research article combined effect of buoyancy force and...

Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 725643 8 pageshttpdxdoiorg1011552013725643

Research ArticleCombined Effect of Buoyancy Force and Navier Slip onMHD Flow of a Nanofluid over a Convectively Heated VerticalPorous Plate

Winifred Nduku Mutuku-Njane1 and Oluwole Daniel Makinde2

1 Mechanical Engineering Department Cape Peninsula University of Technology PO Box 1906 Bellville 7635 South Africa2 Faculty of Military Science Stellenbosch University Private Bag X2 Saldanha 7395 South Africa

Correspondence should be addressed to Winifred Nduku Mutuku-Njane winnieronnie1yahoocom

Received 18 May 2013 Accepted 19 June 2013

Academic Editors E Konstantinidis and L Nobile

Copyright copy 2013 W N Mutuku-Njane and O D MakindeThis is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We examine the effect ofmagnetic field on boundary layer flow of an incompressible electrically conducting water-based nanofluidspast a convectively heated vertical porous plate with Navier slip boundary condition A suitable similarity transformation isemployed to reduce the governing partial differential equations into nonlinear ordinary differential equations which are solvednumerically by employing fourth-order Runge-Kutta with a shooting techniqueThree different water-based nanofluids containingcopper (Cu) aluminium oxide (Al

2O3) and titanium dioxide (TiO

2) are taken into consideration Graphical results are presented

and discussed quantitatively with respect to the influence of pertinent parameters such as solid volume fraction of nanoparticles(120593) magnetic field parameter (Ha) buoyancy effect (Gr) Eckert number (Ec) suctioninjection parameter (119891

119908) Biot number (Bi)

and slip parameter (120573) on the dimensionless velocity temperature skin friction coefficient and heat transfer rate

1 IntroductionMagnetohydrodynamic (MHD) boundary layer flow of anelectrically conducting viscous incompressible fluid with aconvective surface boundary condition is frequently encoun-tered in many industrial and technological applications suchas extrusion of plastics in the manufacture of Rayon andNylon the cooling of reactors purification of crude oil textileindustry polymer technology and metallurgy As a resultthe simultaneous occurrence of buoyancy and magnetic fieldforces on fluid flowhas been investigated bymany researchers[1ndash5] In their investigations all the authors mentionedabove assumed the no-slip boundary conditions Howevermore recently researchers have investigated the flow problemtaking slip flow condition at the boundary [6ndash9]

On the other hand with the advent of nanofluids therehas been wide usage of recently discovered smart fluidin many industrial and biomedical applications Nanofluidconcept is employed to designate a fluid in which nanometer-sized particles are suspended in conventional heat transferbase fluids to improve their thermal physical properties

Nanoparticles are made from various materials such asmetals (Cu Ag Au Al and Fe) oxide ceramics (Al

2O3 CuO

and TiO2) nitride ceramics (AlN SiN) carbide ceramics

(SiC tiC) semiconductors carbonnanotubes and compositematerials such as alloyed nanoparticles or nanoparticle core-polymer shell composites It is well known that conventionalheat transfer fluids such as oil water and ethylene glycol ingeneral have poor heat transfer properties compared to thoseof most solids Nanofluids have enhanced thermophysicalproperties such as thermal conductivity thermal diffusivityviscosity and convective heat transfer coefficients comparedwith those of base fluids like oil or water [10] Severalauthors [11ndash14] have conducted theoretical and experimen-tal investigations to demonstrate that nanofluids distinctlyexhibit enhanced heat transfer properties which goes upwith increasing volumetric fraction of nanoparticles Furtherstudies on nanofluids have been currently undertaken byscientists and engineers due to their diverse technical andbiomedical applications such as nanofluid coolant electron-ics cooling vehicle cooling transformer cooling computers

2 The Scientific World Journal

Table 1 Thermophysical properties of water and nanoparticles [23 24]

Materials 120588 (kgm3) 119888119901(JkgK) 119896 (WmK) 120590 (Sm)

Pure water 9971 4179 0613 55 times 10minus6

Copper (Cu) 8933 385 401 596 times 106

Alumina (Al2O3) 3970 765 40 35 times 106

Titania (TiO2) 4250 6862 89538 26 times 106

cooling and electronic devices cooling medical applicationsmagnetic drug targeting cancer therapy and safer surgeryby cooling process industries and materials and chemicalsdetergency food and drink oil and gas paper and printingand textiles

According to Aziz [15] the concept of no-slip conditionat the boundary layer is no longer valid for fluid flows inmicroelectromechanical systems and must be replaced byslip condition The slip flow model states a proportionalrelationship between the tangential components of the fluidvelocity at the solid surface to the shear stress on the fluid-solid interface [16] The proportionality is called the sliplength which describes the slipperiness of the surface [7]Many researchers studied the effect of linear momentumand nonlinear slip on the MHD boundary layer flow withheatmass transfer of freeforcedcombined convection pastdifferent geometries [17ndash20] In spite of the importance ofMHD related studies on boundary layer flow problems thepossibility of fluid exhibiting apparent slip phenomenon onthe solid surface has received little attention

The aim of the present study is to investigate the com-bined effects of buoyancy magnetic field suction Navier slipand convective heating on a steady boundary layer flow overa flat surface In the subsequent sections the boundary layerpartial differential equations first transformed into a systemof nonlinear ordinary differential equations before beingsolved numerically using a shooting method together withthe fourth-order Runge-Kutta-Fehlberg integration schemeA graphical representation of the pertinent parameters onthe flow field and heat transfer characteristics is displayedand thoroughly discussed To our best of knowledge theinvestigations of the proposed problem are new and theresults have not been published before

2 Model FormulationThe steady laminar incompressible two-dimensional MHDboundary layer flow of an electrically conducting water-based nanofluid past a convectively heated porous verticalsemiinfinite flat plate under the combined effects of buoyancyforces and Navier slip is considered The nanofluids containthree different types of nanoparticles Cu Al

2O3 and TiO

2

Let the 119909-axis be taken along the direction of plate and let119910-axis be normal to it The left side of the plate is assumedto be heated by convection from a hot fluid at temperature119879119891 which provides a heat transfer coefficient ℎ

