The Shape Effect of Gold Nanoparticles on Squeezing ...Jul 01, 2020 · ResearchArticle The Shape Effect of Gold Nanoparticles on Squeezing Nanofluid Flow and Heat Transfer between

Research ArticleThe Shape Effect of Gold Nanoparticles on Squeezing NanofluidFlow and Heat Transfer between Parallel Plates

Umair Rashid1 Thabet Abdeljawad 234 Haiyi Liang 15 Azhar Iqbal 6

Muhammad Abbas 7 and Mohd Junaid Siddiqui6

1CAS Key Laboratory of Mechanical Behavior and Design of Materials Department of Modern MechanicsUniversity of Science and Technology of China Hefei Anhui 230026 China2Department of Mathematics and General Sciences Prince Sultan University Riyadh 11586 Saudi Arabia3Department of Medical Research China Medical University Taichung 40402 Taiwan4Department of Computer Science and Information Engineering Asia University Taichung 40402 Taiwan5IAT-Chungu Joint Laboratory for Additive ManufacturingAnhui Chungu 3D Printing Institute of Intelligent Equipment and Industrial Technology Wuhu Anhui 241200 China6Mathematics and Natural Sciences Prince Mohammad Bin Fahd University Al Khobar 31952 Saudi Arabia7Department of Mathematics University of Sargodha Sargodha 40100 Pakistan

Correspondence should be addressed to abet Abdeljawad tabdeljawadpsuedusa and Haiyi Liang hyliangustceducn

Received 1 July 2020 Accepted 5 August 2020 Published 2 September 2020

Academic Editor Marin Marin

Copyright copy 2020 Umair Rashid et alis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e focus of the present paper is to analyze the shape effect of gold (Au) nanoparticles on squeezing nanofluid flow and heattransfer between parallel plates e different shapes of nanoparticles namely column sphere hexahedron tetrahedron andlamina have been examined using water as base fluid e governing partial differential equations (PDEs) are transformed intoordinary differential equations (ODEs) by suitable transformations As a result nonlinear boundary value ordinary differentialequations are tackled analytically using the homotopy analysis method (HAM) and convergence of the series solution is ensurede effects of various parameters such as solid volume fraction thermal radiation Reynolds number magnetic field Eckertnumber suction parameter and shape factor on velocity and temperature profiles are plotted in graphical form For various valuesof involved parameters Nusselt number is analyzed in graphical form e obtained results demonstrate that the rate of heattransfer is maximum for lamina shape nanoparticles and the sphere shape of nanoparticles has performed a considerable role intemperature distribution as compared to other shapes of nanoparticles

1 Introduction

Nanotechnology has recently emerged and has become aworldwide revolution to obtain exceptional qualities and fea-tures over the last few decades It developed at such a fast paceand is still going through a revolutionary phase Nanotech-nology is coming together to play a crucial and commercial rolein our future world Gold nanoparticles are one of the utmoststablemetal nanoparticles and their current fascinating featuresinclude assembly of several types inmaterial science individualnanoparticles behaviors magnetic nanocytotoxic opticalproperties size-related electronic significant catalysis and

biological applications Gold nanoparticles have attracted re-search attention due to their properties and various potentialapplications is progression would go to the later generationof nanotechnology that requires products of gold nanoparticleswith precise shape controlled size large production facilitiesand pureness Gold nanoparticles are widely used as preferredmaterials in numerous fields because of their unique opticaland physical properties that is surface plasmon oscillation forlabeling sensing and imaging Recently significant develop-ments have been made in biomedical fields with superiorbiocompatibility in therapeutics and treatment of variousdiseases Gold nanoparticles can be prepared and conjugate

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 9584854 12 pageshttpsdoiorg10115520209584854

with numerous functionalizing agents such as dendrimersligands surfactants RNA DNA peptides polymers oligo-nucleotides drugs and proteins [1]

Squeezing nanofluid flow with the effect of thermal ra-diation and magnetohydrodynamics (MHD) has importantuses in the development of the real world It has gained theconsideration of researchers due to its extensive usagesSqueezing flow has increasing usages in several areas par-ticularly in the food industry and chemical engineering eundertakings and properties of the squeezing flow of nano-fluid for industrial usages such as electronic transportationbiomechanics foods and nuclear reactor have been explainedin many publications in the open literature ere are variousexamples concerning squeezing flow but the most significantones are injection compression and polymer preparation

e squeezing flow of nanofluid has gained significantconsideration due to the valuable verities of applications in thephysical and biophysical fields [2] Hayat et al [3] discussed theMHD in squeezing flow by using two disks Dib et al [4]examined the analytical solution of squeezing nanofluid flowDuwairi et al [5] addressed the heat transfer on the viscoussqueezed flow between parallel plates Domairry and Hatami[6] examined the time-dependent squeezing of nanofluid flowbetween two surfaces by applying differential transformationtechniques Sheikholeslami and Ganji [7] studied the heattransfer in squeezed nanofluid flow based on homotopyperturbation method e thermal radiation effect in two-dimensional and time-dependent squeezing flow was inves-tigated using homotopy analysis method by Khan et al [8]Sheikholeslami et al [9] presented the effect of MHD onsqueezing nanofluid flow in a rotating system Gupta and SahaRay [10] investigated the unsteady squeezing nanofluid flowbetween two parallel plates by using the Chebyshev waveletexpansion e effects of MHD on alumina-kerosene nano-fluid and heat transfer within two horizontal plates were ex-amined by Mahmood and Kandelousi [11]

In the fields of engineering and science there are variousmathematical problems to find but the exact solution isalmost complicated Homotopy analysis method (HAM) is awell-known and critical method for solving mathematics-related problems e main advantage of the homotopyanalysis method is finding the approximate solution to thenonlinear differential equation without linearization anddiscretization Earlier time in 1992 Liao [12ndash16] introducedthis technique to find out the analytical results of nonlinearproblems e author concluded that homotopy analysismethod (HAM) quickly converges to an approximate so-lution e homotopy analysis method gives us a series ofsolutions e approximate solution by homotopy analysismethod is quite perfect since it contained all the physicalparameters involved in a problem Due to the effectivenessand quick convergence of the solution various researchersnamely Rashidi et al [17 18] and Abbasbandy and Shirzadi[19 20] used homotopy analysis method (HAM) to find thesolutions of highly nonlinear and coupled equations Hus-sain et al [21] presented the bioconvection model forsqueezing flow using homotopy analysis method with theeffect of thermal radiation heat generationabsorption

