reveal that the inclination angle of the channel can alter substantially the fluid flow and the heat transfer mechanisms, espe-

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cially at high Richardson and Darcy numbers. The maximum and minimum global Nusselt numbers (Nu) are reached for α = +90◦ and α = −90◦ . The importance of periodic heat input in convective fluid motion in engineering applications has made it to receive a lot of attention by researchers. Examples are seen in automatic control systems, in the air cooling in the coil of an air conditioner, in the cylinder of a combustion engine and in the steam generator of a boiler etc. Flow behaviour of fluids in channels subjected to time-dependent surface temperature has been studied by many authors. Sparrow and Gregg (1960) and Chung and Anderson (1961) studied the case in which the surface temperature varies slightly about a mean level, which is higher than the ambient temperature using a perturbation expansion, but their results were restricted to small amplitudes. However, Yang et al. (1974) used a finite difference approach in their study of laminar natural convection with oscillatory surface temperature and were able to overcome the restriction on the amplitude. Fully developed convection between two periodically heated parallel plates was studied by Bar-Cohen and Rohsenow (1984), while Wang (1988) investigated the effect of Strouhal number on the development of boundary layers in a vertical channel whose boundary is subjected to periodic heating and periodic heat flux. Kwak et al. (1998) analysed numerically the steady-periodic natural convection in a square enclosure having both the upper and lower walls insulated, the left vertical wall kept at a constant temperature, and the right vertical wall kept at uniform temperature which varies in time with a sinusoidal law, while Lage and Bejan (1993), Antohe and Lage (1996) considered the case in which right vertical wall is kept at uniform temperature which varies periodicaly in time with square wave pulses. Resonance phenomenon was predicted in the works of Lage and Bejan (1993), Antohe and Lage (1996) and Kwak et al. (1998) in which the heat flux through a vertical surface fluctuates with an amplitude that, for fixed values of the other parameters, reaches a maximum for a given value of angular frequency called the resonance frequency. Nanda and Sharma (1963) were also able to circumvent the limitation of the result for only small amplitude by separating the temperature and velocity into steady and oscillatory components. Barletta and Zanchini (2003) 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 exists a resonance frequency that maximises the amplitude of the friction factor oscillations at the unsteady temperature wall. Nawaf (2006) investigated the effect of periodic oscillations of the surface temperature with time on the mixed convection flow in a square porous cavity with an oscillating wall temperature. It is found that at the resonance frequency, the peak average Nusselt number is observed. Oscillatory flow and heat transfer in a composite porous medium channel were studied by Umavathi et al. (2006). Mixed convection flow and heat transfer in a vertical wavy channel containing porous and fluid layer with travelling thermal waves were analysed by Umavathi and Shekar (2011). They found that the flow reversal near the walls increases with Darcy number and inertial effect. The aim of this work is to generalise the work of Barletta and Zanchini (2003) to accommodate porous medium applications which is frequently encountered in petroleum technology, geophysics and energy-related engineering problem. Amongst the numerous applications of our work, the results from this work will serve as basis for engineers to design, for instance, heat exchange enhancement techniques based on flow with regular harmonic heat input and maintain electronic components at an acceptable operating temperature. The governing Darcy–Brinkman equations for the problem are separated into steady and oscillatory components using separation of variable method; this gives a pair of independent

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Fig. 1 Schematic diagram

boundary value problems. This is then solved along with its boundary conditions and constraint equations using the method of undetermined coefficient. Closed-form solutions were derived for the velocity and temperature profiles which help in describing the hydrodynamic behaviour of the fluid flow.

2 Mathematical Analysis Considering a two-dimensional flow of viscous incompressible fluid which steadily flows in between two infinitely wide plane parallel walls filled with porous material. The fluid is bounded by two parallel walls separated by a distance 2L, which are inclined with respect to the gravitational acceleration g. (See Fig. 1). The flow is assumed to be fully developed, laminar and parallel such that U has the non-vanishing component U along the X -axis. So that the mass balance equation drops to ∂U ∂ X = 0. The X -axis, the Y -axis, the gravitational acceleration g lie on the same plane. The wall at Y = −L is kept isothermally with a temperature T1 , while the wall at Y = L is subjected to an oscillating temperature. T (X, L , t) = T2 + T cos(ωt).

