Unsteady/SteadyFreeConvectiveCouetteFlowof ...downloads.hindawi.com/archive/2012/794741.pdf ·...

International Scholarly Research NetworkISRN ThermodynamicsVolume 2012, Article ID 794741, 10 pagesdoi:10.5402/2012/794741

Research Article

Unsteady/Steady Free Convective Couette Flow ofReactive Viscous Fluid in a Vertical Channel Formed byTwo Vertical Porous Plates

Basant K. Jha,1 Ahmad K. Samaila,1, 2 and Abiodun O. Ajibade1

1 Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria2 Department of Mathematics, Usmanu Danfodiyo University, P.M.B +234-2346, Sokoto, Nigeria

Correspondence should be addressed to Ahmad K. Samaila, [email protected]

Received 28 March 2012; Accepted 15 May 2012

Academic Editors: M. Appell, G. L. Aranovich, A. Ghoufi, and B. Merinov

Copyright © 2012 Basant K. Jha et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper presents unsteady as well as steady-state free convection Couette flow of reactive viscous fluid in a vertical channelformed by two infinite vertical parallel porous plates. The motion of the fluid is induced due to free convection caused by thereactive nature of viscous fluid as well as the impulsive motion of one of the porous plates. The Boussinesq assumption is applied,and the nonlinear governing equations of motion and energy are developed. The time-dependent problem is solved using implicitfinite difference method, and steady-state problem is solved by applying regular perturbation technique. During the course ofcomputation, an excellent agreement was found between the well-known steady-state solutions and transient solutions at largevalue of time.

1. Introduction

The study of convection processes in porous channelsis a well-developed field of investigation because of itsimportance to a variety of situations. For example, thermalinsulations, geothermal system, surface catalysis of chemi-cal reactions, solid matrix heat exchanger, microelectronicheat transfer equipment, porous flat collectors, coal andgrain storage, petroleum industries, dispersion of chemicalcontaminants in various processes, nuclear waste material,and catalytic beds. Some of the examples mentioned aboveinvolve two or more fluids, multidimensional and unsteadyflows [1].

The literature on the topic of unsteady as well as steady-state free convection flow problems is well surveyed bySingh et al. [2], Florio and Harnoy [3], Jha and Ajibade[4], Paul et al. [5], and Langellotto et al. [6]. Muhuri[7] presented natural convection Couette flow between twovertical porous plates when one of the plates moves withuniform acceleration. Singh [8] investigated the motionof the fluid induced by the impulsive motion of one ofthe plates in the presence of convection currents due to

asymmetric thermal condition on the boundaries. Abdulazizand Hashim [9] considered free convection flow betweenporous vertical plates with asymmetric wall temperaturesand concentrations and used homotopy analysis methodto solve boundary-value problems. Fang [10, 11] analysedunsteady velocity and temperature profiles for Couette flowand pressure-driven flow in a channel with porous walls.Unsteady flow investigations with porous boundaries includethe works of Wang et al. [12], Oxarango et al. [13], andMakinde and Ogulu [14]. Makinde and Maserumule [15]presented thermal criticality and entropy analysis for avariable viscosity Couette flow.

In all previous studies the working fluid is considered asnonreactive viscous incompressible fluid. However, unsteadyas well as steady-state free convection Couette flow of areactive viscous fluid in a vertical channel formed by twovertical porous plates can be of importance for the designof equipment used in several types of engineering systems.There are many chemical reactions with important practicalapplications. A common configuration for such reactionsis for the reactants to be made to flow over solid catalyst, withthe reaction taking place on the surface of the catalyst. A full

2 ISRN Thermodynamics

discussion of catalysis and description of many of its practicalapplications is given by Chaudhary and Merkin [16], Merkinand Chaudhary [17], and Chaudhary et al. [18]. Theseauthors studied flow configuration and assume that thereaction takes place only on the catalytic surface and can berepresented schematically by the single first-order Arrheniuskinetics. According to these authors there is a three-waycoupling between fluid flow, fluid/surface temperature, andreactant species concentration. Ayeni [19] and Dainton[20] studied a realistic mathematical description of thermalexplosion which include the effects of Arrhenius temperaturedependence with variable preexponential factor. Recently Jhaet al. [21] investigated transient natural convection flow ofreactive viscous flow in a vertical channel. Most recentlyHazem Attia [22] studied the effect of suction and injectionon unsteady Couette flow.