Heat transfer can be increased by using several meth-odologies and techniques such as increasing the heat transfercoefficient or heat transfer surface which allows for a higherheat transfer rate in small volume fraction Cooling is amajor technical challenge faced by increasing numbers ofindustries involving microelectronics transportationmanufacturing and solid-state lighting So there is an es-sential requirement for innovative coolant with a betterachievement that would be employed for enhanced prop-erties [22] Recently nanotechnology has contributed toimproving the new and innovative class of heat transfernanofluid Base fluids are embedded with nanosize materialsto obtain nanofluids (nanofibers nanoparticles nanotubesnanorods nanowires droplet or nanosheet) [23] Signifi-cantly nanofluids have the ability to enhance heat transferrate in several areas like nuclear reactors solar power plantstransportation industry (trucks automobiles and airplanes)electronics and instrumentation biomedical applicationsmicroelectromechanical system and industrial cooling us-ages (cancer therapeutics cryopreservation and nanodrugdelivery) [24] ere are several studies to show the appli-cations of nanofluid heat transfer Kristiawan et al [25]studied the convective heat transfer in a horizontal circulartube using TiO2-water nanofluid Turkyilmazoglue and Popdiscussed the heat and mass transfer of convection flow ofnanofluids containing nanoparticles of Ag Cu TiO2 Al2O3and CuO [26] Sheikholeslami and Ganji [7] presented theanalytical results of heat transfer in water-Cu nanofluidQiang and Yimin [27] investigated the experimental studiesof convective heat transfer in water-Cu nanofluid Elgazery[28] examined the studies of Ag-Cu-Al2O3-TiO2-waternanofluid over a vertical permeable stretching surface with anonuniform heat sourcesink Rea et al [29] studied theviscous pressure and convective heat transfer in a verticalheated tube of Al2O3-ZrO2-water nanofluids Salman et al[30] discussed by using a numerical technique the concept ofaffecting convective heat transfer of nanofluid in microtubeusing different categories of nanoparticles such as Al2O3Cuo SiO2 and ZnO Sheikholeslami et al [31 32] studiedhybrid nanofluid for heat transfer expansion Hassan et al[33] discussed convective heat transfer in Ag-Cu hybridnanofluid flow Bhatti et al [34] examined numerically hallcurrent and heat transfer effects on the sinusoidal motion ofsolid particles Furthermore many researchers did work onheat transfer and thermal radiation see [35ndash39]

In light of the above literature study it has been observedthat Cu Ag Al2O3 SiO2 Cuo and ZnO are mostly used tofind the heat transfer e gold (Au) was rarely used to findthe heat transfer rate due to mixed convection [40] eshape of nanoparticles is very significant in the enhancementof heat transfer It is necessary to find the heat transfer rate innanofluid under the exact shapes of nanoparticles [41] Fromthe literature survey it is observed that no effort has beenmade on gold (Au) nanoparticles shape effect on squeezingflow e basic purpose of the present study is to analyze theshape effect of gold (Au) nanoparticles on squeezingnanofluid flow and heat transfer Various types of nano-particles are under deliberation column sphere hexahe-dron tetrahedron and lamina e effects of various

2 Mathematical Problems in Engineering

physical parameters on velocity and temperature distribu-tions are analyzed through plotted graphs

2 Problem Description

Consider heat transfer in the incompressible two-dimen-sional laminar and stable squeezing nanofluid between twohorizontal plates at y 0 and y h e lower plate is fixedby two forces which are equal and opposite Both the platesare separated by distance h A uniform B magnetic field isapplied along y-axis Moreover the effect of nonlinearthermal radiation is also considered e thermophysicalproperties of gold nanoparticles and water are presented inTable 1 e values of nanoparticles shapes-related param-eters are presented in Table 2 e partial governingequations of the problem are modeled as [42]

zu

zx+

zv

zy 0 (1)

uzu

zx+ v

zu

zy minus

1ρnf

zp

zx+μnf

ρnf

z2u

zx2 +

z2u

zy21113888 1113889 minus

σnfB2u

ρnf

(2)

uzv

zy minus

1ρnf

zp

zy+μnf

ρnf

z2u

zx2 +

z2u

zy21113888 1113889 (3)

uzT

zx+ v

zT

zy

knf

(ρCp)nf

z2T

zx2 +

z2T

zy21113888 1113889 +

μnf

(ρCp)nf

middot 2zu

zx1113888 1113889

2

+zv

zy1113888 1113889

2⎡⎣ ⎤⎦ +

zv

zx1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭

+16σlowast

3(ρCp)nfKlowast

z2T

zy2

(4)

e reverent boundary value conditions are

f 0 fprime 1 θ 1 at η 0

f v0

ah fprime 0 θ 0 at η 1

(5)

e following similarity variables are induced to non-dimensionalize governing equations (1)ndash(4)

u axfprime(η)

v minus ahf(η)

η y

h

θ(η) T minus T1

T2 minus T1

(6)

Equation (1) is identically satisfied Eliminating thepressure and by using equation (6) into equations (2) (3)and (4) one has the following nonlinear coupled boundaryvalue problems

fPrimePrime minus RA1

A2fprimefPrime minus ff

1113874 1113875 minus

1A2

MfPrime 0 (7)

(1 + Rd )θPrime + PrA3

A4Rfθprime + 4

A2

A4Ecfprime21113888 1113889 0 (8)

f(0) 0

fprime(0) 1

f(1) A

fprime(1) 0

θ(0) 1

θ(infin) 0

(9)

e dimensionless quantities are

A v0

ah

R ah

2

vf

Pr μf(ρCp)f

ρfkf

Ec ρfa

2h2

(ρCp)f Tw minus Tinfin( 1113857

M σnf

B2h2

vfρf

Rd 16σlowastT3

infin3knf

klowast

(10)

Here A Pr M R Ec and Rd represent the suctionparameter Prandtl number magnetic parameter Reynoldsnumber Eckert number and thermal radiation parameterrespectively One has

Table 1 ermophysical properties of gold (Au) and pure water as[40 43]

Physical properties Gold (Au) Pure waterρ (kgm3) 19300 9983Cp (JkgmiddotK) 129 4182k (WmmiddotK) 318 060

Table 2 e values of nanoparticles shapes-related parameters as[44]

Shapes Column Sphere Hexahedron Tetrahedron Laminaϕ 04710 1 08060 07387 01857m 63698 3 37221 40613 161576

Mathematical Problems in Engineering 3

A1 ρnf

ρf

A2 μnf

μf

A3 (ρCp)nf

(ρCp)f

A4 knf

kf

(11)

Here A1 A2 A3 and A4 represent the ratio of densityviscosity heat capacitances and thermal conductivityrespectively

In this study we considerρnf

ρf

(1 minus ϕ) +ρs

ρf

ϕ

μnf

μf

1

(1 minus ϕ)2middot5

(ρCp)nf

(ρCp)f

(1 minus ϕ) +(ρCp)s

(ρCp)f

ϕ

knf

kf

ks +(m minus 1)kf1113960 1113961 minus (m minus 1)ϕ kf minus ks1113872 1113873

kS +(m minus 1)kf1113960 1113961 + ϕ kf minus ks1113872 1113873

(12)

where kf μf ρf and (ρCp)f represent thermal conduc-tivity dynamic viscosity density and specific heat of thefluid respectively whereas ks μs ρs and (ρCp)s denote thethermal conductivity dynamic viscosity density and spe-cific heat of the solid respectively m and ϕ are the shapefactor and volume fraction of nanoparticles respectively

e physical quantity of Nu (Nusselt number) is definedas

Nu A4θprime(0)1113868111386811138681113868

1113868111386811138681113868 (13)

3 Solution by HAM

e auxiliary linear operators are selected as follows

Lf(f) d4fdη4

Lθ(θ)d2θdη2

(14)

ese auxiliary operators satisfy the following properties

Lf C1η3

6+ C2

η2

2+ C3η + C41113890 1113891 0

Lθ C5eη

+ C6eminus η

1113858 1113859 0

(15)

e initial guesses are chosen as

f0(η) (1 minus 2A)η3 +(3A minus 2)η2 + η

θ0(η) 1 minus η(16)