(1)

∂T Heat flow is assumed to occur only in the transverse direction, therefore = 0. ∂X A steady mass flow rate is prescribed: therefore, the average velocity in a channel section, defined as L 1 U dY (2) U0 = 2L −L is time independent. The porous medium is assumed to be isotropic and homogeneous. The thermophysical properties of the solid matrix and the fluid are also assumed to be constant except the density variation in the buoyancy term in the momentum equations, i.e the Boussinessq approximation.

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Ignoring both viscous and Darcy dissipations in the fluid. The stream wise and the transverse momentum balance equations using the Darcy–Brinkman model Nield and Bejan (2013) ∂P U ∂U ∂ 2U = 0 gβ(T − T0 ) cos ϕ − + μeff −μ 2 ∂t ∂X ∂Y K ∂P = 0. 0 gβ(T − T0 ) sin ϕ + ∂Y

0

(3) (4)

where P = p + 0 g(X cos ϕ − Y sin ϕ). Differentiating both sides of Eq. (3) with respect to X gives ∂2 P =0 ∂ X2

(5)

Moreover, if Eq. (4) is differentiated with respect to X , one gets ∂2 P =0 ∂ X ∂Y

(6)

P(X, Y, t) = A(Y, t) − B(t)X

(7)

It is noted that Eqs. (5) and (6) give

The energy balance equation is given by ∂2T ∂T =α 2 ∂t ∂Y

(8)

Invoking the equation of state = (T ) which is considered as linear function of T , where β is the thermal expansion coefficient, 0 is the mass density at T0 and T0 is the average temperature in the channel. = 0 [1 − β(T − T0 )] (9) where T0 is an average temperature both with respect to the interval −L ≤ Y ≤ L and to 2π (Barletta and Zanchini 1999, 2003) the period 0 ≤ t ≤ ω 2π L ω ω T0 = dt T dY (10) 4π L 0 −L The non-dimensional quantities in the above equations are defined as: T − T0 U D2 B Y U0 D , u= , y = , η = ωt, λ = , Re = T U0 D μU0 ν gβT D 3 cos ϕ T1 − T0 ν T2 − T1 Gr = , ξ= , Pr = , χ = ν2 T α T ωD 2 K νeff Da = 2 , Ω = , γ = . D ν ν θ =

(11)

Pr represents the Prandtl number which is inversely proportional to the thermal diffusivity of the working fluid, Da is the Darcy number, λ is the pressure gradient in the fluid, Ω is the frequency of the oscillating temperature, ρ represents the fluid-saturated porous media density, μ is fluid viscosity and μeff denotes the effective fluid viscosity of the porous medium. All other physical quantities used are defined in the nomenclature.

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Substituting Eq. (11) into Eqs. (3) and (8), we have the momentum and energy equations in dimensionless form as: ∂u u Gr ∂ 2u =λ+ θ +γ 2 − . ∂η Re ∂y Da ∂θ ∂ 2θ Ω Pr = 2. ∂η ∂y Ω

(12) (13)

The hydrodynamic and thermal boundary conditions are given as 1 1 u − ,η = 0 = u ,η 4 4 1 1 , η = ξ + χ + eiΩ θ − , η = ξ, θ 4 4

(14) (15)

The constraints imposed on the function u(y, η) and θ (y, η) are

1 4

− 41

u(y, η)dy =

2π

dη

1 4

− 14

0

1 2

(16)

θ (y, η)dy = 0.