The objective of the present work is to analyze the effectof suction/injection on time dependent unsteady as well assteady-state free convection Couette flow of viscous reactivefluid in a vertical channel formed by two infinite verticalparallel porous plates.

2. Mathematical Analysis

Consider the transient free convective Couette flow ofviscous reactive fluid in a vertical channel formed by twoinfinite vertical parallel porous plates. The system underconsideration is sketched in Figure 1. The flow is assumedtransient and fully developed. The x′-axis is taken alongthe direction of the flow and parallel to the infinite verticalporous plates and y′-direction perpendicular to the flowdirection. At time t′ ≤ 0, the fluid and the porous plates areassumed to be same temperature T0, and there is no fluidmotion. At t′ > 0, one of the porous plates (y′ = H) startsmoving with constant velocity and fluid generate heat dueto its reactive nature causes to set up fluid motion inside thechannel. In addition, at the same time the flow is subjectedto suction of the fluid from one porous plate and at thesame rate fluid is being injected through the other porousplate.

The fluid is assumed to be Newtonian and obeys theBoussinesq’s approximation. Under the previous assump-tions the energy and momentum equations in dimensionalform are

∂u′

∂t′− v0

∂u′

∂y′= υ

∂2u′

∂y′2+ gβ(T′ − T0),

∂T′

∂t′− v0

∂T′

∂y′= α

∂2T′

∂y′2+QCoA

ρCpexp( −ERT′

),

(1)

where T0 the initial fluid and wall temperature, T′ dimen-sional temperature of fluid, Q the heat of reaction, A therate constant, E the activation energy, R the universal gasconstant, Co the initial concentration of the reactant species,v kinematic viscosity, and H gap between the porous plates.

T0

V0

g

T0

V0

u′ = 0u′ = U

x′

y′

y′ = Hy′ = 0

Figure 1: Schematic diagram of the problem.

The initial and boundary conditions to be satisfied are

t′ ≤ 0 : u′ = 0, T′ = T0 for 0 ≤ y′ ≤ h,

t′ > 0

{u′ = 0, T′ = T0, at y′ = 0,

u′ = U , T′ = T0, at y′ = H.

(2)

Equations (1) and (2) can be nondimensionalized using thevariables

y = y′

H, t = t′v

H2,

u = u′

U, θ = E

RT02 (T′ − T0),

λ = QCoAEH2

κRT02 exp

( −ERT0

), ε = RT0

E,

s = V0H

υ, δ = gβRT0

2h2

EUυ, Pr = υ

α.

(3)

The physical quantities used in (3) are defined in thenomenclature.

Using the expressions (3) and (1) the dimensionlessmomentum and energy equations are

∂u

∂t− s

∂u

∂y= ∂2u

∂y2+ δθ,

∂θ

∂t− s

∂θ

∂y= 1

Pr∂2θ

∂y2+

λ

Prexp(

θ

1 + εθ

),

(4)

while the initial and boundary conditions in dimensionlessform are

t ≤ 0 : u = θ = 0 for 0 ≤ y ≤ 1,

t > 0 :

{u = 0, θ = 0 at y = 0,

u = 1, θ = 0 at y = 1.

(5)

ISRN Thermodynamics 3

0

0.2

0.4

0.6

0.8

1

1.2

y

Vel

ocit

y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t = 0.2

s = −0.5

t = 0.3t = 0.4

Steady state

t= 0.2

s = 0.5

t= 0.4Steady state

t= 0.3

Figure 2: Velocity profile (ε = 0.01, Pr = 7.0, λ = 1.0).

3. Analytical Solutions

The governing equations (4) presented in the previoussection are highly nonlinear and exhibit no analytical solu-tions. The importance of analytical solutions which refersto steady free convection Couette flow of viscous reactivefluid relies on the chance to obtain nontrivial benchmarksto test the reliability of numerical codes developed formore complex physical situations. Analytical solutions areoften an opportunity to inspect the internal consistency ofmathematical models and of the approximations adopted, aswell as to develop new theoretical results. The mathematicalmodel representing the steady-state free convection Couetteflow can be obtained by setting ∂u/∂t = 0 and ∂θ/∂t = 0 in(4) to get

d2u

dy2+ s

du

dy+ δθ = 0,

d2θ

dy2+ Pr s

dθ

dy+ λ exp

(θ

1 + εθ

)= 0.