31 Zeroth-Order Deformation e corresponding zerothdeformation problem is defined as follows

(1 minus q)Lf1113954f(η q) minus f0(η)1113960 1113961 qZfNf

1113954f(η q) 1113954θ0(η q)1113960 1113961

1113954f(0 q) 0

1113954fprime(0 q) 1

1113954f(1 q) A

1113954fprime(1 q) 0

(1 minus q)Lθ1113954θ(η q) minus θ0(η)1113960 1113961 qZθNθ

1113954θ0(η q) 1113954f(η q)1113960 1113961

1113954θ(0 q) 1

1113954θ(1 q) 0

(17)

in which qϵ[0 1] is called an embedding parameter andZf ne 0 and Zθ ne 0 are the convergence control parameterssuch that 1113954f(η 0) f0(η) 1113954θ(η 0) θ0(η) and1113954f(η 1) f(η) 1113954θ(η 1) θ(η) it means that when q variesfrom 0 to 1 1113954f(η q) varies from initial guess f0(η) to thefinal solution f(η) and 1113954θ(η q) varies from initial guessesθ0(η) to θ(η)

e nonlinear operators Nf and Nθ are given by

Nf[1113954f(η q) 1113954θ(η q)] z41113954f(η q)

zη4minus

A1

A2R

z1113954f(η q)

zηz21113954f(η q)

zη2minus 1113954f(η q)

z31113954f(η q)

zη31113888 1113889 minus

1A2

Mz2 1113954f(η q)

zη2 0

Nθ[1113954θ(η q) 1113954f(η q)] 1 + Rd( 1113857

z21113954θ(η q)

zη2+ Pr

A3

A4R1113954f(η q)

z1113954θ(η q)

zη+ 4

A2

A4Ec

z2 1113954f(η q)

zη21113888 1113889 0

(18)


Expanding 1113954f(η q) and 1113954θ(η q) with respect to qMaclaurinrsquos series and q 0 we obtain

f(η q) f0(η) + 1113944infin

m1fm(η)q

m

θ(η q) θ0(η) + 1113944infin

m1θm(η)q

m

(19)

where fm(η) (1m)((zmf(η q))(zηm))|q0 and θm(η)

(1m)((zmθ(η q))(zηm))|q0

32 Higher-Order Deformation Problem e higher-orderproblems are as follows

Lf fm(η) minus χmfmminus 1(η)1113858 1113859 ZfRmf fmminus 1(η) θmminus 1(η)( 1113857

fm(0) 0

fmprime (0) 0

fmprime (1) 0

fmprime (1) 0

LLf θm(η) minus χmθmminus 1(η)1113858 1113859 ZθRmθ θmminus 1(η) fmminus 1(η)( 1113857

θm(0) 0

θm(1) 0

(20)

where

χm

0 whenmle 1

1 mgt 1

⎧⎪⎨

⎪⎩

Rmf fm(η) fmminus 1Prime (η) minus

A1

A2R

middot 1113944mminus 1

z0fzprimefmminus 1minus zPrime minus 1113944

mminus 1

z0fzfmminus 1minus zPrime⎛⎝ ⎞⎠ minus Mfmminus 1Prime (η)

Rmf θm(η) 1 + Rd( 1113857θmminus 1Prime + Pr

A3

A4R


z0fzθmminus 1minus zprime + 4

A2

A4Ec 1113944

mminus 1

z0fzPrimefmminus 1minus zPrime⎛⎝ ⎞⎠

(21)

e mth-order solutions are

fm(η) flowastm + C1

η3

6+ C2

η2

2+ C3η + C4 (22)

θm(η) θlowastm + C5eη

+ C6eminus η

(23)

where Cmz (z 1 minus 6) are constants to be determined by

using the boundary conditions

33 Convergence of Series Solutions Zeroth- and higher-order deformation problems are given in equations (7) and

(8) which clearly show that the series solutions containnonzero auxiliary parameters Zf and Zθ e convergence ofthe solutions is checked through plotting Z-curves Zf and Zθas displayed in Figures 1 and 2 It is evident that the seriessolutions (22) and (23) converge when minus 18 ge Zf le minus 02and minus 18 ge Zθ le minus 02

4 Results and Discussion

e physical insight of the problem is discussed in thispresent portion e schematic model of squeezing nano-fluid is shown in Figure 3 e dynamics of heat transfer inthe squeezing nanofluid fluid flow are described under thevariation of dimensionless solid volume fraction thermalradiation Reynolds number magnetic field Eckert numbersuction parameter and shape factor e analysis is carriedout using the following range of parameters 01ge ϕle 02

05geAle 1005geMle 4005geRle 10001geEcle 09 and05geRdle 20 It is evident from Figures 4ndash7 that nano-particles which participate in heat transfer arelaminagt columngt tetrahedrongt hexahedrongt sphere

ϕ is a very important parameter for squeeze flow ofnanofluid From Figure 8 it is noted that the impact of ϕ onprimary velocity seemed ineffective It is also observed thatthe primary velocity is increased with increase of R asdisplayed in Figure 9 It is because that the inertia with theviscous ratio is dominant From Figure 10 it is observedthat the effect ofM on the primary velocity is decreased dueto the Lorentz force produced byM e Lorentz force actsagainst the motion of squeeze flow of nanofluid evariation of A on the dimensionless primary velocity isshown in Figure 11 From Figure 11 it can be seen that theprimary velocity is intensifying with the increase of Aphysically the wall shear stress increases with the increaseof A

e secondary velocity decreases in the half of the regionas shown in Figure 12 In Figures 13 and 14 it is distin-guished that secondary velocity changed in half of the region(the region above the central line between the plates) Ithappened due to the constraint of law of conservation ofmass e variation of A is plotted in Figure 15 secondaryvelocity is also increased with the increase of A

e shape effects of nanoparticles on dimensionlesstemperatures profiles are shown in Figure 16 It is notedfrom Figure 16 that spheregt hexahedrongt tetrahedrongtcolumn gt lamina It is also observed that lamina nano-particles have minimum temperature because of maximumviscosity while sphere shape nanoparticles have maximumtemperature because of minimum viscosity From Figure 17it is observed that the temperature profile has a direct re-lation with ϕ the reason is that increasing the volumefraction causes enhanced thermal conductivity of thenanofluid which turns to increase the boundary layerthickness From Figure 17 it is also observed that the sphereshape nanoparticles show a prominent role in temperaturedistribution Figure 18 depicts the influence of R on thedimensionless temperature profile with increasing R thedimensionless temperature profile decreases because ofdecreasing the thermal layer thickness Figure 18 showed


ndash3 ndash2 ndash1 0 1

50

55

60

65

70

ћf

fPrime (0

)

Figure 1 e Zfndashcurve for fPrime(0)

ndash3 ndash2 ndash1 0 1ndash6

ndash4

ndash2

0

2

4

6

θprime (0

)

ћθ

ColumnSphereHexahedron

TetrahedronLamina

Figure 2 e Zθndashcurve for θprime(0)

y-axis

Column

Sphere

Tetrahedron

Hexahedron

Nanofluid

x-axis

Lamina

Figure 3 Schematic model of squeezing nanofluid

010 012 014 016 018 02010

15

20

25

30

uN

ϕ

Figure 4 Nu for values of R andM A Rd 05 and Ec 03 Bluecolumn Green sphere Red hexahedron Brown tetrahedronBlack lamina R 05 solid line R 10 dash line

05 06 07 08 09 10

15

10

20

25

30

35

A

Nu

Figure 5 Nu for values of Ec andM R Rd 05 and ϕ 02 Bluecolumn Green sphere Red hexahedron Brown tetrahedronBlack lamina Ec 001 solid line Ec 003 dash line

10 12 14 16 18 20

15

20

25

30

35

Rd

Nu

Figure 6 Nu for values of R and M A 05 Ec 03 and ϕ 02Blue column Green sphere Red hexahedron Brown tetrahe-dron Black lamina R 05 solid line R 10 dash line


05 06 07 08 09 10

15

20

25

30

35

R

Nu

Figure 7 Nu for values of Rd andM A 05 Ec 03 and ϕ 02Blue column Green sphere Red hexahedron Brown tetrahe-dron Black lamina Rd 10 solid line Rd 20 dash line