(17)

Differentiating with respect to η, both sides of the integral constraint on u(y, η) expressed in Eq. (16) becomes 1 4 ∂u(y, η) dy = 0 (18) ∂η − 41 1 1 By integrating Eq. (12) with respect to y, in the range [− , ] we have 4 4 1 1 1 4 4 4 ∂u Gr Ω dy = λdy + θ dy ∂η Re − 41 − 14 − 41 1 1 2 4 4 ∂ u 1 +γ dy − u(y, η)dy 2 ∂y Da − 14 − 41 ⏐ ⏐ 1 ∂u ⏐ ∂u ⏐ λ 1 1 Gr 1 4 ⏐ ⏐ − = θ (y, η)dy − + ∂ y ⏐ y=− 1 ∂ y ⏐ y= 1 2γ Re γ − 41 Da 2γ 4

(19) (20)

4

The fanning friction factors f 1 and f 2 at the walls Y = −L and Y = L, respectively, are defined as ⏐ ⏐ ⏐ ⏐ 2ν ∂U ⏐ 2 ∂u ⏐ 2ν ∂U ⏐ 2 ∂u ⏐ ⏐ ⏐ ⏐ ⏐ f1 = 2 = , f = − = − (21) 2 Re ∂ y ⏐ y=− 1 Re ∂ y ⏐ y= 1 U0 ∂Y ⏐Y =−L U02 ∂Y ⏐Y =L 4

4

From Eq. (20), the dimensionless pressure drop and the fanning friction factor are related as follows: 1 4 λ 1 2 Gr f 1 Re + f 2 Re = + θ (y, η)dy − (22) γ γ Re − 41 Daγ

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3 Analytical Solution for Steady-Periodic Regime Defining complex-valued functions u ∗ (y, η), θ ∗ (y, η), λ∗ (η) can fulfil the equations below Gr ∗ ∂ 2u∗ u∗ ∂u ∗ = λ∗ + θ +γ . − ∂η Re ∂ y2 Da ∂θ ∗ ∂ 2θ ∗ Ω Pr = . ∂η ∂ y2 1 1 u∗ − , η = 0 = u∗ ,η 4 4 1 1 ∗ ∗ , η = ξ + χ + eiΩ θ − , η = ξ, θ 4 4 1 4 1 u ∗ (y, η)dy = 2 − 41 1 2π 4 dη θ ∗ (y, η)dy = 0. Ω

− 41

0

(23) (24) (25) (26) (27) (28)

with u = e(u ∗ ), θ = e(θ ∗ ), λ = e(λ∗ ). The solution of Eqs. (23–28) can be written in the form for a steady-periodic regime, Gr ∗ u (y)eiη Re b θ ∗ (y, η) = θa∗ (y) + θb∗ (y)eiη Gr ∗ iη λ e λ∗ (η) = λa∗ + Re b

u ∗ (y, η) = u a∗ (y) +

(29) (30) (31)

Two distinct boundary value problems were obtained by substituting Eqs. (29–31) into Eqs. (23–28). The first boundary value problem is expressed as d 2 u a∗ u∗ Gr ∗ − a = −λa∗ − θ . 2 dy Da Re a d 2 θa∗ =0 dy 2 1 ∗ ∗ 1 ua − = 0 = ua 4 4 1 1 = ξ, θa∗ =ξ +χ θa∗ − 4 4 1 4 1 u a∗ (y)dy = 1 2 −4 1 4 θa∗ (y)dy = 0. γ

− 41

123

(32) (33) (34) (35) (36) (37)


while the second is given as

u∗ d 2 u ∗b − iΩu ∗b − b = −λ∗b 2 dy Da d 2 θb∗ − iΩ Pr θb∗ = 0 dy 2 1 ∗ ∗ 1 ub − = 0 = ub 4 4 1 1 θb∗ − = 0, θb∗ =1 4 4 1 4 u ∗b (y)dy = 0.