(6)

The boundary conditions are

u = 0, θ = 0 at y = 0,

u = 1, θ = 0 at y = 1.(7)

In order to construct an approximate solution to (6) subjectto (7), we employed a regular perturbation method bytaking a power series expansion in the reactant consumptionparameter λ:

θ =∞∑i=0

θiλi,

u =∞∑i=0

uiλi.

(8)

Substituting (8) into (6) and collecting the coefficients oflike powers of λ, the solution of the governing equations isobtained as

θ = λ

[1sPr

(1− exp

(−sPry))

(1− exp(−sPr)

) − y

sPr

]

+ λ2

{1

2sPr

(y2

sPr− 2y

(sPr)2 +2

(sPr)3

)

− y

(sPr)2

((1 + exp

(−sPry))

(1− exp(−sPr)

))

+1

(sPr)31(

1− exp(−sPr)) +

k1

sPr

+ k2 exp(−sPry

)},

u =[1− exp

(−sy)][1− exp(s)

]

+ λ

[δ

{1

2sPr

(y2

s− 2

s

{y

s− 1

s2

})

− 1z1

{y

s− 1

s2

}− 1

z1

exp(−sPry

)s2Pr(1− Pr)

}

+c1

s+ c2 exp

(−sy)]

+ λ2

[G1

s− δ

6(sPr)2

{y3

s− 3y2

s2+

6ys3− 6

s4

}

+ δ

{1

2(sPr)3 +1

2(sPr)2(1− exp(−sPr))}

×{y2

s− 2y

s2+

2s3

}

− δ

(sPr)4

(y

s− 1

s2

)− δ

(sPr)3(1− exp(−sPr))

×{

y

s(1− Pr)− 1

s2(1− Pr)2

}exp(−sPry

)

− δ

(sPr)41(

1− exp(−sPr)) exp

(−sPry)

s(1− Pr)

− δ

(sPr)3(1− exp(−sPr)){y

s− 1

s2

}

− k1

sPr

(y

s− 1

s2

)+k2 exp

(−sPry)

s2Pr(1− Pr)

+G2 exp(−sy)

].

(9)


Steady-state skin frictions on the boundary plates is

τ0 = du

dy

∣∣∣∣∣y=0

= s[1− exp(−s)] + λ

[δ(

Prz1s(1− Pr)

− 1s3Pr

)− c2s

]

− λ2[δ

s5

(1

Pr2+

1Pr3

+1

Pr4

)

+δ

s4Pr2(1− exp(−sPr)

)(

1 +1

(1− Pr)2 +1Pr

)

+k1

s2Pr− k2

s(1− Pr)−G2s

],

τ1 = du

dy

∣∣∣∣∣y=1

=[s exp(−s)][1− exp(s)

]

+ λ[δ{

12sPr

(2s− 2

s2

)+sPrz1

exp(−sPr)s2Pr(1− Pr)

}

− c2s exp(−s)]

− λ2

[δ

6(sPr)2

{3s− 6

s2+

6s3

}

+ δ

{1

2(sPr)3 +1

2(sPr)2(1− exp(−sPr))}

×{

2s− 2

s2

}

− δ

s5(Pr)4 +δsPr exp(−sPr)

(sPr)3(1− exp(−sPr)) 1s(1− Pr)

+δ

(sPr)41(

1− exp(−sPr)) sPr exp(−sPr)

s(1− Pr)

− δ

s2(Pr)3(1− exp(−sPr))

− k1

s2Pr− k2sPr exp(−sPr)

s2Pr(1− Pr)−G2s exp(−s)

].

(10)

The steady-state rate of heat transfer on the boundary platesis

Nu0 = dθ

dy

∣∣∣∣∣y=0

= λ

[1(

1− exp(−sPr)) − 1

sPr

]

− λ2

[1

(sPr)3 +2

(sPr)2(1− exp(−sPr)) + sPr k2

],

Nu1 = dθ

dy

∣∣∣∣∣y=1

= λ

[exp(−sPr)(

1− exp(−sPr)) − 1

sPr

]

+ λ2

[1

(sPr)3 (sPr− 1)− 1

(sPr)2(1− exp(−sPr))

× {1 + (1− sPr) exp(−sPr)}

−sPrk2 exp(−sPr)

].