119891 while

the right surface is subjected to a stream of an electricallyconducting cold nanofluid at temperature119879

infinin the presence

of a transverse magnetic field of strength 1198610applied parallel

to the 119910-axis as shown in Figure 1 The induced magneticfield due to the motion of the electrically conducting fluid is

x

g

T = Tinfin

Bo

y

u

Nanofluid = Vw

120582u = 120583f120597u

120597y120582

minuskf120597T

120597y= hf(Tf minus T)

u = Uinfin

Figure 1 Flow configuration and coordinate system

negligible It is also assumed that the external electrical fieldis zero and that the electric field due to the polarization ofcharges is negligible (see Table 1)

Assuming a Boussinesq incompressible fluid model thecontinuity momentum and energy equations describing theflow can be written as

120597119906

120597119909+120597V120597119910

= 0

119906120597119906

120597119909+ V

120597119906

120597119910= 119880infin

119889119880infin

119889119909+120583119899119891

120588119899119891

1205972119906

1205971199102+ 120573119899119891119892 (119879 minus 119879

infin)

minus1205901198991198911198612

0(119906 minus 119880

infin)

120588119899119891

119906120597119879

120597119909+ V

120597119879

120597119910=

119896119899119891

(120588119888119901)119899119891

1205972119879

1205971199102+

120583119899119891

(120588119888119901)119899119891

(120597119906

120597119910)

2

+1205901198991198911198612

0

(120588119888119901)119899119891

(119906 minus 119880infin)2

(1)

The boundary conditions at the plate surface and at the freestream may be written as

120582119906 (119909 0) = 120583119891

120597119906

120597119910(119909 0) V (119909 0) = 119881

119908

minus119896119891

120597119879

120597119910(119909 0) = ℎ

119891[119879119891minus 119879 (119909 0)]

119906 (119909infin) = 119880infin(119909) 119879 (119909infin) = 119879

infin

(2)

The Scientific World Journal 3

where (119906 V) are the velocity components of the nanofluidin the 119909- and 119910-directions respectively 119879 is the nanofluidtemperature 119880

infin(119909) = 119886119909 is the free stream velocity (which

implies that the free stream fluid velocity is increasing withaxial distance along the plate surface) 119879

infinis the free stream

temperature 119892 is acceleration due to gravity 120582 is the slipcoefficient 120583

119899119891is dynamic viscosity of the nanofluid 120588

119899119891

is density of the nanofluid 119896119899119891

is thermal conductivity ofthe nanofluid 120590

119899119891is electrical conductivity of the nanofluid

(120588119888119901)119899119891

is heat capacity at constant pressure of the nanofluidand 120573

119899119891is volumetric expansion coefficient of the nanofluid

which are defined as [21 22]

120583119899119891=

120583119891

(1 minus 120593)25 120588

119899119891= (1 minus 120593) 120588

119891+ 120593120588119904

120573119899119891= (1 minus 120593) 120573

119891+ 120593120573119904 120572

119899119891=

119896119899119891

(120588119888119901)119899119891

119896119899119891

119896119891

=(119896119904+ 2119896119891) minus 2120593 (119896

119891minus 119896119904)

(119896119904+ 2119896119891) + 120593 (119896

119891minus 119896119904)

(120588119888119901)119899119891= (1 minus 120593) (120588119888

119901)119891+ 120593(120588119888

119901)119904

120590119899119891= (1 minus 120593) 120590

119891+ 120593120590119904

(3)

where 120593 is the nanoparticle volume fraction (120593 = 0 corre-spond to a regular fluid) 120588

119891and 120588

119904are the densities of the

base fluid and the nanoparticle respectively 120573119891and 120573

119904are

the thermal expansion coefficients of the base fluid and thenanoparticle respectively 119896

119891and 119896119904are the thermal conduc-

tivities of the base fluid and the nanoparticles respectively(120588119888119901)119891and (120588119888

119901)119904are the heat capacitance of the base fluid

and the nanoparticle respectively and 120590119904and 120590

119891are the

electrical conductivities of the base fluid and the nanofluidrespectively

In order to simplify the mathematical analysis of the pro-blem we introduce the following dimensionless variables

120578 = (119886

120592119891

)

12

119910 120595 = (119886120592119891)12

119909119891 (120578)

120579 (120578) =119879 minus 119879infin

119879119891minus 119879infin

(4)

where 120578 is the similarity variable and120595 is the stream functiondefined as

119906 =120597120595

120597119910 V = minus

120597120595

120597119909 (5)

After introducing (5) into (1) and (2) we obtain the followingordinary differential equations

119891101584010158401015840+ (1 minus 120593)

25

(1 minus 120593 +120593120588119904

120588119891

)11989111989110158401015840

minus (1 minus 120593)25

(1 minus 120593 +120593120588119904

120588119891

)(1198911015840)2

+ (1 minus 120593)25

(1 minus 120593 +120593120588119904

120588119891

)

+ Gr(1 minus 120593)25 (1 minus 120593 +120593120588119904

120588119891

)(1 minus 120593 +120593120573119904

120573119891

)120579

minusHa(1 minus 120593)25 (1 minus 120593 +120593120590119904

120590119891

)(1198911015840minus 1) = 0

12057910158401015840+

Pr 119896119891[1 minus 120593 + 120593(120588119888

119901)119904(120588119888119901)119891]

119896119899119891

1198911205791015840

+Pr Ec 119896

119891

119896119899119891(1 minus 120593)

25(11989110158401015840)2

+HaPr Ec 119896

119891

119896119899119891

times (1 minus 120593 +120593120590119904

120590119891

)(1198911015840minus 1)2

= 0

(6)

Taking into account the variable plate surface permeabilityand the hydrodynamic slip boundary functions definedrespectively as

119881119908= minus119891119908(119886120592119891)12

120582119906 (119909 0) = 120583119891

120597119906

120597119910(119909 0) (7)

the boundary conditions are

119891 (0) = 119891119908 119891

1015840(0) = 120573119891

10158401015840(0)

1205791015840(0) = Bi [120579 (0) minus 1]

1198911015840(infin) = 1 120579 (infin) = 0

(8)

where a prime symbol denotes derivative with respect to 120578119891119908is a constant with 119891

