Ten

Samuel O. Adesanya

D nc

R ed 4 N

KEYWORDS

Oscillatory ow; uid through a porous vertical channel with velocity slip and temperature jump. Exact solution

presented and discussed.

open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

heat transfer on the peristaltic ow through an asymmetric

channel. Hayat et al. [2] investigated the slip effects on the ow

More interesting result on slip ow can be seen in [35] to men-li [6] (and refer-ndary condition

ver hydrophobicAlso in mechan-channel

cial heartblood, pa

subjected to periodic cooling several investigations on oscilla-tory ow problems have been conducted. For example, Jhaand Ajibade [79] presented some results on the free convective

motion of a viscous incompressible uid between two periodi-cally heated innite vertical parallel plates. In all the studies in[79], the effects of velocity slip and temperature jump were

neglected by assuming that the uid velocity is zero relativeto the solid boundary. However, with the inux of microelec-tronic devices over the last few years, all these studies may not

* Tel.: +234 8055161181.

E-mail address: adesanyas@run.edu.ng

Peer review under responsibility of Ain Shams University.

Production and hosting by Elsevier

Ain Shams Engineering Journal (2015) xxx, xxxxxx

Ain Shams

Ain Shams Engin

www.elsevier.cowww.sciencesolutions of momentum and energy equations with slip and foam to mention just a few.

In order to improve the cooling of electronic componentsThe study of convective heat transfer to viscous incompressibleslip ow through a channel has gained a lot of attention

because of its importance in physiological ows, electroniccooling, drying processes, heat exchangers and many more.An appreciable number of studies have been reported on theseows under different ow conditions by Hayat and his collab-

orators. For instance, Hayat et al. [1] obtained closed form

tion just a few. Moreover, Mehmood and Aences therein) showed that the no-slip bou

may not be suitable for hydrophilic ows oboundaries at both the micro- and nanoscale.ical engineering, partial slip can occur in a

coated or polished surface like polished artiThe phenomenon also common in the ow ofhttp://dx.doi.org/10.1016/j.asej.2014.12.0082090-4479 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: Adesanya SO, Free convective ow of heat generating uid through a porous vertical channel with velocity slip and temjump, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2014.12.008with a

valves.int and1. Introduction and heat transfer of a third grade uid past a porous plate. 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is anVelocity slip;

Temperature jump;

Micro-channel

of the oscillatory ow problem is obtained in the slip ow regime through a microchannel. The

effects of various ow parameters on the temperature and velocity proles together with the inu-

ence of the velocity slip and temperature jump on the rate of heat transfer and the skin friction areeceived 19 June 2014; revisAes, Redeemers University, Redemption City, Ogun State, Nigeria

ovember 2014; accepted 14 December 2014

bstract This paper investigates the unsteady free convective ow of heat generating/absorbingepartment of Mathematical ScieENGINEERING PHYSICS AND MATHEMA

Free convective ow of heat gthrough a porous vertical chaslip and temperature jump

*ICS

nerating uidnel with velocity

University

eering Journal

m/locate/asejdirect.comperature

to macroscopic velocity slip and temperature jump [17]. The

ow is induced by the periodic heating introduced on bothwalls. The governing equations for the fully developed ow

Q is the term due to internal heat generation

k represents the thermal conductivity/ is the specic heat ratioT uid temperatureT0;T1 and T2 referenced uid temperature respectivelykn is the Knudsen numberSt is the Strouhal numbers is the suction/injection parameter

Pr is the Prandtl numberd is the heat generating parameterc is the Navier slip parameter

2 S.O. Adesanyaaccurately guarantee the efcient and effective removal of heat

generated within the high performance electronic devices.According to Dharaiya and Kandlikar [10], Hung and Ru[11] micro-channels are the most efcient way of high heat uxremoval from small areas.

It is known that for ows through a micro-channel, kn 0represents the no slip condition and kn < 0:001 is valid for con-tinuum ow while 0:001 < kn < 0:1 is the slip regime whichcan be modelled using the Navier stokes equation taking theslip velocity and temperature jump into consideration for accu-rate results. Motivated by Zheng et al. [12], the main objective

of this paper was to investigate the combined effects of partialslip and temperature jump on the free convective ow of heatgenerating and absorbing uid through a micro-channelthereby extending the work done in [8] to a micro-channel.

More details on velocity slip and temperature jump can befound in the work by Haddad et al. [13,14], Hooman et al.[15], Chen [16] and Aziz [17].

To achieve this objective, the problem is formulated and thedimensionless analysis is performed in the Section 2 of thepaper. Based on the oscillatory nature of the ow, exact solu-

tion of the problem is presented. It is interesting to note thatwhen velocity slip and temperature jump are neglected in thepresent problem, the result coincides with what is obtained

in [8]. The real part of the results is presented and discussedin Section 4 of the paper while Section 5 concludes the work.