(11)

The constants c1, c2, k1, k2, A, B, A1, B1, G1, G2 are definedin the appendix section.

4. Numerical Solutions

To solve the time-dependent equation (4), the differentialequations have been transformed into the correspondingfinite difference equation. The procedure involves discretiza-tion of the momentum and energy equations into thefinite difference equations at the grid point (i, j) in whichthe time derivatives are approximated by the backwarddifference while the spatial derivatives are replaced by thecentral difference formula. The above equations are solvedby Thomas algorithm by manipulating into a system of linearalgebraic equations in the tridiagonal form.

In each time step, firstly the temperature field has beensolved and then the velocity field is evaluated using thealready known values of the temperature field. The processof computation is advanced until a solution is approached bysatisfying the following convergence criterion:

∑∣∣∣Ai, j+1 − Ai, j

∣∣∣M|A|max

< 10−5 (12)

with respect to temperature and velocity fields. Here Ai, j

stands for the velocity or temperature fields, M is the numberof interior grid points, and |A|max is the maximum absolutevalue of Ai, j .

5. Results and Discussion

The basic parameters that governed this flow are the reactantconsumption parameter (λ), the activation energy parameter(ε), the Grashof number (δ), the Prandtl number (Pr)which is inversely proportional to the thermal diffusivityof the working fluid, and suction/injection parameter (s),which were simultaneously applied each to opposite porousplates of the channel at the same rate. For the purpose ofdiscussion, some numerical calculations are carried out fordimensionless velocity (u), temperature (θ), skin friction (τ),and the rate of heat transfer (Nu). In the entire computationthe value of Grashof number (δ) is taken to be unity. Theeffect of various parameter on the flow are presented in


0

0.2

0.4

0.6

0.8

1

1.2

Vel

ocit

y

y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

λ = 1

t = 0.1t = 0.2

Steady state

λ = 2t = 0.1t = 0.2Steady state

t = 0.1t = 0.2Steady state

λ = 3

Figure 3: Velocity profile (ε = 0.01, Pr = 7.0, s = 0.5).

Steady state Steady state

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

Vel

ocit

y

y

Pr = 0.71 Pr = 7

t = 0.1t = 0.2t = 0.3

t = 0.1t = 0.2t = 0.3

Figure 4: Velocity profile (ε = 0.01, s = 0.5, λ = 1.0).

graphical form in Figures 2 to 17 so as to clearly reveal theinfluence of each governing parameter on the flow. Figure 2describes the influence of s and t on the velocity profilewithin the channel. This figure reveals that as time increases,the velocity increases. This is the consequence of temperatureincrease with time. This figure also reflects that when suctiontakes place on the porous plate at y = 0 (s > 0) fluid velocityis higher compared to a situation when injection takes placeon the porous plate at y = 0 (s < 0). This is physicallytrue since suction on the porous plate y = 0 happensconcurrently with injection on the moving porous plate(y = 1) which acts in support of the motion of the porousplate (y = 1), and hence an increase in fluid velocity is theconsequence. On the other hand (s < 0) means injectionon the plate y = 0 which translates to a correspondingsuction on the plate (y = 1) and this act against the influenceof the impulsive motion of the porous plate. The effectof reactant consumption parameter and time on velocity

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Tem

pera

ture


y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t = 0.6t = 0.8t = 1

t = 0.6t = 0.8t = 1

s = −0.5

Figure 5: Temperature profile (ε = 0.01, Pr = 7.0, λ = 1.0).