119908gt 0 representing suction rate at

the plate surface 119891119908

lt 0 corresponds to injection 119891119908

=

0 shows an impermeable surface 120582 = 0 represents highlylubricated surface and 120582 = infin corresponds to a normalsurface The local Reynolds number (Re

119909) Grashof number

(Gr) Hartmann number (Ha) Prandtl number (Pr) Eckertnumber (Ec) slip parameter (120573) and Biot number (Bi) aredefined as

Re119909=119880infin119909

120592119891

Gr =120573119891119892 (119879119891minus 119879infin)

119880infin119886

Ha =120590119891119861119900

2

120588119891119886 Pr =

120592119891

120572119891

Ec =1198802

infin

119862119901119891(119879119891minus 119879infin) 120573 =

120583119891

120582radic

119886

120592119891

Bi =ℎ119891

119896119891

radic120592119891

119886

(9)


The physical quantities of practical significance in this workare the skin friction coefficient 119862

119891and the local Nusselt

number Nu which are expressed as

119862119891=

120591119908

1205881198911198802infin

Nu =119909119902119908

119896119891(119879119891minus 119879infin) (10)

where 120591119908is the skin friction and 119902

119908is the heat flux from the

plate which are given by

120591119908= 120583119899119891

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

119902119908= minus119896119899119891

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

(11)

Putting (11) into (10) we obtain

Re12119909119862119891=

1

(1 minus 120593)2511989110158401015840(0)

Reminus12119909

Nu = minus119896119899119891

119896119891

1205791015840(0)

(12)

The set of (6) and together with the boundary conditions(8) are coupled nonlinear boundary value problems whichare solved numerically using a shooting algorithm witha Runge-Kutta Fehlberg integration scheme This methodinvolves transforming (6) and (8) into a set of initial valueproblems which contain unknown initial values that needto be determined by guessing after which a fourth orderRunge-Kutta iteration scheme is employed to integrate theset of initial valued problems until the given boundaryconditions are satisfied The entire computation procedure isimplemented using a program written and carried out usingMaple computer language From the process of numericalcomputation the fluid velocity the temperature the skinfriction coefficient and the Nusselt number are proportionalto 1198911015840(120578) 120579(120578) 11989110158401015840(120578) and 1205791015840(120578) respectively

3 Results and Discussion

Physically realistic numerical values were assigned to thepertinent parameters in the system in order to gain an insightinto the flow structure with respect to velocity temperatureskin friction and Nusseltrsquos number The results were pre-sented graphically in Figures 2ndash13 and conclusions are drawnfor the flow field The Prandtl number is kept constant at 62[21] Ha = 0 corresponds to absence of magnetic field and120593 = 0 is regular fluid

31 Dimensionless Velocity Profiles Figures 2ndash4 illustratethe effects of various thermophysical parameters on thenanofluids velocity profiles Generally it is noted that the fluidvelocity increases gradually from zero at the plate surfaceto the free stream prescribed value far away from the platesatisfying the boundary conditions Figure 2 shows that themomentum boundary layer thickness for Cu-water nanofluidis smaller than the rest of the nanofluids consequently Cu-water nanofluid tends to flow closer to the convectively heatedplate surface and serves as a better coolant than the othernanofluids It is observed in Figures 3 and 4 that an increase

0 1 2 3120578

1

09

08

07

06

05

04

03

02

Cu-waterAl2O3-waterTiO2-water

fw = 02 Ha = 10minus12Gr = Bi = Ec = 120593 = 120573 = 01

f998400 (120578)

Figure 2 Velocity profiles for different nanofluids

Gr = Bi = 120573 = 01 fw = 02

Ha = 10minus15 10minus12 10minus11

Ec = 3 5 7

120593 = 0 002 008

0 1 2 3120578

1

09

08

07

06

05

04

03

02

f998400 (120578)

Figure 3 Velocity profiles with increasing Ha 120593 and Ec

in the magnetic field intensity (Ha) nanoparticle volumefraction (120593) Eckert number (Ec) Grashof number (Gr) andthe suctioninjection parameter (119891

119908) causes an overshoot of

the fluid velocity towards the plate surface hence decreasingboth the momentum boundary layer thickness and the fluidvelocity From the physics of the problem an increase in the


0 1 2 3120578

1

09

08

07

06

05

04

03

02

f998400 (120578)

fw = minus04 0 03

120601 = Ec = Bi = 01 Ha = 10minus12

Gr = 0 2 4

120573 = 05 1 3

Figure 4 Velocity profiles with increasing Gr 120573 and 119891119908

magnetic field intensity leads to an increase in the Lorentzforce which is a retarding force to the transport phenomenaThis retarding force can control the nanofluids velocity whichis useful in numerous applications such as magneto hydrody-namic power generation and electromagnetic coating ofwiresand metal We also note that the fluid velocity at the platesurface increases with an increase in the slip parameter (120573)This is in agreement with the fact that higher 120573 implies anincrease in the lubrication and slipperiness of the surface

32 Dimensionless Temperature Profiles Figures 5ndash7 showthe effects of various parameters on the temperature profileIn general the maximum fluid temperature is achievedat the plate surface due to the convectional heating butdecreases exponentially to zero far away from the platesurface satisfying the free stream conditions As expected atthe plate surface Cu-water has the highest temperature anda greater thermal boundary layer thickness than the othertwo nanofluids as seen in Figure 5This is in accordance withthe earlier observation since the Cu-water nanofluid is morelikely to absorb more heat from the plate surface owing toits close proximity to the hot surface It is observed fromFigure 6 that increasing Ha 120593 Bi and Ec leads to an increasein both the fluid temperature and the thermal boundary layerthicknessThis can be attributed to the additional heating dueresistance of fluid flow as a result of the magnetic field thepresence of the nanoparticle the increased rate at which theheat moves from the hot fluid to the plate and the additionalheating as a result of the viscous dissipation

On the other hand it is evident that surface slipperinessand suction affect the temperature of the fluid inversely This


120579(120578)

0 05 1 15 2

025

02

015

01

005

120578


Figure 5 Temperature profiles for different nanofluids

120579(120578)