γ

(38) (39) (40) (41) (42)

− 41

The solution of Eqs. (32) and (33) under conditions Eqs. (34–37) is

θa∗ (y) = 2χ y λa∗ =

N 4

N cosh N N Da N cosh − 4 sinh 4 4 ⎞ ⎛

(43)

(44) ⎛

⎞

⎜ ⎜ 1 Gr cosh(N y) ⎟ sinh(N y) ⎟ ⎟ ⎟ u a∗ (y) = λa∗ Da ⎜ + χ Da ⎜ 4y − ⎠ ⎝1 − ⎝ N N ⎠ 2 Re cosh sinh 4 4

(45)

1 γ Da while the solution of Eqs. (38) and (39) under conditions Eqs. (40–42) is

where N = √

1 sinh Γ +y 4 θb∗ (y) = Γ sinh 2 N1 Γ Γ N1 2N12 N1 cosh sinh − Γ cosh sinh 4 4 4 4 λ∗b =

N1 Γ 2 N1 2 Γ − N1 N1 cosh − 4 sinh Γ cosh 4 4 4

(46)

(47)

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√ iΩ Pr,

iΩ 1 + γ Daγ ⎞ ⎛ ⎞⎞ ⎛⎛ 1 1 +y +y sinh N1 sinh Γ ⎟ ⎜ ⎟⎟ ⎜⎜ 1 4 4 ⎟−⎜ ⎟⎟ ⎜⎜ u ∗b (y) = ⎠ ⎠⎠ ⎝ ⎝ ⎝ 2 2 N Γ γ (Γ − N1 ) 1 sinh sinh 2 2 ⎛ ⎛ ⎞⎞

where Γ =

+

N1 =

⎜ λ∗b ⎜ 1 y) ⎟⎟ ⎟ ⎜1 − ⎜ cosh(N ⎟ ⎝ ⎝ 2 N1 ⎠⎠ γ N1 cosh 4

(48)

The Fanning friction factor can be written as Gr ∗ ∗ f 1 Re = e f 1a Re + f 1b Re eiη . Re Gr ∗ ∗ f 2b Re + Re eiη . f 2 Re = e f 2a Re

(49) (50)

∗ Re, f ∗ Re, f ∗ Re and f ∗ Re are, respectively, given by where f 1a 2a 1b 2b ⏐ ⏐ ∗⏐ du ∗b ⏐ du ∗ ∗ ⏐ f 1a Re = 2 a ⏐ , f Re = 2 1b dy ⏐ y=− 1 dy ⏐ y=− 1 4 ⏐ 4 ∗⏐ ∗⏐ ⏐ du du a b ∗ ∗ ⏐ ⏐ Re = −2 , f 2b Re = −2 . f 2a dy ⏐ y= 1 dy ⏐ y= 1 4

(51) (52)

4

4 Results and Discussion The present problem investigates the effect of porosity on mixed convection flow in an inclined channel filled with porous material. To validate the results of the present model, the results are compared with Barletta and Zanchini (2003) in Table 1 for 0.1 ≤ Da ≤ 2.0, Pr = 0.71 and Ω = 10. As can be seen from the table, the solutions of the present work perfectly agree with those of Barletta and Zanchini (2003) for large value of Da. The expressions for the steady dimensionless temperature and the steady dimensionless heat transfer (Nusselt number) are not given in this problem as they are already discussed in the work of Barletta and Zanchini Table 1 Comparison of |u ∗b | for Pr = 0.71 and Ω = 10 with Barletta and Zanchini (2003) y

Barletta and Zanchini (2003)

Present work Da = 0.1

Da = 0.5

Da = 1.0

Da = 2.0 0.00148256

−0.2

0.00148649

0.00141226

0.00147093

0.00147866

−0.1

0.00174646

0.00163556

0.00172316

0.00173474

0.00174058

0

0.00011365

0.00011005

0.00011292

0.00011329

0.00011347

0.1

0.00174351

0.00163250

0.00172019

0.00173177

0.00173762

0.2

0.00150983

0.00143548

0.00149426

0.00150199

0.00150590

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Fig. 2 The variation of the modulus of λ∗b with Ω a Da = 0.001 b Da = 0.1

∗ Re versus Ω in the range 0 Ω ≤ 100 a Da = 0.1 b Fig. 3 Plots of the amplitude of oscillation of f 1b Da = 0.001