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

y

Tem

pera

ture

Steady state

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

s = −0.5t = 0.6t = 0.8t = 1

s = −1.5

Steady state

t = 0.6t = 0.8t = 1

Figure 6: Temperature profile (ε = 0.01, Pr = 7.0, λ = 1.0).

profile is depicted in Figure 3. This figure shows that as λincreases the velocity increases. This physical fact can beexplained, as reaction increases in the fluid, the temperatureincreases which causes velocity increase. This figure alsoreflects that the time required reaching steady-state velocityincreases as λ increases. From the figure it is observed thatthe influence of λ on velocity is almost negligible whenthe time is small. However, at steady state, the velocity isobserved to increase as reactant consumption parameterincreases. Figure 4 reflects unsteady as well as steady-statevelocity profiles for different value of nondimensional timeand Prandtl number. It is evident from the figure that astime increases the velocity increases. Figure 4 also showsthat the Prandtl number has insignificant effect on unsteadyand steady-state velocity. Figure 5 shows the influence of


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Tem

pera

ture

Steady state Steady state Steady state

y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t = 0.8t = 0.9t = 1

t = 0.8t = 0.9t = 1

t = 0.8t = 0.9t = 1

λ = 1 λ = 2 λ = 3

Figure 7: Temperature profile for (ε = 0.01, s = 0.5, Pr = 7.0).

0

0.1

0.2

0.3

0.4

0.5

0.6

Tem

pera

ture

Steady state Steady state Steady state

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t = 0.2 t = 0.2 t = 0.2t = 0.1 t = 0.1 t = 0.1

y

λ = 1 λ = 2 λ = 3

Figure 8: Temperature profile (ε = 0.01, Pr = 0.71, s = 0.5).

s on the temperature profiles of water (Pr = 7.0) withinthe channel. It is seen from the figure that temperatureincreases with time. In addition, in case of suction at y = 0,velocity is maximum near the left plate (y = 0), and incase of injection the maximum is shifted towards the rightporous plate (y = 1), and hence we conclude that thereis no symmetry in the temperature profile in the presenceof suction/injection. Figure 6 describes the behaviour ofunsteady as well as steady-state temperature profiles as afunction of time for different value of s (s = −0.5,−1.5).In this figure, temperature is observed to be high near theplate with injection (y = 1). Figure 7 reveals the influence ofλ on the temperature profile. It is reflected from this figurethat as time increases the temperature increases. It is evidentfrom Figure 7 that the time required to reach steady statetemperature increases as reactant consumption parameter

0

0.5

1

1.5

2

2.5

0.5 1.5

s

Skin

fric

tion


−0.5

Pr = 0.71Pr = 7

t = 0.4

t = 0.2t = 0.3

t = 0.4

t = 0.2t = 0.3

Figure 9: Variation of skin friction (ε = 0.01, λ = 1.0) at y = 0.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 2 3

Skin

fric

tion


λ

Pr = 0.71Pr = 7

t = 0.1t = 0.2t = 0.3

t = 0.1t = 0.2t = 0.3

Figure 10: Variation of skin friction (ε = 0.01, s = 0.5, λ = 1.0) aty = 0.

increases. Variation of temperature as a function of reactantconsumption parameter and time is illustrated in Figure 8.It is clear from the figure that as λ increases the temperatureincreases. It is also evident that the time required to reachsteady state is strongly dependent on λ (i.e., time required toreach steady state is high for high value of λ). Variation ofskin friction for water (Pr = 7.0) and air (Pr = 0.71) at theplate y = 0 and y = 1 is shown in Figures 9, 10, and 11.These figures reflect that as time increases the skin frictionincreases and finally attains its steady-state value. The effectof suction/injection parameter and the nondimensional timeon the skin friction is shown in Figure 9. It is clear fromthe figure that the skin friction is directly proportionalto nondimensional time and suction/injection parameter


01 2 3

Skin

fric

tion


−0.2

−1

−0.6

−0.4

λ

−0.8

−1.2

Pr = 0.71Pr = 7

t = 0.4

t = 0.2t = 0.3

t = 0.4

t = 0.2t = 0.3

Figure 11: Variation of skin friction (ε = 0.01, s = 0.5) at y = 1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−0.5 0.5 1.5

s

Steady state

Nu

ssel

t n

um

ber

(τ)

t = 1.5 t = 2.5t = 2

Figure 12: Variation of Nusselt number (ε = 0.01, λ = 1.0, Pr =7.0) at y = 0.