00

05 1 15 2120578

07

06

05

04

03

02

01

120573 = Gr = 01 fw = 02

Ha = 10minus15 10minus12 10minus11120593 = 0 001 003 Bi = 0 03 1

Ec = 02 0215 03

Figure 6 Temperature profiles with increasing 120593 Ha Bi and Ec

is clearly seen from Figure 7 where both temperature andthermal boundary layer decrease as 119891

119908and 120573 increase

33 Effects of Parameters Variation on the Skin Friction andNusselt Number Figures 8ndash13 demonstrate the effects of thevarious pertinent parameters at the plate surface for boththe skin friction coefficient and the local Nusselt number


0 05 1 15 2120578

120579(120578)

0

03

02

01

fw = minus001 01 02

120573 = 001 02 03

120593 = Gr = Ec = Bi = 01 Ha = 10minus12

Figure 7 Temperature profiles with increasing 120573 Gr and 119891119908

fw = 02 Ha = 10minus12Ec = 120573 = Gr = Bi = 01

0 005 01 015 02120601

3

28

26

24

22

2

18

16

14


radicRe

xCf

Figure 8 Local skin friction profiles for different nanofluids

(rate of heat transfer) The presence of nanoparticle in theconvectional fluid leads to an increase in the skin friction asseen in Figure 8 where increasing the nanoparticle volumefraction increases the skin friction for the three nanoparticles(Cu Al

2O3 and TiO

2) used with Cu-water exhibiting the

highest incrementThis is as expected since Cu-water movescloser to the plate surface leading to an elevation in the

0 005 01 015 02

3

25

35

2

15

Gr = 1 3 4

Ec = 05 2 3

120573 = Bi = Gr = 01 fw = 02

120593


radicRe

xCf

Figure 9 Effects of increasing Gr Ha and Ec on local skin friction

fw = minus04 01 04

120573

120593 = Ec = Bi = Gr = 01 Ha = 10minus1226

24

22

2

18

16

14

12

1

08

radicRe

xCf

0 05 1 15

Figure 10 Effects of increasing 120573 and 119891119908on local skin friction

velocity gradient at the plate surface As expected increasingHa Gr Ec and 119891

119908leads to an increase in the skin friction

coefficient while an increase in 120573 reduces the skin frictioncoefficient as shown in Figures 9 and 10 There is an increasein the rate of heat transfer with an increase in 120593 Bi and 119891

119908

as seen in Figures 11-12 with Al2O3exhibiting the highest

incrementThe converse is seen with increasing Ha as shownin Figure 13


Cu-waterAl2O3-water

TiO2-water

0 005 01 015 02120593

Ec = 120573 = Gr = Bi = 01

fw = 02 Ha = 10minus12012

011

01

009

NuradicRe

x

Figure 11 Local Nusselt number for different nanofluids

Ec = 120573 = Gr = 01 Ha = 10minus12

fw = minus01 001 03

Bi = 007 01 012

014

013

012

011

01

009

008

007

006

0 005 01 015 02120593

NuradicRe

x

Figure 12 Effects of increasing 120593 Bi and Ha on local Nusseltnumber

4 Conclusions

The problem of hydromagnetic boundary layer flow of anincompressible electrically conducting water-based nanoflu-ids past a convectively heated vertical porous plate withNavier slip boundary condition was studied The governing

004

002

0

minus002

01 02 03 04 05Ec

012

01

008

006

120601 = 120573 = Gr = Bi = 01 fw = 02


NuradicRe

x

Figure 13 Effects of increasing 120593 Ha and Ec on local Nusseltnumber

nonlinear partial differential equationswere transformed intoa self-similar form and numerically solved using shootingtechnique with a fourth-order Runge-Kutta-Fehlberg inte-gration scheme putting into consideration the enhancedelectrical conductivity of the convectional base fluid due tothe presence of the nanoparticles Our results showed that thefluid velocity increases while the local skin friction decreaseswith the increase in the slip parameter (120573) but the reverseis observed with the increase in the magnetic field intensity(Ha) nanoparticle volume fraction (120593) Eckert number (Ec)Grashof number (Gr) and the suctioninjection parameter(119891119908) Both the temperature and the thermal boundary layer

thickness are enhanced by increasing the magnetic fieldintensity (Ha)nanoparticle volume fraction (120593) Eckert num-ber (Ec) and the intensity of Newtonian heating (Bi) whilethe cooling effect on the convectively heated plate surfaceis enhanced by increasing the velocity slip (120573) and suctionparameter (119891

119908)

Acknowledgments

The authors would like to thank the Organization forWomenin Science for the DevelopingWorld (OWSDW) the SwedishInternationalDevelopmentCooperationAgency (SIDA) andthe African Union Science and Technology Commission forfinancial support

References

[1] K R Singh and T G Cowling ldquoEffect of magnetic field on freeconvective flow of electrically conducting fluids past a semi-infinite flat platerdquo Quarterly Journal of Mechanics and AppliedMathematics vol 16 pp 1ndash15 1963

[2] S F Ahmmed andM S AlamSarker ldquoMHDnatural convectionflow of viscous incompressible fluid from a vertical flat platerdquoJournal of Physical Sciences vol 13 pp 77ndash85 2009


[3] S P Anjali Devi and M Kayalvizhi ldquoAnalytical solution ofMHDflowwith radiation over a stretching sheet embedded in aporous mediumrdquo International Journal of Applied Mathematicsand Mechanics vol 6 no 7 pp 82ndash106 2010

[4] V Rajesh ldquoRadiation effects onMHD free convection flow neara vertical plate with ramped wall temperaturerdquo InternationalJournal of Applied Mathematics and Mechanics vol 6 no 21pp 60ndash677 2010

[5] M G Reddy and N B Reddy ldquoSoret and Dufour effects onsteady MHD free convection flow past a semi-infinite movingvertical plate in a porous medium with viscous dissipationrdquoInternational Journal of Applied Mathematics and Mechanicsvol 6 no 1 pp 1ndash12 2010

[6] S P Anjali Devi and J Wilfred Samuel Raj ldquoThermo-diffusioneffects on unsteady hydromagnetic free convection flow withheat and mass transfer past a moving vertical plate with timedependent suction and heat source in a slip flow regimerdquoInternational Journal of Applied Mathematics and Mechanicsvol 7 no 21 pp 20ndash51 2010