(2003). In order to investigate the effect of the governing parameters on the hydrodynamic behaviour of the fluid, numerical simulations are carried out on the analytical solutions obtained in Eqs. (47, 48, 51, 52) for variations in the governing parameters. The results of the simulation are presented graphically in the Figs. 2, 3, 4, 5, 6, 7, 8, 9 and 10. Figure 2 depicts the variation of the modulus of λ∗b with Ω in the range 0 ≤ Ω ≤ 100 for Pr = 0.7, 7 and 100. From Eq. (47), λ∗b is a function of the dimensionless angular frequency Ω, Darcy number, Da and the Prandtl number Pr . It is found that the amplitude of the oscillation dimensionless pressure drop |λ∗b | decreases with the increase in the dimensionless angular frequency, Ω,

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∗ Re versus Ω in the range 0 ≤ Ω ≤ 100 a Da = 0.1 b Da = 0.01 Fig. 4 Plots of the modulus of f 2b

∗ Re versus Ω in the range 0 ≤ Ω ≤ 100 for Pr = 0.71 Fig. 5 Plots of the modulus of f 2b

for every Pr and the decay is faster for higher values of Pr . This is attributed to the fact that increasing dimensionless angular frequency (Ω) and Prandtl number (Pr ) reduces the intensity of heating on the channel wall and the thermal boundary layer, respectively, which reduce the buoyancy effect as well as the convection current which invariably reduces the velocity leading to the reduction in the oscillation dimensionless pressure drop. Likewise, as Darcy number increases, |λ∗b | decreases; since increasing Da reduces the impediment in the flow, thereby reducing the pressure required to drive the flow. Figure 3 shows the variation of ∗ Re| versus Ω. It is found that lower Darcy number reduces the amplitude of the oscillation | f 1b dimensionless fanning friction on the wall at Y = −L. This is physical true because lower Da implies reduced porosity which increases resistance to the flow, thereby reducing the fluid ∗ Re|. It is evident that the amplitude flow velocity which in turn leads to the reduction in | f 1b

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Fig. 6 Plots of the modulus of |u ∗b | for different y with Ω = 1 a Da = 0.1 b Da = 0.001


of the oscillation dimensionless fanning friction factor reduces as the dimensionless angular frequency increases. In Fig. 4, the variation of amplitude of the oscillation dimensionless ∗ Re|) versus Ω is presented. It is seen that there exists a frequency fanning friction factor (| f 2b of the oscillation temperature which maximises the amplitude of the oscillation dimensionless ∗ Re| for every Prandtl number and this frequency reduces as Prandtl number increases. In | f 2b ∗ Re|) versus Ω Fig. 5, the variation of amplitude of the oscillation fanning friction factor (| f 2b is presented for different Da. It is seen that increasing Darcy number increases the amplitude of the oscillation fanning friction factor on this wall Y = L. It is also noticed that as

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Fig. 9 Plot of velocity (u) versus distance (y) for Ω = 100, Pr = 0.7 and Da = 0

Darcy increases, the resonance frequency decreases that maximises the amplitude of the ∗ Re|). For Da = 0.001, 0.01 and 0.1, the resonance oscillation fanning friction factor (| f 2b frequencies are Ω = 127, 81 and 63 respectively for Pr = 0.71. Figure 6 reports the plot of the modulus of u ∗b within the channel when Ω = 1 and for Pr = 0.7 and Pr = 100. When the working fluid is water (Pr = 7.0), the behaviour of the velocity profile is similar to that of air (Pr = 0.7) when Ω = 1. For Pr = 0.7, the dimensionless velocity oscillation profile is almost exactly symmetric with respect to the midplane of the channel and there is a perceived stagnation in the fluid flow along the centre plane, while for Pr = 100 the oscillation dimensionless velocity profile breaks the symmetry about the center plane as the