until steady-state value is attained. This figure also revealsmonotonic increase in the skin friction as sincreases. This isdue to the fact that as s increases the velocity increases andconsequently there is higher skin friction at the boundary.Figures 10, and 11 show the influence of time and reactantconsumption parameter on the skin-friction at y = 0 andy = 1. From the two figures it is clear that the skin-frictionincreases as time increases until steady-state value is attained.Figures 9, 10, and 11 also reflect that the skin friction isalways higher in the case of air (Pr = 0.71) than water (Pr =7.0). The rate of heat transfer which is expressed as Nusseltnumber (Nu) at y = 0 and y = 1 is shown in Figures 12,13, 14, 15, 16, and 17. Figures 12 and 13 show the influenceof s on the rate of heat transfer. Figure 12 reveals that therate of heat transfer increases as nondimensional time and

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.5 1.5

s

Nu

ssel

t n

um

ber

Steady state

−0.5

t = 1.5t = 2

t = 2.5

Figure 13: Variation of Nusselt number (ε = 0.01, λ = 1.0, Pr =7.0) at y = 1.

0

0.5

1

1.5

2

2.5

3

3.5

1 2 3

Nu

ssel

t n

um

ber

Steady state

λ

t = 1.5

t = 2t = 2.5

t = 3

Figure 14: Variation of Nusselt number (s = 0.5, ε = 0.01, Pr =7.0) at y = 0.

suction/injection parameter increases until a steady-statevalue is attained. Figure 13 shows monotonic decrease inrate of heat transfer as s increases. It is reflected from thisfigure that both the transient and steady-state rate of heattransfer coincide at large values of s and t. Figures 14 and15 show the variation of reactant consumption parameteron the rate of heat transfer. These two figures reveal thatthere is a monotonic increase in the rate of heat transfer at(y = 0, y = 1) corresponding to water (Pr = 7.0) and air(Pr = 0.71). This is due to the higher temperature gradientbetween fluid and porous plates. It is also observed from thefigure that as time increases the rate of heat transfer increasesuntil steady-state Nusselt number is reached. Figures 16 and17 show the influence of reactant consumption parameterwith time on the Nusselt number. It is reflected from thesefigures that the rate of heat transfer is directly proportional to


0

0.5

1

1.5

2

2.5

3

1 2 3

Nu

ssel

t n

um

ber

Steady state

λ

t = 0.2

t = 0.3t = 0.4


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2 3

Nu

ssel

t n

um

ber

Steady state

λ

t = 1.5

t = 2

t = 2.5


the reactant consumption parameter. These two figures alsoshow that as time increase the rate of heat transfer increasesuntil steady-state value is reached.

6. Conclusion

The problem of unsteady as well as steady-state naturalconvection Couette flow of reactive viscous fluid in avertical channel formed by two vertical porous plates hasbeen presented. The velocity field and temperature fieldare obtained analytically by perturbation series method forsteady free convective Couette flow of viscous reactive fluidin a vertical channel formed by two vertical porous platesand numerically by implicit finite difference technique forunsteady free convective Couette flow of viscous reactivefluid in a vertical channel formed by two vertical porousplates. Graphical results for the velocity, temperature, skin

0

0.5

1

1.5

2

2.5

1 2 3

Nu

ssel

t n

um

ber

Steady state

λ

t = 0.2t = 0.3

t = 0.4


friction, and the Nusselt number variations were presentedand discussed for various physical parametric values.

The main findings are as follows.

(i) Skin friction is always higher in the case of air (Pr =0.71) than water (Pr = 7.0).

(ii) It is also seen that increase in λ increases heat transferon the porous plates.

(iii) The heat transfer is higher at left porous plate y =0 where injection takes place in comparison to rightplate where suction takes place y = 1.

(iv) The introduction of suction/injection has distortedthe symmetric nature of the flow.

(v) During the course of computation it is observed thatat steady state the effect of Pr on the fluid flow is toincrease the velocity and temperature as Pr increaseswhen injection is considered (s < 0) at y = 0while fluid velocity and temperature decreases as Princreases in the presence of suction (s > 0) at y = 0,which is not true in transient case (i.e., velocity andtemperature decreases as Pr increases).