[7] O D Makinde ldquoComputational modelling of MHD unsteadyflow and heat transfer toward a flat plate with navier slip andnewtonian heatingrdquo Brazilian Journal of Chemical Engineeringvol 29 no 1 pp 159ndash166 2012

[8] S Mukhopadhyay S Md Uddin and G C Layek ldquoLie groupanalysis on MHD boundary layer slip flow past a heatedstretching sheet in presence of heat sourcesinkrdquo InternationalJournal of Applied Mathematics and Mechanic vol 8 no 1 pp51ndash66 2012

[9] I M Eldesoky ldquoInfluence of slip condition on peristaltic tra-nsport of a compressiblemaxwell fluid through porousmediumin a tuberdquo International Journal of Applied Mathematics andMechanics vol 8 no 2 pp 99ndash117 2012

[10] K VWong andO de Leon ldquoApplications of nanofluids currentand futurerdquo Advances in Mechanical Engineering vol 2010Article ID 519659 11 pages 2010

[11] S U S Choi Z G Zhang W Yu F E Lockwood and EA Grulke ldquoAnomalous thermal conductivity enhancement innanotube suspensionsrdquo Applied Physics Letters vol 79 no 14pp 2252ndash2254 2001

[12] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001

[13] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000

[14] X Wang X Xu and S U S Choi ldquoThermal conductivity ofnanoparticle-fluid mixturerdquo Journal of Thermophysics and HeatTransfer vol 13 no 4 pp 474ndash480 1999

[15] A Aziz ldquoHydrodynamic and thermal slip flow boundary layersover a flat plate with constant heat flux boundary conditionrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 15 no 3 pp 573ndash580 2010

[16] C L M H Navier ldquoMemoire sur les lois du mouvement desfluidesrdquoMemoires de lrsquoAcademie Royale des Sciences de lrsquoInstitutde France vol 6 pp 389ndash440 1823

[17] G Singh and O D Makinde ldquoMHD slip flow of viscous fluidover an isothermal reactive stretching sheetrdquo Annals of FacultyEngineering HunedoramdashInternal Journal of Engineering 2013Toxe 11 Fascile 2

[18] J Md Uddin I Pop and A I M Ismail ldquoFree convectionboundary layer flow of a nanofluid from a convectively heated

vertical plate with linear momentum slip boundary conditionrdquoSains Malaysiana vol 41 no 11 pp 1475ndash1482 2012

[19] M J Martin and I D Boyd ldquoFalkner-Skan flow over a wedgewith slip boundary conditionsrdquo AIAA Journal of Thermophysicsand Heat Transfer vol 24 no 2 pp 263ndash270 2010

[20] K Bhattacharyya S Mukhopadhyay and G C Layek ldquoMHDboundary layer slip flow and heat transfer over a flat platerdquoChinese Physics Letters vol 28 no 2 Article ID 024701 2011

[21] S Ahmad A M Rohni and I Pop ldquoBlasius and Sakiadis pro-blems in nanofluidsrdquo Acta Mechanica vol 218 no 3-4 pp 195ndash204 2011

[22] R K Tiwari and M K Das ldquoHeat transfer augmentation in atwo-sided lid-driven differentially heated square cavity utilizingnanofluidsrdquo International Journal of Heat and Mass Transfervol 50 no 9-10 pp 2002ndash2018 2007

[23] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010

[24] S E Ahmed and A Mahdy ldquoNatural convection flow and heattransfer enhancement of a nanofluid past a truncated cone withmagnetic field effectrdquo World Journal of Mechanics vol 2 pp272ndash279 2012

International Journal of

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Control Scienceand Engineering

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



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Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


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DistributedSensor Networks



Table 1 Thermophysical properties of water and nanoparticles [23 24]

Materials 120588 (kgm3) 119888119901(JkgK) 119896 (WmK) 120590 (Sm)

Pure water 9971 4179 0613 55 times 10minus6

Copper (Cu) 8933 385 401 596 times 106

Alumina (Al2O3) 3970 765 40 35 times 106

Titania (TiO2) 4250 6862 89538 26 times 106

cooling and electronic devices cooling medical applicationsmagnetic drug targeting cancer therapy and safer surgeryby cooling process industries and materials and chemicalsdetergency food and drink oil and gas paper and printingand textiles

According to Aziz [15] the concept of no-slip conditionat the boundary layer is no longer valid for fluid flows inmicroelectromechanical systems and must be replaced byslip condition The slip flow model states a proportionalrelationship between the tangential components of the fluidvelocity at the solid surface to the shear stress on the fluid-solid interface [16] The proportionality is called the sliplength which describes the slipperiness of the surface [7]Many researchers studied the effect of linear momentumand nonlinear slip on the MHD boundary layer flow withheatmass transfer of freeforcedcombined convection pastdifferent geometries [17ndash20] In spite of the importance ofMHD related studies on boundary layer flow problems thepossibility of fluid exhibiting apparent slip phenomenon onthe solid surface has received little attention

The aim of the present study is to investigate the com-bined effects of buoyancy magnetic field suction Navier slipand convective heating on a steady boundary layer flow overa flat surface In the subsequent sections the boundary layerpartial differential equations first transformed into a systemof nonlinear ordinary differential equations before beingsolved numerically using a shooting method together withthe fourth-order Runge-Kutta-Fehlberg integration schemeA graphical representation of the pertinent parameters onthe flow field and heat transfer characteristics is displayedand thoroughly discussed To our best of knowledge theinvestigations of the proposed problem are new and theresults have not been published before

2 Model FormulationThe steady laminar incompressible two-dimensional MHDboundary layer flow of an electrically conducting water-based nanofluid past a convectively heated porous verticalsemiinfinite flat plate under the combined effects of buoyancyforces and Navier slip is considered The nanofluids containthree different types of nanoparticles Cu Al

2O3 and TiO

2

Let the 119909-axis be taken along the direction of plate and let119910-axis be normal to it The left side of the plate is assumedto be heated by convection from a hot fluid at temperature119879119891 which provides a heat transfer coefficient ℎ