Nomenclature

t0 timeu0 velocityv0 constant horizontal velocityq uid densitym kinematic viscosityrT is the thermal accommodation coefcientk is the molecular mean free pathn is the tangent momentum accommodation coef-

cient intensityg gravitational accelerationb volumetric expansionCp is the specic heat capacity at constant pressure2. Mathematical analysis

Consider the laminar free convective ow of a viscous incom-pressible heat generating/absorbing uid in a vertical channeldue to heating of the porous channel plates with slip and tem-

perature jump at the lower wall. The micro-channel walls aretaken vertically and parallel to the x-axis at y=h. It isassumed that on one plate (y= h), uid is injected into the

channel with certain constant velocity v0 and that it is suckedoff from the other plate (y= h) at the same rate (see Fig. 1).It is further assumed that interfacial interaction between the

gas molecules and the surface atoms exists. In other words,the gas molecules are assumed to interact with the surface ofthe solid via a long range attractive force. As a result of this

interaction, the gas molecules can be adsorbed onto the surfacewhich are then reected after some time lag. This time lag leads

Please cite this article in press as: Adesanya SO, Free convective ow of heat generajump, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2014.12.008and heat transfer can be written as [8]:

@u0@t0 v0 @u

0@y0 m @

2u0@y02 gbT T0

@T@t0 v0 @T@y0 kqCp @

2T@y02 QqCp T0 T

9=; 1

together with appropriate initial condition

u0t0; y0 0; Tt0; y0 T0 t0 0 2For rareed ow with temperature jump, the appropriate

initial-boundary conditions can be written as [8,12 and 17]

u0t0; y0 2 nn

kdu0

dy0;

Ty0; t0 T1 T2Cosxt 2 rTrT2/

/ 1kPr

dT

dy0;

y0 h t0 > 0 3and the non-moving wall and isothermal condition give

u0t0; y0 0; Ty0; t0 T1 T2Cosxt y0 h t0 > 04Figure 1 Channel geometry.

ting uid through a porous vertical channel with velocity slip and temperature

The following expressions are used in [8] to split the velocitym1 m2

Free convective ow of heat generating uid 3and temperature into steady and unsteady part respectively.

u0t; y gbh2m T1 T0Ay T2ByeiwtTt; y T0 T1 T0Fy T2Gyeixt

)5

together with the following dimensionless quantities and

variables

y y0h; St h2xm ; Pr lCpk ; s hV0m

d Q0h2k; kn kh 2rTrT

2//1 ; c 2nknh

6

gives

iStT2Byeiwt T1 T0sA0y T2sB0yeiwt T1 T0A00y T2B00yeiwt T1 T0Fy T2Gyeixt 7

and

iPrStT2Gyeixt T1 T0sPrF0y T2sPrG0yeixt T1 T0F00y T2G00yeixt dT1 T0Fy T2dGyeixt 8

Similarly, the boundary conditions yield

T1 T0Ay T2Byeiwt cknT1 T0Ay cknT2Byeiwt; y 1

T1 T0Ay T2Byeiwt 0 y 19

together with the boundary conditions

T1 T0Fy T2Gyeixt T1 T0 T2Cosxt kn

PrT1 T0F0y T2G0yeixt; y 1

T1 T0Fy T2Gyeixt T1 T0 T2Cosxt; y 110

equating orders of eixt, in the steady regime we have

A00y sA0y Fy; A1 cA01; A1 0F00y sPrF0y dFy 0; F1 1 kn

PrF01; F1 1

11While in the periodic regime, we get

B00y sB0y iStBy Gy; B1 cB01;B1 0G00y sPrG0y d iPrStGy 0;G1 1 kn

PrG01; G1 1

12Evidently, in the asymptotic case as c; kn ! 0 the boundary

valued problem (11), (12) reduced to that in [8]. Observe thatEq. (12) is a generalized form of (11) and as such will givethe same solution as (12) whenever St 0. Hence, the solutionof the (12) when St 0 will be sufcient to describe the peri-odic uid ow behaviour.

3. Method of solution

The exact solution of the boundary valued problem (12) can bewritten asPlease cite this article in press as: Adesanya SO, Free convective ow of heat generajump, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2014.12.008Gy a1e 2 y a2e 2 yBy a3e

m32 y a4e

m42 y 1

n0a5e

m12 y a6e

m22 y

13The rate of heat transfer is given by

Nu @G@y

a1m12

em1y a2m22

em2y 14

while the shear stress is given by

Sf @B@y

a3m32

em32 y a4m4

2em42 y 1

n0

a5m12

em12 y a6m2

2em22 y

15

the constants are all dened in the Appendix.