Appendix

Consider

c1 = δ[1− exp(−s)]

{1z1

+1

s2Pr− 1

2sPr− 1

s2Pr2(1− Pr)

}

− δ{

1s3Pr

+1z1s

− 1z1sPr(1− Pr)

},


c2 = δ[1− exp(−s)]

{1

2s2Pr+

1s3Pr2(1− Pr)

}

− δ

s[1− exp(−s)]

{1z1

+1

s2Pr

},

k1 = − 12

{1sPr

− 2

(sPr)2 +2

(sPr)3

}1[

1− exp(−sPr)]

+1sPr

[1 + exp(−sPr)

][1− exp(−sPr)

]2 −1

(sPr)21[

1− exp(−sPr)]2

+exp(−sPr)

(sPr)2[1− exp(−sPr)]2 +

1

(sPr)3exp(−sPr)[

1− exp(−sPr)] ,

k2 = − k1

sPr− 1

(sPr)3[1− exp(−sPr)] − 1

(sPr)4 ,

A = 1s6Pr2

+1

s6Pr3+

1s6Pr4

k1

s3Pr+

k2

s2Pr(1− Pr),

B = 1[1− exp(−sPr)

]

×{

1

(1− Pr)2s5Pr3+

1s5Pr2

− 1s5Pr4(1− Pr)

+1

s5Pr3

},

A1 = 1

6(sPr)2

{1s− 3

s2+

6s3− 6

s4

}

−{

1

2(sPr)3 +1

2(sPr)2[1− exp(−sPr)]}

×{

1s− 2

s2+

2s3

},

B1 = 1

(sPr)4

{1s− 1

s2

}+

1

(sPr)31[

1− exp(−sPr)]

×{

1s(1− Pr)

− 1

s2(1− Pr)2

}exp(−sPr)

+1

(sPr)41[

1− exp(−sPr)] exp(−sPr)s(1− Pr)

+1

(sPr)3

{1s− 1

s2

}1[

1− exp(−sPr)]

+k

sPr

{1s− 1

s2

}− k2 exp(−sPr)

s2Pr(1− Pr),

G1 = −δ[A + B]s− sG2,

G2 = δ[(A + B) + (A1 + B1)][

exp(−s)− 1] .

(A.1)

Nomenclature

Cp: Specific heat of the fluid at constant pressureE: Activation energyg: Acceleration due to gravity

H : Gap between the channelsNu0: Nusselt number at y = 0Nu1: Nusselt number at y = 1Pr: Prandtl numberQ: Heat reaction parameterR: Universal gas constants: Suction/injection parametert′: Dimensional timet: Dimensionless timeT′: Dimensional temperature of the fluidT0: Initial temperature of the fluidu′: Dimensional velocity of the fluidu: Dimensionless velocity of the fluidx′: Dimensional coordinate parallel to channely′: Dimensional coordinate perpendicular to channely: Dimensionless coordinate.

Greek Letters

β: Volumetric coefficient of thermal expansionρ: Density of the fluidτ0: Dimensionless skinfriction at y = 0τ1: Dimensionless skin friction at y = 1α: Thermal diffusivityθ: Dimensionless temperatureλ: Reactant consumption parameterv: Kinematic viscosityε: Activation energy parameterδ: Grashof number.

Acknowledgment

The author A. K. Samaila is thankful to Usmanu DanfodiyoUniversity, Sokoto, for financial support.

References

[1] S. Mahmud and R. Andrew Fraser, “Flow, thermal, andentropy generation characteristics inside a porous channelwith viscous dissipation,” International Journal of ThermalSciences, vol. 44, no. 1, pp. 21–32, 2005.

[2] A. K. Singh, H. R. Gholami, and V. M. Soundalgekar, “Tran-sient free convection flow between two vertical parallel plates,”Heat and Mass Transfer/Waerme- und Stoffuebertragung, vol.31, no. 5, pp. 329–331, 1996.

[3] L. A. Florio and A. Harnoy, “Augmenting natural convectionin a vertical flow path through transverse vibrations of anadiabatic wall,” Numerical Heat Transfer A, vol. 52, no. 6, pp.497–530, 2007.

[4] B. K. Jha and A. O. Ajibade, “Free convective flow betweenvertical porous plates with periodic heat input,” ZAMMZeitschrift fur Angewandte Mathematik und Mechanik, vol. 90,no. 3, pp. 185–193, 2010.