119891 while

the right surface is subjected to a stream of an electricallyconducting cold nanofluid at temperature119879

infinin the presence

of a transverse magnetic field of strength 1198610applied parallel

to the 119910-axis as shown in Figure 1 The induced magneticfield due to the motion of the electrically conducting fluid is

x

g

T = Tinfin

Bo

y

u

Nanofluid = Vw

120582u = 120583f120597u

120597y120582

minuskf120597T

120597y= hf(Tf minus T)

u = Uinfin

Figure 1 Flow configuration and coordinate system

negligible It is also assumed that the external electrical fieldis zero and that the electric field due to the polarization ofcharges is negligible (see Table 1)

Assuming a Boussinesq incompressible fluid model thecontinuity momentum and energy equations describing theflow can be written as

120597119906

120597119909+120597V120597119910

= 0

119906120597119906

120597119909+ V

120597119906

120597119910= 119880infin

119889119880infin

119889119909+120583119899119891

120588119899119891

1205972119906

1205971199102+ 120573119899119891119892 (119879 minus 119879

infin)

minus1205901198991198911198612

0(119906 minus 119880

infin)

120588119899119891

119906120597119879

120597119909+ V

120597119879

120597119910=

119896119899119891

(120588119888119901)119899119891

1205972119879

1205971199102+

120583119899119891

(120588119888119901)119899119891

(120597119906

120597119910)

2

+1205901198991198911198612

0

(120588119888119901)119899119891

(119906 minus 119880infin)2

(1)

The boundary conditions at the plate surface and at the freestream may be written as

120582119906 (119909 0) = 120583119891

120597119906

120597119910(119909 0) V (119909 0) = 119881

119908

minus119896119891

120597119879

120597119910(119909 0) = ℎ

119891[119879119891minus 119879 (119909 0)]

119906 (119909infin) = 119880infin(119909) 119879 (119909infin) = 119879

infin

(2)








119899119891




(120588119888119901)119899119891




120583119899119891=

120583119891

(1 minus 120593)25 120588

119899119891= (1 minus 120593) 120588

119891+ 120593120588119904

120573119899119891= (1 minus 120593) 120573

119891+ 120593120573119904 120572

119899119891=

119896119899119891

(120588119888119901)119899119891

119896119899119891

119896119891

=(119896119904+ 2119896119891) minus 2120593 (119896

119891minus 119896119904)

(119896119904+ 2119896119891) + 120593 (119896

119891minus 119896119904)

(120588119888119901)119899119891= (1 minus 120593) (120588119888

119901)119891+ 120593(120588119888

119901)119904

120590119899119891= (1 minus 120593) 120590

119891+ 120593120590119904

(3)


119891and 120588



119904are






119891are the



120578 = (119886

120592119891

)

12

119910 120595 = (119886120592119891)12

119909119891 (120578)

120579 (120578) =119879 minus 119879infin

119879119891minus 119879infin

(4)


119906 =120597120595

120597119910 V = minus

120597120595

120597119909 (5)


119891101584010158401015840+ (1 minus 120593)

25

(1 minus 120593 +120593120588119904

120588119891

)11989111989110158401015840


(1 minus 120593 +120593120588119904

120588119891

)(1198911015840)2

+ (1 minus 120593)25

(1 minus 120593 +120593120588119904

120588119891

)

+ Gr(1 minus 120593)25 (1 minus 120593 +120593120588119904

120588119891

)(1 minus 120593 +120593120573119904

120573119891

)120579


120590119891

)(1198911015840minus 1) = 0

12057910158401015840+

Pr 119896119891[1 minus 120593 + 120593(120588119888

119901)119904(120588119888119901)119891]

119896119899119891

1198911205791015840

+Pr Ec 119896

119891

119896119899119891(1 minus 120593)

25(11989110158401015840)2

+HaPr Ec 119896

119891

119896119899119891

times (1 minus 120593 +120593120590119904

120590119891

)(1198911015840minus 1)2

= 0

(6)


119881119908= minus119891119908(119886120592119891)12

120582119906 (119909 0) = 120583119891

120597119906

120597119910(119909 0) (7)


119891 (0) = 119891119908 119891

1015840(0) = 120573119891

10158401015840(0)

1205791015840(0) = Bi [120579 (0) minus 1]

1198911015840(infin) = 1 120579 (infin) = 0

(8)





=




Re119909=119880infin119909

120592119891

Gr =120573119891119892 (119879119891minus 119879infin)

119880infin119886

Ha =120590119891119861119900

2

120588119891119886 Pr =

120592119891

120572119891

Ec =1198802

infin

119862119901119891(119879119891minus 119879infin) 120573 =

120583119891

120582radic

119886

120592119891

Bi =ℎ119891

119896119891

radic120592119891

119886

(9)





119862119891=

120591119908

1205881198911198802infin

Nu =119909119902119908

119896119891(119879119891minus 119879infin) (10)




120591119908= 120583119899119891

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

119902119908= minus119896119899119891

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

(11)


Re12119909119862119891=

1

(1 minus 120593)2511989110158401015840(0)

Reminus12119909

Nu = minus119896119899119891

119896119891

1205791015840(0)

(12)





0 1 2 3120578

1

09

08

07

06

05

04

03

02



f998400 (120578)


Gr = Bi = 120573 = 01 fw = 02


Ec = 3 5 7

120593 = 0 002 008

0 1 2 3120578

1

09

08

07

06

05

04

03

02

f998400 (120578)






0 1 2 3120578

1

09

08

07

06

05

04

03

02

f998400 (120578)

fw = minus04 0 03

120601 = Ec = Bi = 01 Ha = 10minus12

Gr = 0 2 4

120573 = 05 1 3






120579(120578)

0 05 1 15 2

025

02

015

01

005

120578



120579(120578)

00

05 1 15 2120578

07

06

05

04

03

02

01

120573 = Gr = 01 fw = 02


Ec = 02 0215 03






0 05 1 15 2120578

120579(120578)