4. Results and discussion

In this section, variation of parameters are carried out for dif-

ferent values of parameters such as the heat generation/absorption, temperature jump, suction/injection, Prandtl num-ber, Knudsen number and Navier slip. To show the accuracyof the solution, Tables 1 and 2 show the comparison of the

present result with previously obtained result in the literaturein the asymptotic limit. Note that d < 0 is the internal heatgeneration while d > 0 implies the heat absorption. Fig. 2shows the temperature prole at different values of the temper-ature jump parameter. It is observed from the plot that duringuid absorption, the temperature prole is decreased with an

increase in the values of the temperature jump parameter kn.This is due to the fact that the molecular distance of the uidincreases as kn increases hence heat ux decreases within the

channel.Figs. 3 and 4, represent the effects of variation of heat gen-

eration/absorption parameter on the temperature prole. Theresult shows that as the internal generation parameter

increases there is corresponding rise in the uid temperature.This is due to the fall in thermal conductivity of the uid.On the other hand if the uid chemical interaction is endother-

mic then the uid temperature decreases as seen in Fig. 4. InFig. 5, the inuence of uid suction and injection on the uidtemperature is shown. From the gure, one observed that in

the absence of suction/injection the temperature is symmetricalabout the channel half width, further increase in this parame-ter shows that the temperature increases but the prole is

observed to be skewed towards the wall with suction. How-ever, a decrease in the uid temperature is observed in Fig. 6as the Strouhal number (oscillation parameter) increases. Thisis attributed to the reduction in the intensity of heating of the

boundary plates as the frequency of heating increases. InFig. 7, it is observed that an increase in the uid Prandtl num-ber decreases the uid temperature. This is due to decrease in

thermal conductivity of the uid within the micro-channel.As observed in Fig. 8, the rate of heat transfers at the plate

with suction decreases with increase in the temperature jump

parameter. This is physically true since the uidplate interac-tion at the suction wall decreases. This consequently decreasesthe heat ux on the boundary plates as the temperature jumpincreases. In Fig. 9, it is observed that an increase in the Navier

slip parameter enhances the ow at the plate with suction thisis due to increased gasmolecule interaction with the suctionwall. On the other hand as observed in Fig. 10, an increaseting uid through a porous vertical channel with velocity slip and temperature

Table 1 Showing convergence of solution of Gy when c 0; kn 0; St 1 s Pr d.

4 S.O. Adesanyay Previous result [8] for Gy1 10.75 0.773980.5 0.6480870.25 0.5969080 0.602555

0.25 0.652739

0.5 0.739168

0.75 0.856242

1 1in the temperature jump is seen to decrease the uid velocity

close to the suction wall as well as decreasing the uid temper-ature maximum. This is physically correct since an increase inkn implies an increase in the molecular distance in the uid and

this decreases the uid temperature as shown in Fig. 2. This inturn weakens the convection currents and it is expected todecrease the uid ow velocity.

While in Fig. 11, increasing the heat generation parameter

is observed to improve the uid ow. This is due to the phys-ical fact that temperature within the channel grows withincrease in heat generation parameter (Fig. 3) due to chemical

interaction of the uid particles and this strengthens the con-vection currents which consequently increases the uid velocity

Table 2 Showing convergence of solution of By when c 0; kn y Previous result [8] for By1 2:22045 10160.75 0.1414660.5 0.2139660.25 0.2431390 0.242830

0.25 0.218953

0.5 0.172309

0.75 0.100621

1 6:66134 1016

Figure 2 Temperature prole for different values of temperature

jump parameter.

Please cite this article in press as: Adesanya SO, Free convective ow of heat generajump, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2014.12.008Present result for Gy Absolute error1 4:44089 10160.77398 1:11022 10160.648087 2:22045 10160.596908 1:11022 10160.602555 2:22045 10160.652739 5:55112 10160.739168 7:77156 10160.856242 7:77156 10161 4:44089 1016while the reverse is the case during uid absorption as shownin Fig. 12. In Fig. 13, it is observed that an increase in Prandtlnumber leads to a decrease in the ow velocity. This is true due

to uid thickening associated with rise in uid viscosity as thePrandtl number increases. As seen in Fig. 14, an increase inuid suction/injection increases the ow velocity and the

increase breaks the symmetry in the ow, which is skewedtowards the plate with suction. While in Fig. 15, the effect ofoscillation on the ow velocity is seen to reduce the ow veloc-

ity. Finally, the effect of the velocity slip on the skin friction ispresented in Fig. 16. From the graph it is observed that anincrease in the Navier slip parameter weakens the skin frictionat the suction wall.