00 02 04 06 08 1000

02

04

06

08

10

f (η)

η

Figure 8 fprime(η) for values of ϕ and R 03 M 05 andA 10Blue ϕ 01 Green ϕ 02

00 02 04 06 08 1000

02

04

06

08

10

f (η)

η

Figure 9 fprime(η) for values of R and M 05 A 10 and ϕ 02Blue R 05 Green R 10

00 02 04 06 08 1000

02

04

06

08

10

f (η)

η

Figure 10 fprime(η) for values of M and R 03 A 10 and ϕ 02Blue M 05 Green M 40

00 02 04 06 08 1000

02

04

06

08

10f (η)

η

Figure 11 fprime(η) for values of A and R 03 M 05 and ϕ 02Blue A 05 Green A 10

00 02 04 06 08 1000

02

04

06

08

10

12

14

f (η

)

η

Figure 12 fprime(η) for values of M and R 03 A 10 and ϕ 02Blue ϕ 01 Green ϕ 02


00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η

Figure 16 θ(η) for effect of the nanoparticles shapes and R 03Rd M 05 Ec 07 A 10 and ϕ 02 Blue column Greensphere Red hexahedron Brown tetrahedron Black lamina

00 02 04 06 08 1000

05

10

15

20

25

30θ

(η)

η

Figure 17 θ(η) for values of ϕ and Ec 07 Rd M 05 R 03and A 10 Blue column Green sphere Red hexahedronBrown tetrahedron Black lamina ϕ 01 dot line ϕ 02 dashline

00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η

Figure 18 θ(η) for values of R and Ec 07 Rd M 05 A 10and ϕ 02 Blue column Green sphere Red hexahedron Browntetrahedron Black lamina R 05 dot line R 10 dash line


that the effect of sphere shape nanoparticles is more sig-nificant than other shapes of nanoparticles under the in-fluence of R Figure 19 describes the impact ofM on thermalboundary layer thickness From Figure 19 it is noted that thetemperature increases with the increase of M e reason isthat M tends to increase a dragging force which producesheat in temperature profile Figure 19 depicts that the sphereshape nanoparticles in Au-water play a leading role in thetemperature profile Figure 20 shows that the temperatureprofile increases with A physically the heated nanofluid ispushed towards the wall where the buoyancy forces canintensify the viscosity at is why it decreases the wall shearstress Figure 21 shows the effect of Rd on temperatureprofile from this figure it is illustrated that the Rd has aninverse relation with temperature profile Due to this thegreater value of Rd corresponds to an increase in thedominance of conduction over radiation and hence re-duction in the buoyancy force and the thermal boundarylayer thickness Under the effect of Rd sphere shapenanoparticles have an important role in temperature dis-tribution Figure 22 displays that the squeezed nanofluid flow

temperature increases with the increase of the Ec the reason isthat the frictional heat is deposited in squeezed nanofluidhowever thermal boundary layer thickness of sphere shapenanoparticles seems to be more animated in squeezednanofluid by Ec effect

5 Conclusion

In the present paper the effect of gold (Au) nanoparticles onsqueezing nanofluid flow has been thoroughly examined eanalytical solution was obtained by using homotopy analysismethod (HAM) for a range of pertinent parameters such asshape factor solid volume fraction thermal radiation Rey-nolds number magnetic field suction parameter and Eckertnumber e effects of various parameters have been illus-trated through graphse Pr keeps fixed at 62 In the view ofresults and discussions the following deductions have arrived

(i) e nanoparticles of sphere shape show a re-markable role in the disturbance of temperatureprofiles

00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η

Figure 19 θ(η) for values of M and R 03 Rd 05 Ec 07A 10 and ϕ 02 Blue column Green sphere Red hexahe-dron Brown tetrahedron Black lamina M 10 dot line M 30dash line

00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η

Figure 20 θ(η) for values of A and R 03 Rd M 05 Ec 07and ϕ 02 Blue column Green sphere Red hexahedron Browntetrahedron Black lamina A 07 dot line A 10 dash line

00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η

Figure 21 θ(η) for values of Rd and Ec 07 R 03 M 05A 10 and ϕ 02 Blue column Green sphere Red hexahe-dron Brown tetrahedron Black lamina Rd 05 dot line Rd 10dash line

00 02 04 06 08 100

1

2

3

4

θ (η

)

η

Figure 22 θ(η) for values of Ec and R 03 Rd M 05 A 10and ϕ 02 Blue column Green sphere Red hexahedron Browntetrahedron Black lamina Ec 06 dot line Ec 09 dash line


(ii) e nanoparticles of tetrahedron shape show amoderate role in the disturbance of temperatureprofiles

(iii) e nanoparticles of lamina shape play a lower rolein the disturbance of temperature profiles

(iv) e nanoparticles of lamina shape play a principalrole in the heat transfer rate

(v) e nanoparticles of tetrahedron shape show amoderate role in the heat transfer rate

(vi) e nanoparticles of tetrahedron shape show alower role in the heat transfer rate

(vii) Performances of lamina and sphere shapesnanoparticles in the forms of disturbance ontemperature profiles and the heat transfer areopposite to each other

(viii) Performances of hexahedron and tetrahedronshapes nanoparticles in forms of the disturbancetemperature profiles and the heat transfer areopposite to each other

Abbreviations

PDEs Partial differential equationsODEs Ordinary differential equationsRNA Ribonucleic acidDNA Deoxyribonucleic acidMHD MagnetohydrodynamicHPM Homotopy perturbation methodHAM Homotopy analysis method

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare no conflicts of interest

Authorsrsquo Contributions

Umair Rashid Azhar Iqbal and Mohd Junaid Siddiquicontributed to conceptualization Umair Rashid abetAbdeljawad and Azhar Iqbal contributed to methodologyUmair Rashid Haiyi Liang Azhar Iqbal and MuhammadAbbas contributed to software Haiyi Liang Umair Rashidabet Abdeljawad Azhar Iqbal Mohd Junaid Siddiquiand Muhammad Abbas contributed to validation HaiyiLiang Umair Rashid abet Abdeljawad Azhar IqbalMohd Junaid Siddiqui and Muhammad Abbas contributedto formal analysis Haiyi Liang Umair Rashid abetAbdeljawad Azhar Iqbal Mohd Junaid Siddiqui andMuhammad Abbas contributed to investigation abetAbdeljawad and Muhammad Abbas contributed to re-sources Umair Rashid Azhar Iqbal and Muhammad Abbasare responsible for writing and original draft preparationHaiyi Liang Umair Rashid Azhar Iqbal and MuhammadAbbas are responsible for writing review and editing of thepaper Azhar Iqbal and Muhammad Abbas contributed to

visualization Haiyi Liang abet Abdeljawad andMuhammad Abbas contributed to supervision abetAbdeljawad and Muhammad Abbas contributed to fundingacquisition All authors have read and agreed to the pub-lished version of the manuscript

Acknowledgments

e authors thank Dr Muhammad Kashif Iqbal GC Uni-versity Faisalabad Pakistan for his assistance in proof-reading the manuscript e authors would like toacknowledge Prince Sultan University for funding this workthrough the research group Nonlinear Analysis Methods inApplied Mathematics (NAMAM) (Group no RG-DES-2017-01-17)

References

[1] M Das K H Shim S S A An and D K Yi ldquoReview on goldnanoparticles and their applicationsrdquo Toxicology and Envi-ronmental Health Sciences vol 3 no 4 pp 193ndash205 2011