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Fig. 10 Plot of velocity (u) versus distance (y) for Ω = 100, Pr = 0.7 and Da = 0.1

flow is continuous at the midplane of the channel. |u ∗b | decreases as Darcy number decreases. This is because the permeability of the porous matrix decreases as Darcy number decreases. Variation of modulus of u ∗b versus distance (y) for Ω = 10 is displayed in Fig. 7. It is evident that when the working fluid is air (Pr = 0.7), the symmetric nature of the velocity profile is still maintained, whereas the velocity profile at Pr = 7.0 is similar to that at Pr = 100 for Ω = 1; this is seen that a large dimensionless angular frequency tends to suppress the thermal diffusivity at the boundary plate. Figure 8 shows the variation of the modulus of u ∗b versus distance (y) and Ω = 100 for Pr = 0.7, 7 and 100, respectively. It is found that for higher dimensionless angular frequency, the velocity profile for Pr = 0.7 loses its symmetry nature at the midplane. It is also seen that for Pr = 100, it has a lower average value velocity compared to when the working fluid is air or water. Figures 9 and 10 report the velocity profile (real (u ∗ )) versus distance (y) for Ω = 100. The flow is reversal type, and the tendency of reversal has increasing trend with increasing Gr 1 positive values on the lower inclined channel wall (y = − ), whereas the trend increased Re 4 Gr 1 with increasing negative values on the sinusoidally heated wall (y = ). Re 4 Gr after which flow reversal sets in near each bounding Table 2 shows the critical values of Re channel walls. It is observed from the table that as Darcy number decreases, the tendency Gr for a reversed flow is lowered as it requires higher values of to achieve flow reversal on Re 1 both channel walls. Also, it is noticed that on the lower inclined channel wall (y = − ), for 4 small Ω, 1 ≤ Ω ≤ 10, the tendency of reversal flow has increasing trend as Ω increases, whereas an opposite trend is noticed for Ω in the range 50 ≤ Ω ≤ 200 on increasing Ω.

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Table 2 The critical

Gr for several values of Ω with Pr = 0.7 Re Pr = 0.7 Gr Re y=− 1

Ω

4

Da = 0.1

Da = 0.01

Da = 0.1

4

1

182.3034

−181.1755

5

152.0258

−159.8686

8

122.9936

−95.3019

1

310.2866

−308.0803

5

259.4809

−274.2512

8

208.1448

−152.5726

50

218.3015

−186.9956

100

251.4394

−173.3897

200 Da = 0.01

Gr Re y= 1

50 100 200

1100.400 378.1378 436.1740 1329.400

−145.0183 −306.6607 −275.2100 −201.7228

An increase in flow reversal is ignited by an increase in Ω on the sinusoidally heated wall 1 (y = ). 4

5 Conclusion Analytical solution has been obtained for the steady-periodic regime of a fully developed mixed convection flow in an inclined parallel-plate channel filled with saturated porous material. The governing equations were non-dimensionalised and divided into both steady and oscillating parts using separation of variables. A pair of independent boundary value problems is obtained. The method of undetermined coefficient was then employed to solve independent boundary value problems together with the boundary condition and the constraint equations to get a closed-form solution of the governing momentum and energy equations from which the effect of the Prandtl number, dimensionless frequency of the surface temperature Ω, and Darcy number Da on the hydrodynamic behaviour were evaluated. Favourable comparisons with the obtained analytical results and previously published work were performed. The following conclusions are drawn from the study: 1. As Ω increases, the value of maxima of the amplitude of the dimensionless oscillation velocity reduces while the value of the minima increases. 2. Increasing Ω reduces the amplitude of dimensionless velocity oscillation. Of worthy to note is a point of stagnation in the inclined plane when the working fluid is air (Pr = 0.7). 3. There exists a resonance frequency which maximises the oscillation amplitude of the dimensionless fanning friction factor on the boundary channel wall with oscillating temperature for every Pr , and this increases with increasing Prandtl number (Pr ). 4. Increasing the Darcy number (Da) suppresses the amplitude of the dimensionless oscillation pressure drop, but increases the amplitude of oscillation dimensionless velocity |u ∗b |.

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5. It is seen that an increase in Darcy number results in an increase in the amplitude of the oscillation fanning friction factors, while increasing Darcy number (Da) reduces dimensionless frequency of the surface temperature at which a resonance occurs. 6. An absolute maximum occurs close to the sinusoidally heated channel wall because of an increase in thermal diffusivity on this channel wall. 7 Reversal flow can be checked by the use of appropriate value of both Ω and Da.

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