0; St 1 s Pr d.Present result for By Absolute error4:44089 1016 2:22045 10160.141466 7:77156 10160.213966 2:22045 10160.243139 2:22045 10160.242830 1:66533 10160.218953 55:55112 10160.172309 1:11022 10160.100621 7:77156 10162:22045 1016 4:44089 1016

Figure 3 Temperature prole for different values of internal heat

generation parameter.

ting uid through a porous vertical channel with velocity slip and temperature

Figure 4 Temperature prole for different values of internal heat

absorption parameter.

Figure 5 Temperature prole for different values of wall

injection parameter.

Figure 6 Temperature prole for different values of oscillation

parameter.

Figure 7 Temperature prole for different values of Prandtl

number.

Figure 8 Heat transfer rate for different values of temperature

jump parameter.

Figure 9 Velocity prole for different values of velocity slip

parameter.

Free convective ow of heat generating uid 5

Please cite this article in press as: Adesanya SO, Free convective ow of heat generating uid through a porous vertical channel with velocity slip and temperaturejump, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2014.12.008

Figure 10 Velocity prole for different values of temperature

jump parameter.

Figure 11 Velocity prole for different values of internal heat

generation parameter.

Figure 12 Velocity prole for different values of internal heat

absorption parameter.

Figure 13 Velocity prole for different values of Prandtl

number.

Figure 14 Velocity prole for different values of wall injection

parameter.

Figure 15 Velocity prole for different values of Strouhal

number.

6 S.O. Adesanya

Please cite this article in press as: Adesanya SO, Free convective ow of heat generating uid through a porous vertical channel with velocity slip and temperaturejump, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2014.12.008

a5 4 4iSta1 2sa1m2 a1m22

a6 4 2sa2m1 a2m21 4iSta2

n0 ss2 4iSt

pm1

s

s2 4iSt

pm1

s

s2 4iSt

pm2

s

s2 4iSt

pm2

n1 n0 2em3 2em4 cm3em4 cm4em3

References

Free convective ow of heat generating uid 75. Conclusion

In this paper, the free convective ow of heat generating/absorbing uid through porous vertical micro-channel withvelocity slip and temperature jump is studied. Exact solutionsare obtained for the velocity and temperature proles. The

present result reduced to that obtained in [8] in the absenceof Navier slip and temperature jump. In addition, an increasein the slip parameter is shown to increase the ow velocity and

decrease the shear stress at the suction wall. Moreover, anincrease in the temperature jump parameter increases the uidtemperature and decreases the rate of heat transfer at the suc-

tion wall. The result obtained here is signicant in the coolingof electronic devices.

Acknowledgement

The author would like to thank the anonymous reviewers for

their useful contributions and suggestions.

Appendix A

m1 Pr sPr2 s2 4i Pr St 4d

p

2 2p

Figure 16 Wall shear stress for different values of slip

parameter.m2 Pr s Pr s 4i Pr St 4d

a1 em12 2Pr 2em2 Pr knm2

2em1 Pr 2em2 Pr em2knm1 em1knm2

a2 em22 2Pr 2em2 Pr knm2

2em1 Pr 2em2Pr em2knm1 em1knm2

a3 4e

12m1m2m3 em4

12m22 m1 em112m2m4 2

a5

nn1

a4 4e

12m1m2m4 em112m2cm3 2 em312m22 cm1a5 e

nn1

Please cite this article in press as: Adesanya SO, Free convective ow of heat generajump, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2014.12.008em212m1a6m4 2 em412m1 2 m2

a6

o

m212m1cm3 2 em312m12 cm2a6

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Samuel Olumide Adesanya received his BSc in

Mathematics at Ogun State University, Ago-

Iwoye, Ogun State in 1998 and MSc in

Mathematics in 2007 at Olabisi Onabanjo

University. He got his PhD in Mathematics at

Ladoke Akintola University of Technology,

Ogbomosho, Oyo State, Nigeria in 2012. He is

presently a Lecturer at the Department of

Mathematical Sciences, Redeemers Univer-

sity, Nigeria. SOA has published many jour-

nal articles on Adomian decomposition Method, reactive ows and

nonlinear systems.

8 S.O. AdesanyaPlease cite this article in press as: Adesanya SO, Free convective ow of heat generajump, Ain Shams Eng J (2015), http://dx.doi.org/10.1016/j.asej.2014.12.008ting uid through a porous vertical channel with velocity slip and temperature

app8Free convective flow of heat generating fluid through a porous vertical channel with velocity slip and temperature jump1 Introduction2 Mathematical analysis3 Method of solution4 Results and discussion5 ConclusionAcknowledgementAppendix AReferences