[2] S Muhammad S Shah G Ali M Ishaq S Hussain andH Ullah ldquoSqueezing nanofluid flow between two parallelplates under the influence of MHD and thermal radiationrdquoAsian Research Journal of Mathematics vol 10 no 1 pp 1ndash202018

[3] T Hayat T Abbas M Ayub T Muhammad and A AlsaedildquoOn squeezed flow of jeffrey nanofluid between two paralleldisksrdquo Applied Sciences vol 6 pp 1ndash15 2016

[4] A Dib A Haiahem and B Bou-said ldquoApproximate ana-lytical solution of squeezing unsteady nanofluid flowrdquo PowderTechnology vol 269 pp 193ndash199 2015

[5] H M Duwairi B Tashtoush and R A Damseh ldquoOn heattransfer effects of a viscous fluid squeezed and extrudedbetween two parallel platesrdquo Heat and Mass Transfer vol 41pp 112ndash117 2004

[6] G Domairry and M Hatami ldquoSqueezing Cu-water nanofluidflow analysis between parallel plates by DTM-Pade methodrdquoJournal of Molecular Liquids vol 193 pp 37ndash44 2014

[7] M Sheikholeslami and D D Ganji ldquoHeat transfer of Cu-water nanofluid flow between parallel platesrdquo Powder Tech-nology vol 235 pp 873ndash879 2013

[8] S I Khan U Khan N Ahmed and S T Mohyud-Dinldquoermal radiation effects on squeezing flow casson fluidbetween parallel disksrdquo Communications in NumericalAnalysis vol 2016 no 2 pp 92ndash107 2016

[9] M Sheikholeslami M Hatami and D D Ganji ldquoNanofluidflow and heat transfer in a rotating system in the presence of amagnetic fieldrdquo Journal of Molecular Liquids vol 190pp 112ndash120 2014

[10] A K Gupta and S Saha Ray ldquoNumerical treatment for in-vestigation of squeezing unsteady nanofluid flow between twoparallel platesrdquo Powder Technology vol 279 pp 282ndash2892015

[11] M Mahmoodi and S Kandelousi ldquoKeroseneminus aluminananofluid flow and heat transfer for cooling applicationrdquoJournal of Central South University vol 23 no 4 pp 983ndash9902016

[12] S Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in Non-linear Science and Numerical Simulation vol 15 no 8pp 2003ndash2016 2010


[13] S Liao ldquoAn analytic solution of unsteady boundary-layerflows caused by an impulsively stretching platerdquo Communi-cations in Nonlinear Science and Numerical Simulationvol 11 no 3 pp 326ndash339 2006

[14] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147no 2 pp 499ndash513 2004

[15] S-J Liao ldquoOn the analytic solution of magnetohydrodynamicflows of non-Newtonian fluids over a stretching sheetrdquoJournal of Fluid Mechanics vol 488 pp 189ndash212 2003

[16] S-J Liao ldquoAn explicit totally analytic approximate solutionfor Blasiusrsquo viscous flow problemsrdquo International Journal ofNon-linear Mechanics vol 34 no 4 pp 759ndash778 1999

[17] M M Rashidi A M Siddiqui and M Asadi ldquoApplication ofhomotopy analysis method to the unsteady squeezing flow ofa second-grade fluid between circular platesrdquo MathematicalProblems in Engineering vol 2010 Article ID 70684018 pages 2010

[18] M M Rashidi and S A Mohimanian Pour ldquoAnalytic ap-proximate solutions for unsteady boundary-layer flow andheat transfer due to a stretching sheet by homotopy analysismethodrdquo Nonlinear Analysis Modelling and Control vol 15no 1 pp 83ndash95 2010

[19] S Abbasbandy ldquoHomotopy analysis method for heat radia-tion equationsrdquo International Communications in Heat andMass Transfer vol 34 no 3 pp 380ndash387 2007

[20] S Abbasbandy and A Shirzadi ldquoA new application of thehomotopy analysis method solving the Sturm-Liouvilleproblemsrdquo Communications in Nonlinear Science and Nu-merical Simulation vol 16 no 1 pp 112ndash126 2011

[21] S Hussain S Muhammad G Ali et al ldquoA bioconvectionmodel for squeezing flow between parallel plates containinggyrotactic microorganisms with impact of thermal radiationand heat generationabsorptionrdquo Journal of Advances inMathematics and Computer Science vol 27 no 4 pp 1ndash222018

[22] O Manca Y Jaluria G Lauriat K Vafai and L Wang ldquoHeattransfer in nanofluids 2013rdquo Advances in Mechanical Engi-neering vol 2014 Article ID 832415 2 pages 2014

[23] W Yu and H Xie ldquoA review on nanofluids preparationstability mechanisms and applicationsrdquo Journal of Nano-materials vol 2012 Article ID 435873 17 pages 2012

[24] Y Xuan and Q Li ldquoHeat transfer enhancement of nano-fluidsrdquo International Journal of Heat and Fluid Flow vol 21no 1 pp 58ndash64 2000

[25] B Kristiawan B Santoso A T Wijayanta M Aziz andT Miyazaki ldquoHeat transfer enhancement of TiO2waternanofluid at laminar and turbulent flows a numerical ap-proach for evaluating the effect of nanoparticle loadingsrdquoEnergies vol 11 no 6 Article ID 1584 2018

[26] M Turkyilmazoglu and I Pop ldquoHeat and mass transfer ofunsteady natural convection flow of some nanofluids past avertical infinite flat plate with radiation effectrdquo InternationalJournal of Heat and Mass Transfer vol 59 pp 167ndash171 2013

[27] L Qiang and X Yimin ldquoConvective heat transfer and flowcharacteristics of Cu-water nanofluidrdquo Science in ChinaSeries E Technological Sciences vol 45 no 4 pp 408ndash4162002

[28] N S Elgazery ldquoNanofluids flow over a permeable unsteadystretching surface with non-uniform heat sourcesink in thepresence of inclined magnetic fieldrdquo Journal of the EgyptianMathematical Society vol 27 no 1 pp 1ndash26 2019

[29] U Rea T McKrell L-w Hu J Buongiorno andJ Buongiorno ldquoLaminar convective heat transfer and viscous

pressure loss of alumina-water and zirconia-water nano-fluidsrdquo International Journal of Heat and Mass Transfervol 52 no 7-8 pp 2042ndash2048 2009

[30] B H Salman H A Mohammed and A S Kherbeet ldquoHeattransfer enhancement of nanofluids flow in microtube withconstant heat fluxrdquo International Communications in Heatand Mass Transfer vol 39 no 8 pp 1195ndash1204 2012

[31] M Sheikholeslami R Ellahi and C Fetecau ldquoCuOndashwaternanofluid magnetohydrodynamic natural convection inside asinusoidal annulus in presence of melting heat transferrdquoMathematical Problems in Engineering vol 2017 Article ID5830279 9 pages 2017

[32] S Manikandan and K S Rajan ldquoNew hybrid nanofluidcontaining encapsulated paraffin wax and sand nanoparticlesin propylene glycol-water mixture potential heat transferfluid for energy managementrdquo Energy Conversion andManagement vol 137 pp 74ndash85 2017

[33] M HassanMMarin R Ellahi and S Z Alamri ldquoExplorationof convective heat transfer and flow characteristics synthesisby Cu-Agwater hybrid-nanofluidsrdquo Heat Transfer Researchvol 49 no 18 pp 1837ndash1848 2018

[34] M M Bhatti R Ellahi A Zeeshan M Marin and N IjazldquoNumerical study of heat transfer and Hall current impact onperistaltic propulsion of particle-fluid suspension withcompliant wall propertiesrdquo Modern Physics Letters B vol 33no 35 Article ID 1950439 2019