0

03

02

01

fw = minus001 01 02

120573 = 001 02 03

120593 = Gr = Ec = Bi = 01 Ha = 10minus12



0 005 01 015 02120601

3

28

26

24

22

2

18

16

14


radicRe

xCf



2O3 and TiO



0 005 01 015 02

3

25

35

2

15

Gr = 1 3 4

Ec = 05 2 3

120573 = Bi = Gr = 01 fw = 02

120593


radicRe

xCf


fw = minus04 01 04

120573

120593 = Ec = Bi = Gr = 01 Ha = 10minus1226

24

22

2

18

16

14

12

1

08

radicRe

xCf

0 05 1 15





119908




Cu-waterAl2O3-water

TiO2-water

0 005 01 015 02120593

Ec = 120573 = Gr = Bi = 01

fw = 02 Ha = 10minus12012

011

01

009

NuradicRe

x


Ec = 120573 = Gr = 01 Ha = 10minus12

fw = minus01 001 03

Bi = 007 01 012

014

013

012

011

01

009

008

007

006

0 005 01 015 02120593

NuradicRe

x


4 Conclusions


004

002

0

minus002

01 02 03 04 05Ec

012

01

008

006

120601 = 120573 = Gr = Bi = 01 fw = 02


NuradicRe

x




119908)

Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















119899119891




(120588119888119901)119899119891




120583119899119891=

120583119891

(1 minus 120593)25 120588

119899119891= (1 minus 120593) 120588

119891+ 120593120588119904

120573119899119891= (1 minus 120593) 120573

119891+ 120593120573119904 120572

119899119891=

119896119899119891

(120588119888119901)119899119891

119896119899119891

119896119891

=(119896119904+ 2119896119891) minus 2120593 (119896

119891minus 119896119904)

(119896119904+ 2119896119891) + 120593 (119896

119891minus 119896119904)

(120588119888119901)119899119891= (1 minus 120593) (120588119888

119901)119891+ 120593(120588119888

119901)119904

120590119899119891= (1 minus 120593) 120590

119891+ 120593120590119904

(3)


119891and 120588



119904are






119891are the



120578 = (119886

120592119891

)

12

119910 120595 = (119886120592119891)12

119909119891 (120578)

120579 (120578) =119879 minus 119879infin

119879119891minus 119879infin

(4)


119906 =120597120595

120597119910 V = minus

120597120595

120597119909 (5)


119891101584010158401015840+ (1 minus 120593)

25

(1 minus 120593 +120593120588119904

120588119891

)11989111989110158401015840


(1 minus 120593 +120593120588119904

120588119891

)(1198911015840)2

+ (1 minus 120593)25

(1 minus 120593 +120593120588119904

120588119891

)

+ Gr(1 minus 120593)25 (1 minus 120593 +120593120588119904

120588119891

)(1 minus 120593 +120593120573119904

120573119891

)120579


120590119891

)(1198911015840minus 1) = 0

12057910158401015840+

Pr 119896119891[1 minus 120593 + 120593(120588119888

119901)119904(120588119888119901)119891]

119896119899119891

1198911205791015840

+Pr Ec 119896

119891

119896119899119891(1 minus 120593)

25(11989110158401015840)2

+HaPr Ec 119896

119891

119896119899119891

times (1 minus 120593 +120593120590119904

120590119891

)(1198911015840minus 1)2

= 0

(6)


119881119908= minus119891119908(119886120592119891)12

120582119906 (119909 0) = 120583119891

120597119906

120597119910(119909 0) (7)


119891 (0) = 119891119908 119891

1015840(0) = 120573119891

10158401015840(0)

1205791015840(0) = Bi [120579 (0) minus 1]

1198911015840(infin) = 1 120579 (infin) = 0

(8)





=




Re119909=119880infin119909

120592119891

Gr =120573119891119892 (119879119891minus 119879infin)

119880infin119886

Ha =120590119891119861119900

2

120588119891119886 Pr =

120592119891

120572119891

Ec =1198802

infin

119862119901119891(119879119891minus 119879infin) 120573 =

120583119891

120582radic

119886

120592119891

Bi =ℎ119891

119896119891

radic120592119891

119886

(9)





119862119891=

120591119908

1205881198911198802infin

Nu =119909119902119908

119896119891(119879119891minus 119879infin) (10)




120591119908= 120583119899119891

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

119902119908= minus119896119899119891

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

(11)


Re12119909119862119891=

1

(1 minus 120593)2511989110158401015840(0)

Reminus12119909

Nu = minus119896119899119891

119896119891

1205791015840(0)

(12)





0 1 2 3120578

1

09

08

07

06

05

04

03

02



f998400 (120578)


Gr = Bi = 120573 = 01 fw = 02


Ec = 3 5 7

120593 = 0 002 008

0 1 2 3120578

1

09

08

07

06

05

04

03

02

f998400 (120578)






0 1 2 3120578

1

09

08

07

06

05

04

03

02

f998400 (120578)

fw = minus04 0 03

120601 = Ec = Bi = 01 Ha = 10minus12

Gr = 0 2 4

120573 = 05 1 3






120579(120578)

0 05 1 15 2

025

02

015

01

005

120578



120579(120578)

00

05 1 15 2120578

07

06

05

04

03

02

01

120573 = Gr = 01 fw = 02


Ec = 02 0215 03






0 05 1 15 2120578

120579(120578)

0

03

02

01

fw = minus001 01 02

120573 = 001 02 03

120593 = Gr = Ec = Bi = 01 Ha = 10minus12



0 005 01 015 02120601

3

28

26

24

22

2

18

16

14


radicRe

xCf



2O3 and TiO



0 005 01 015 02

3

25

35

2

15

Gr = 1 3 4

Ec = 05 2 3

120573 = Bi = Gr = 01 fw = 02

120593


radicRe

xCf


fw = minus04 01 04

120573

120593 = Ec = Bi = Gr = 01 Ha = 10minus1226

24

22

2

18

16

14

12

1

08

radicRe

xCf

0 05 1 15





119908




Cu-waterAl2O3-water

TiO2-water

0 005 01 015 02120593

Ec = 120573 = Gr = Bi = 01

fw = 02 Ha = 10minus12012

011

01

009

NuradicRe

x


Ec = 120573 = Gr = 01 Ha = 10minus12

fw = minus01 001 03

Bi = 007 01 012

014

013

012

011

01

009

008

007

006

0 005 01 015 02120593

NuradicRe

x


4 Conclusions


004

002

0

minus002

01 02 03 04 05Ec

012

01

008

006

120601 = 120573 = Gr = Bi = 01 fw = 02


NuradicRe

x




119908)

Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119862119891=

120591119908

1205881198911198802infin

Nu =119909119902119908

119896119891(119879119891minus 119879infin) (10)




120591119908= 120583119899119891

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

119902119908= minus119896119899119891

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

(11)


Re12119909119862119891=

1

(1 minus 120593)2511989110158401015840(0)

Reminus12119909

Nu = minus119896119899119891

119896119891

1205791015840(0)

(12)





0 1 2 3120578

1

09

08

07

06

05

04

03

02



f998400 (120578)


Gr = Bi = 120573 = 01 fw = 02


Ec = 3 5 7

120593 = 0 002 008

0 1 2 3120578

1

09

08

07

06

05

04

03

02

f998400 (120578)






0 1 2 3120578

1

09

08

07

06

05

04

03

02

f998400 (120578)

fw = minus04 0 03

120601 = Ec = Bi = 01 Ha = 10minus12

Gr = 0 2 4

120573 = 05 1 3






120579(120578)

0 05 1 15 2

025

02

015

01

005

120578



120579(120578)

00

05 1 15 2120578

07

06

05

04

03

02

01

120573 = Gr = 01 fw = 02


Ec = 02 0215 03






0 05 1 15 2120578

120579(120578)

0

03

02

01

fw = minus001 01 02

120573 = 001 02 03

120593 = Gr = Ec = Bi = 01 Ha = 10minus12



0 005 01 015 02120601

3

28

26

24

22

2

18

16

14


radicRe

xCf



2O3 and TiO



0 005 01 015 02

3

25

35

2

15

Gr = 1 3 4

Ec = 05 2 3

120573 = Bi = Gr = 01 fw = 02

120593


radicRe

xCf


fw = minus04 01 04

120573

120593 = Ec = Bi = Gr = 01 Ha = 10minus1226

24

22

2

18

16

14

12

1

08

radicRe

xCf

0 05 1 15





119908




Cu-waterAl2O3-water

TiO2-water

0 005 01 015 02120593

Ec = 120573 = Gr = Bi = 01

fw = 02 Ha = 10minus12012

011

01

009

NuradicRe

x


Ec = 120573 = Gr = 01 Ha = 10minus12

fw = minus01 001 03

Bi = 007 01 012

014

013

012

011

01

009

008

007

006

0 005 01 015 02120593

NuradicRe

x


4 Conclusions


004

002

0

minus002

01 02 03 04 05Ec

012

01

008

006

120601 = 120573 = Gr = Bi = 01 fw = 02


NuradicRe

x




119908)

Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3120578

1

09

08

07

06

05

04

03

02

f998400 (120578)

fw = minus04 0 03

120601 = Ec = Bi = 01 Ha = 10minus12

Gr = 0 2 4

120573 = 05 1 3






120579(120578)

0 05 1 15 2

025

02

015

01

005

120578



120579(120578)

00

05 1 15 2120578

07

06

05

04

03

02

01

120573 = Gr = 01 fw = 02


Ec = 02 0215 03






0 05 1 15 2120578

120579(120578)

0

03

02

01

fw = minus001 01 02

120573 = 001 02 03

120593 = Gr = Ec = Bi = 01 Ha = 10minus12



0 005 01 015 02120601

3

28

26

24

22

2

18

16

14


radicRe

xCf



2O3 and TiO



0 005 01 015 02

3

25

35

2

15

Gr = 1 3 4

Ec = 05 2 3

120573 = Bi = Gr = 01 fw = 02

120593


radicRe

xCf


fw = minus04 01 04

120573

120593 = Ec = Bi = Gr = 01 Ha = 10minus1226

24

22

2

18

16

14

12

1

08

radicRe

xCf

0 05 1 15





119908




Cu-waterAl2O3-water

TiO2-water

0 005 01 015 02120593

Ec = 120573 = Gr = Bi = 01

fw = 02 Ha = 10minus12012

011

01

009

NuradicRe

x


Ec = 120573 = Gr = 01 Ha = 10minus12

fw = minus01 001 03

Bi = 007 01 012

014

013

012

011

01

009

008

007

006

0 005 01 015 02120593

NuradicRe

x


4 Conclusions


004

002

0

minus002

01 02 03 04 05Ec

012

01

008

006

120601 = 120573 = Gr = Bi = 01 fw = 02


NuradicRe

x




119908)

Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 05 1 15 2120578

120579(120578)

0

03

02

01

fw = minus001 01 02

120573 = 001 02 03

120593 = Gr = Ec = Bi = 01 Ha = 10minus12



0 005 01 015 02120601

3

28

26

24

22

2

18

16

14


radicRe

xCf



2O3 and TiO



0 005 01 015 02

3

25

35

2

15

Gr = 1 3 4

Ec = 05 2 3

120573 = Bi = Gr = 01 fw = 02

120593


radicRe

xCf


fw = minus04 01 04

120573

120593 = Ec = Bi = Gr = 01 Ha = 10minus1226

24

22

2

18

16

14

12

1

08

radicRe

xCf

0 05 1 15





119908




Cu-waterAl2O3-water

TiO2-water

0 005 01 015 02120593

Ec = 120573 = Gr = Bi = 01

fw = 02 Ha = 10minus12012

011

01

009

NuradicRe

x


Ec = 120573 = Gr = 01 Ha = 10minus12

fw = minus01 001 03

Bi = 007 01 012

014

013

012

011

01

009

008

007

006

0 005 01 015 02120593

NuradicRe

x


4 Conclusions


004

002

0

minus002

01 02 03 04 05Ec

012

01

008

006

120601 = 120573 = Gr = Bi = 01 fw = 02


NuradicRe

x




119908)

Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Cu-waterAl2O3-water

TiO2-water

0 005 01 015 02120593

Ec = 120573 = Gr = Bi = 01

fw = 02 Ha = 10minus12012

011

01

009

NuradicRe

x


Ec = 120573 = Gr = 01 Ha = 10minus12

fw = minus01 001 03

Bi = 007 01 012

014

013

012

011

01

009

008

007

006

0 005 01 015 02120593

NuradicRe

x


4 Conclusions


004

002

0

minus002

01 02 03 04 05Ec

012

01

008

006

120601 = 120573 = Gr = Bi = 01 fw = 02


NuradicRe

x




119908)

Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article combined effect of buoyancy force and...

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