[5] T. Paul, B. K. Jha, and A. K. Singh, “Transient natural con-vection in a vertical channel,” International Journal of AppliedMechanics and Engineering, vol. 6, no. 4, pp. 913–922, 2001.

[6] L. Langellotto, O. Manca, and S. Nardini, “Numerical inves-tigation of transient natural convection in air in a convergentvertical channel symmetrically heated at uniform heat flux,”Numerical Heat Transfer A, vol. 51, no. 11, pp. 1065–1086,2007.


[7] P. K. Muhuri, “Flow formation in Couette motion in magne-tohydrodynamics with suction,” Journal of the Physical Societyof Japan, vol. 18, no. 11, pp. 1671–1675, 1963.

[8] A. K. Singh, “Natural convection in unsteady Couette motion,”Defence Science Journal, vol. 38, no. 1, pp. 35–41, 1988.

[9] O. Abdulaziz and I. Hashim, “Fully developed free convectionheat and mass transfer of a micropolar fluid between porousvertical plates,” Numerical Heat Transfer A, vol. 55, no. 3, pp.270–288, 2009.

[10] T. Fang, “Further discussion on the incompressible pressure-driven flow in a channel with porous walls,” InternationalCommunications in Heat and Mass Transfer, vol. 31, no. 4, pp.487–500, 2004.

[11] T. Fang, “A note on the incompressible couette flow withporous walls,” International Communications in Heat and MassTransfer, vol. 31, no. 1, pp. 31–41, 2004.

[12] J. Wang, Z. Gao, G. Gan, and D. Wu, “Analytical solution offlow coefficients for a uniformly distributed porous channel,”Chemical Engineering Journal, vol. 84, no. 1, pp. 1–6, 2001.

[13] L. Oxarango, P. Schmitz, and M. Quintard, “Laminar flowin channels with wall suction or injection: a new model tostudy multi-channel filtration systems,” Chemical EngineeringScience, vol. 59, no. 5, pp. 1039–1051, 2004.

[14] O. D. Makinde and A. Ogulu, “The effect of thermal radiationon the heat and mass transfer flow of a variable viscosityfluid past a vertical porous plate permeated by a transversemagnetic field,” Chemical Engineering Communications, vol.195, no. 12, pp. 1575–1584, 2008.

[15] O. D. Makinde and R. L. Maserumule, “Thermal criticality andentropy analysis for a variable viscosity Couette flow,” PhysicaScripta, vol. 78, no. 1, Article ID 015402, 2008.

[16] M. A. Chaudhary and J. H. Merkin, “Free-convectionstagnation-point boundary layers driven by catalytic surfacereactions: I the steady states,” Journal of Engineering Mathe-matics, vol. 28, no. 2, pp. 145–171, 1994.

[17] J. H. Merkin and M. A. Chaudhary, “Free-convection bound-ary layers on vertical surfaces driven by an exothermicsurface reaction,” Quarterly Journal of Mechanics and AppliedMathematics, vol. 47, no. 3, pp. 405–428, 1994.

[18] M. A. Chaudhary, A. Linan, and J. H. Merkin, “Free convectionboundary layers driven by exothermic surface reactions:critical ambient temperatures,” Mathematical Engineering inIndustry, vol. 5, no. 2, pp. 129–145, 1996.

[19] R. O. Ayeni, “On the explosion of chain-thermal reaction,”Journal of the Australian Mathematical Society, vol. 24, pp.194–202, 1982.

[20] F. S. Dainton, Chain Reaction, An Introduction Wiley, NewYork, NY, USA, 1960.

[21] B. K. Jha, A. K. Samaila, and A. O. Ajibade, “Transient free-convective flow of reactive viscous fluid in a vertical channel,”International Communications in Heat and Mass Transfer, vol.38, no. 5, pp. 633–637, 2011.

[22] A. Hazem Attia, “The effect of suction and injection onunsteady Couette flow,” Kragujevac Journal of Science, vol. 32,pp. 17–24, 2010.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



Advances in

Astronomy

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of

Unsteady/SteadyFreeConvectiveCouetteFlowof ...downloads.hindawi.com/archive/2012/794741.pdf ·...

Documents

Transcript of Unsteady/SteadyFreeConvectiveCouetteFlowof ...downloads.hindawi.com/archive/2012/794741.pdf ·...