[35] G Palani and I A Abbas ldquoFree convection MHD flow withthermal radiation from an impulsively-started vertical platerdquoNonlinear Analysis Modelling and Control vol 14 no 1pp 73ndash84 2009

[36] K Sharma and M Marin ldquoReflection and transmission ofwaves from imperfect boundary between two heat conductingmicropolar thermoelastic solidsrdquo Analele Universitatii ldquoOvi-diusrdquo ConstantamdashSeria Matematica vol 22 no 2pp 151ndash176 2014

[37] S M Atif S Hussain and M Sangheer ldquoEffect of viscousdissipation and Joule heating on MHD radiative tangenthyperbolic nanofluid with convective and slip conditionsrdquoJournal of the Brazilian Society of Mechanical Sciences andEngineering vol 41 no 4 p 189 2019

[38] S M Atif S Hussain and M Sagheer ldquoHeat and masstransfer analysis of time-dependent tangent hyperbolicnanofluid flow past a wedgerdquo Physics Letters A vol 383no 11 pp 1187ndash1198 2019

[39] U Rashid and A Ibrahim ldquoImpacts of nanoparticle shape onAl2O3-water nanofluid flow and heat transfer over a non-linear radically stretching sheetrdquo Advances in Nanoparticlesvol 9 no 1 pp 23ndash39 2020

[40] S Aman I Khan Z Ismail andM Z Salleh ldquoImpacts of goldnanoparticles on MHD mixed convection Poiseuille flow ofnanofluid passing through a porous medium in the presenceof thermal radiation thermal diffusion and chemical reac-tionrdquo Neural Computing and Applications vol 30 no 3pp 789ndash797 2018

[41] R Kandasamy N A bt Adnan and R Mohammad ldquoNano-particle shape effects on squeezedMHDflow of water based CuAl2O3 and SWCNTs over a porous sensor surfacerdquo AlexandriaEngineering Journal vol 57 no 3 pp 1433ndash1445 2018

[42] M Mahmoodi and S Kandelousi ldquoSemi-analytical investi-gation of kerosene-alumina nanofluid between two parallelplatesrdquo Journal of Aerospace Engineering vol 29 no 4 ArticleID 04016001 2016

[43] N A Adnan R Kandasamy and R Mohammad ldquoNano-particle shape and thermal radiation on marangoni water


ethylene glycol and engine oil based Cu Al2O3 andSWCNTsrdquo Journal of Material Sciences amp Engineering vol 6no 4 2017

[44] Y Lin B Li L Zheng and G Chen ldquoParticle shape andradiation effects on Marangoni boundary layer flow and heattransfer of copper-water nanofluid driven by an exponentialtemperaturerdquo Powder Technology vol 301 pp 379ndash386 2016


with numerous functionalizing agents such as dendrimersligands surfactants RNA DNA peptides polymers oligo-nucleotides drugs and proteins [1]

Squeezing nanofluid flow with the effect of thermal ra-diation and magnetohydrodynamics (MHD) has importantuses in the development of the real world It has gained theconsideration of researchers due to its extensive usagesSqueezing flow has increasing usages in several areas par-ticularly in the food industry and chemical engineering eundertakings and properties of the squeezing flow of nano-fluid for industrial usages such as electronic transportationbiomechanics foods and nuclear reactor have been explainedin many publications in the open literature ere are variousexamples concerning squeezing flow but the most significantones are injection compression and polymer preparation

e squeezing flow of nanofluid has gained significantconsideration due to the valuable verities of applications in thephysical and biophysical fields [2] Hayat et al [3] discussed theMHD in squeezing flow by using two disks Dib et al [4]examined the analytical solution of squeezing nanofluid flowDuwairi et al [5] addressed the heat transfer on the viscoussqueezed flow between parallel plates Domairry and Hatami[6] examined the time-dependent squeezing of nanofluid flowbetween two surfaces by applying differential transformationtechniques Sheikholeslami and Ganji [7] studied the heattransfer in squeezed nanofluid flow based on homotopyperturbation method e thermal radiation effect in two-dimensional and time-dependent squeezing flow was inves-tigated using homotopy analysis method by Khan et al [8]Sheikholeslami et al [9] presented the effect of MHD onsqueezing nanofluid flow in a rotating system Gupta and SahaRay [10] investigated the unsteady squeezing nanofluid flowbetween two parallel plates by using the Chebyshev waveletexpansion e effects of MHD on alumina-kerosene nano-fluid and heat transfer within two horizontal plates were ex-amined by Mahmood and Kandelousi [11]

In the fields of engineering and science there are variousmathematical problems to find but the exact solution isalmost complicated Homotopy analysis method (HAM) is awell-known and critical method for solving mathematics-related problems e main advantage of the homotopyanalysis method is finding the approximate solution to thenonlinear differential equation without linearization anddiscretization Earlier time in 1992 Liao [12ndash16] introducedthis technique to find out the analytical results of nonlinearproblems e author concluded that homotopy analysismethod (HAM) quickly converges to an approximate so-lution e homotopy analysis method gives us a series ofsolutions e approximate solution by homotopy analysismethod is quite perfect since it contained all the physicalparameters involved in a problem Due to the effectivenessand quick convergence of the solution various researchersnamely Rashidi et al [17 18] and Abbasbandy and Shirzadi[19 20] used homotopy analysis method (HAM) to find thesolutions of highly nonlinear and coupled equations Hus-sain et al [21] presented the bioconvection model forsqueezing flow using homotopy analysis method with theeffect of thermal radiation heat generationabsorption

Heat transfer can be increased by using several meth-odologies and techniques such as increasing the heat transfercoefficient or heat transfer surface which allows for a higherheat transfer rate in small volume fraction Cooling is amajor technical challenge faced by increasing numbers ofindustries involving microelectronics transportationmanufacturing and solid-state lighting So there is an es-sential requirement for innovative coolant with a betterachievement that would be employed for enhanced prop-erties [22] Recently nanotechnology has contributed toimproving the new and innovative class of heat transfernanofluid Base fluids are embedded with nanosize materialsto obtain nanofluids (nanofibers nanoparticles nanotubesnanorods nanowires droplet or nanosheet) [23] Signifi-cantly nanofluids have the ability to enhance heat transferrate in several areas like nuclear reactors solar power plantstransportation industry (trucks automobiles and airplanes)electronics and instrumentation biomedical applicationsmicroelectromechanical system and industrial cooling us-ages (cancer therapeutics cryopreservation and nanodrugdelivery) [24] ere are several studies to show the appli-cations of nanofluid heat transfer Kristiawan et al [25]studied the convective heat transfer in a horizontal circulartube using TiO2-water nanofluid Turkyilmazoglue and Popdiscussed the heat and mass transfer of convection flow ofnanofluids containing nanoparticles of Ag Cu TiO2 Al2O3and CuO [26] Sheikholeslami and Ganji [7] presented theanalytical results of heat transfer in water-Cu nanofluidQiang and Yimin [27] investigated the experimental studiesof convective heat transfer in water-Cu nanofluid Elgazery[28] examined the studies of Ag-Cu-Al2O3-TiO2-waternanofluid over a vertical permeable stretching surface with anonuniform heat sourcesink Rea et al [29] studied theviscous pressure and convective heat transfer in a verticalheated tube of Al2O3-ZrO2-water nanofluids Salman et al[30] discussed by using a numerical technique the concept ofaffecting convective heat transfer of nanofluid in microtubeusing different categories of nanoparticles such as Al2O3Cuo SiO2 and ZnO Sheikholeslami et al [31 32] studiedhybrid nanofluid for heat transfer expansion Hassan et al[33] discussed convective heat transfer in Ag-Cu hybridnanofluid flow Bhatti et al [34] examined numerically hallcurrent and heat transfer effects on the sinusoidal motion ofsolid particles Furthermore many researchers did work onheat transfer and thermal radiation see [35ndash39]

In light of the above literature study it has been observedthat Cu Ag Al2O3 SiO2 Cuo and ZnO are mostly used tofind the heat transfer e gold (Au) was rarely used to findthe heat transfer rate due to mixed convection [40] eshape of nanoparticles is very significant in the enhancementof heat transfer It is necessary to find the heat transfer rate innanofluid under the exact shapes of nanoparticles [41] Fromthe literature survey it is observed that no effort has beenmade on gold (Au) nanoparticles shape effect on squeezingflow e basic purpose of the present study is to analyze theshape effect of gold (Au) nanoparticles on squeezingnanofluid flow and heat transfer Various types of nano-particles are under deliberation column sphere hexahe-dron tetrahedron and lamina e effects of various





zu

zx+

zv

zy 0 (1)

uzu

zx+ v

zu

zy minus

1ρnf

zp

zx+μnf

ρnf

z2u

zx2 +

z2u

zy21113888 1113889 minus

σnfB2u

ρnf

(2)

uzv

zy minus

1ρnf

zp

zy+μnf

ρnf

z2u

zx2 +

z2u

zy21113888 1113889 (3)

uzT

zx+ v

zT

zy

knf

(ρCp)nf

z2T

zx2 +

z2T

zy21113888 1113889 +

μnf

(ρCp)nf

middot 2zu

zx1113888 1113889

2

+zv

zy1113888 1113889

2⎡⎣ ⎤⎦ +

zv

zx1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭

+16σlowast

3(ρCp)nfKlowast

z2T

zy2

(4)



f v0


(5)


u axfprime(η)

v minus ahf(η)

η y

h

θ(η) T minus T1

T2 minus T1

(6)




1113874 1113875 minus

1A2

MfPrime 0 (7)


A4Rfθprime + 4

A2

A4Ecfprime21113888 1113889 0 (8)

f(0) 0

fprime(0) 1

f(1) A

fprime(1) 0

θ(0) 1

θ(infin) 0

(9)


A v0

ah

R ah

2

vf

Pr μf(ρCp)f

ρfkf

Ec ρfa

2h2


M σnf

B2h2

vfρf

Rd 16σlowastT3

infin3knf

klowast

(10)







A1 ρnf

ρf

A2 μnf

μf

A3 (ρCp)nf

(ρCp)f

A4 knf

kf

(11)



ρf

(1 minus ϕ) +ρs

ρf

ϕ

μnf

μf

1


(ρCp)nf

(ρCp)f


(ρCp)f

ϕ

knf

kf



(12)



Nu A4θprime(0)1113868111386811138681113868

1113868111386811138681113868 (13)

3 Solution by HAM


Lf(f) d4fdη4

Lθ(θ)d2θdη2

(14)


Lf C1η3

6+ C2

η2

2+ C3η + C41113890 1113891 0

Lθ C5eη

+ C6eminus η

1113858 1113859 0

(15)






1113954f(η q) 1113954θ0(η q)1113960 1113961

1113954f(0 q) 0

1113954fprime(0 q) 1

1113954f(1 q) A

1113954fprime(1 q) 0


1113954θ0(η q) 1113954f(η q)1113960 1113961

1113954θ(0 q) 1

1113954θ(1 q) 0

(17)



Nf[1113954f(η q) 1113954θ(η q)] z41113954f(η q)

zη4minus

A1

A2R

z1113954f(η q)

zηz21113954f(η q)


z31113954f(η q)

zη31113888 1113889 minus

1A2

Mz2 1113954f(η q)

zη2 0

Nθ[1113954θ(η q) 1113954f(η q)] 1 + Rd( 1113857

z21113954θ(η q)

zη2+ Pr

A3

A4R1113954f(η q)

z1113954θ(η q)

zη+ 4

A2

A4Ec

z2 1113954f(η q)

zη21113888 1113889 0

(18)



f(η q) f0(η) + 1113944infin

m1fm(η)q

m

θ(η q) θ0(η) + 1113944infin

m1θm(η)q

m

(19)





fm(0) 0

fmprime (0) 0

fmprime (1) 0

fmprime (1) 0


θm(0) 0

θm(1) 0

(20)

where

χm

0 whenmle 1

1 mgt 1

⎧⎪⎨

⎪⎩


A1

A2R



mminus 1



A3

A4R



A2

A4Ec 1113944

mminus 1


(21)



η3

6+ C2

η2

2+ C3η + C4 (22)


+ C6eminus η

(23)













50

55

60

65

70

ћf

fPrime (0

)



ndash4

ndash2

0

2

4

6

θprime (0

)

ћθ


TetrahedronLamina


y-axis

Column

Sphere

Tetrahedron

Hexahedron

Nanofluid

x-axis

Lamina


010 012 014 016 018 02010

15

20

25

30

uN

ϕ


05 06 07 08 09 10

15

10

20

25

30

35

A

Nu


10 12 14 16 18 20

15

20

25

30

35

Rd

Nu



05 06 07 08 09 10

15

20

25

30

35

R

Nu


00 02 04 06 08 1000

02

04

06

08

10

f (η)

η


00 02 04 06 08 1000

02

04

06

08

10

f (η)

η


00 02 04 06 08 1000

02

04

06

08

10

f (η)

η


00 02 04 06 08 1000

02

04

06

08

10f (η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

f (η

)

η



00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30θ

(η)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η





5 Conclusion



00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 100

1

2

3

4

θ (η

)

η










Abbreviations


Data Availability







Acknowledgments


References





















































zu

zx+

zv

zy 0 (1)

uzu

zx+ v

zu

zy minus

1ρnf

zp

zx+μnf

ρnf

z2u

zx2 +

z2u

zy21113888 1113889 minus

σnfB2u

ρnf

(2)

uzv

zy minus

1ρnf

zp

zy+μnf

ρnf

z2u

zx2 +

z2u

zy21113888 1113889 (3)

uzT

zx+ v

zT

zy

knf

(ρCp)nf

z2T

zx2 +

z2T

zy21113888 1113889 +

μnf

(ρCp)nf

middot 2zu

zx1113888 1113889

2

+zv

zy1113888 1113889

2⎡⎣ ⎤⎦ +

zv

zx1113888 1113889

2⎧⎨

⎩

⎫⎬

⎭

+16σlowast

3(ρCp)nfKlowast

z2T

zy2

(4)



f v0


(5)


u axfprime(η)

v minus ahf(η)

η y

h

θ(η) T minus T1

T2 minus T1

(6)




1113874 1113875 minus

1A2

MfPrime 0 (7)


A4Rfθprime + 4

A2

A4Ecfprime21113888 1113889 0 (8)

f(0) 0

fprime(0) 1

f(1) A

fprime(1) 0

θ(0) 1

θ(infin) 0

(9)


A v0

ah

R ah

2

vf

Pr μf(ρCp)f

ρfkf

Ec ρfa

2h2


M σnf

B2h2

vfρf

Rd 16σlowastT3

infin3knf

klowast

(10)







A1 ρnf

ρf

A2 μnf

μf

A3 (ρCp)nf

(ρCp)f

A4 knf

kf

(11)



ρf

(1 minus ϕ) +ρs

ρf

ϕ

μnf

μf

1


(ρCp)nf

(ρCp)f


(ρCp)f

ϕ

knf

kf



(12)



Nu A4θprime(0)1113868111386811138681113868

1113868111386811138681113868 (13)

3 Solution by HAM


Lf(f) d4fdη4

Lθ(θ)d2θdη2

(14)


Lf C1η3

6+ C2

η2

2+ C3η + C41113890 1113891 0

Lθ C5eη

+ C6eminus η

1113858 1113859 0

(15)






1113954f(η q) 1113954θ0(η q)1113960 1113961

1113954f(0 q) 0

1113954fprime(0 q) 1

1113954f(1 q) A

1113954fprime(1 q) 0


1113954θ0(η q) 1113954f(η q)1113960 1113961

1113954θ(0 q) 1

1113954θ(1 q) 0

(17)



Nf[1113954f(η q) 1113954θ(η q)] z41113954f(η q)

zη4minus

A1

A2R

z1113954f(η q)

zηz21113954f(η q)


z31113954f(η q)

zη31113888 1113889 minus

1A2

Mz2 1113954f(η q)

zη2 0

Nθ[1113954θ(η q) 1113954f(η q)] 1 + Rd( 1113857

z21113954θ(η q)

zη2+ Pr

A3

A4R1113954f(η q)

z1113954θ(η q)

zη+ 4

A2

A4Ec

z2 1113954f(η q)

zη21113888 1113889 0

(18)



f(η q) f0(η) + 1113944infin

m1fm(η)q

m

θ(η q) θ0(η) + 1113944infin

m1θm(η)q

m

(19)





fm(0) 0

fmprime (0) 0

fmprime (1) 0

fmprime (1) 0


θm(0) 0

θm(1) 0

(20)

where

χm

0 whenmle 1

1 mgt 1

⎧⎪⎨

⎪⎩


A1

A2R



mminus 1



A3

A4R



A2

A4Ec 1113944

mminus 1


(21)



η3

6+ C2

η2

2+ C3η + C4 (22)


+ C6eminus η

(23)













50

55

60

65

70

ћf

fPrime (0

)



ndash4

ndash2

0

2

4

6

θprime (0

)

ћθ


TetrahedronLamina


y-axis

Column

Sphere

Tetrahedron

Hexahedron

Nanofluid

x-axis

Lamina


010 012 014 016 018 02010

15

20

25

30

uN

ϕ


05 06 07 08 09 10

15

10

20

25

30

35

A

Nu


10 12 14 16 18 20

15

20

25

30

35

Rd

Nu



05 06 07 08 09 10

15

20

25

30

35

R

Nu


00 02 04 06 08 1000

02

04

06

08

10

f (η)

η


00 02 04 06 08 1000

02

04

06

08

10

f (η)

η


00 02 04 06 08 1000

02

04

06

08

10

f (η)

η


00 02 04 06 08 1000

02

04

06

08

10f (η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

f (η

)

η



00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30θ

(η)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η





5 Conclusion



00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 100

1

2

3

4

θ (η

)

η










Abbreviations


Data Availability







Acknowledgments


References


















































A1 ρnf

ρf

A2 μnf

μf

A3 (ρCp)nf

(ρCp)f

A4 knf

kf

(11)



ρf

(1 minus ϕ) +ρs

ρf

ϕ

μnf

μf

1


(ρCp)nf

(ρCp)f


(ρCp)f

ϕ

knf

kf



(12)



Nu A4θprime(0)1113868111386811138681113868

1113868111386811138681113868 (13)

3 Solution by HAM


Lf(f) d4fdη4

Lθ(θ)d2θdη2

(14)


Lf C1η3

6+ C2

η2

2+ C3η + C41113890 1113891 0

Lθ C5eη

+ C6eminus η

1113858 1113859 0

(15)






1113954f(η q) 1113954θ0(η q)1113960 1113961

1113954f(0 q) 0

1113954fprime(0 q) 1

1113954f(1 q) A

1113954fprime(1 q) 0


1113954θ0(η q) 1113954f(η q)1113960 1113961

1113954θ(0 q) 1

1113954θ(1 q) 0

(17)



Nf[1113954f(η q) 1113954θ(η q)] z41113954f(η q)

zη4minus

A1

A2R

z1113954f(η q)

zηz21113954f(η q)


z31113954f(η q)

zη31113888 1113889 minus

1A2

Mz2 1113954f(η q)

zη2 0

Nθ[1113954θ(η q) 1113954f(η q)] 1 + Rd( 1113857

z21113954θ(η q)

zη2+ Pr

A3

A4R1113954f(η q)

z1113954θ(η q)

zη+ 4

A2

A4Ec

z2 1113954f(η q)

zη21113888 1113889 0

(18)



f(η q) f0(η) + 1113944infin

m1fm(η)q

m

θ(η q) θ0(η) + 1113944infin

m1θm(η)q

m

(19)





fm(0) 0

fmprime (0) 0

fmprime (1) 0

fmprime (1) 0


θm(0) 0

θm(1) 0

(20)

where

χm

0 whenmle 1

1 mgt 1

⎧⎪⎨

⎪⎩


A1

A2R



mminus 1



A3

A4R



A2

A4Ec 1113944

mminus 1


(21)



η3

6+ C2

η2

2+ C3η + C4 (22)


+ C6eminus η

(23)













50

55

60

65

70

ћf

fPrime (0

)



ndash4

ndash2

0

2

4

6

θprime (0

)

ћθ


TetrahedronLamina


y-axis

Column

Sphere

Tetrahedron

Hexahedron

Nanofluid

x-axis

Lamina


010 012 014 016 018 02010

15

20

25

30

uN

ϕ


05 06 07 08 09 10

15

10

20

25

30

35

A

Nu


10 12 14 16 18 20

15

20

25

30

35

Rd

Nu



05 06 07 08 09 10

15

20

25

30

35

R

Nu


00 02 04 06 08 1000

02

04

06

08

10

f (η)

η


00 02 04 06 08 1000

02

04

06

08

10

f (η)

η


00 02 04 06 08 1000

02

04

06

08

10

f (η)

η


00 02 04 06 08 1000

02

04

06

08

10f (η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

f (η

)

η



00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30θ

(η)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η





5 Conclusion



00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 100

1

2

3

4

θ (η

)

η










Abbreviations


Data Availability







Acknowledgments


References



















































f(η q) f0(η) + 1113944infin

m1fm(η)q

m

θ(η q) θ0(η) + 1113944infin

m1θm(η)q

m

(19)





fm(0) 0

fmprime (0) 0

fmprime (1) 0

fmprime (1) 0


θm(0) 0

θm(1) 0

(20)

where

χm

0 whenmle 1

1 mgt 1

⎧⎪⎨

⎪⎩


A1

A2R



mminus 1



A3

A4R



A2

A4Ec 1113944

mminus 1


(21)



η3

6+ C2

η2

2+ C3η + C4 (22)


+ C6eminus η

(23)













50

55

60

65

70

ћf

fPrime (0

)



ndash4

ndash2

0

2

4

6

θprime (0

)

ћθ


TetrahedronLamina


y-axis

Column

Sphere

Tetrahedron

Hexahedron

Nanofluid

x-axis

Lamina


010 012 014 016 018 02010

15

20

25

30

uN

ϕ


05 06 07 08 09 10

15

10

20

25

30

35

A

Nu


10 12 14 16 18 20

15

20

25

30

35

Rd

Nu



05 06 07 08 09 10

15

20

25

30

35

R

Nu


00 02 04 06 08 1000

02

04

06

08

10

f (η)

η


00 02 04 06 08 1000

02

04

06

08

10

f (η)

η


00 02 04 06 08 1000

02

04

06

08

10

f (η)

η


00 02 04 06 08 1000

02

04

06

08

10f (η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

f (η

)

η



00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30θ

(η)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η





5 Conclusion



00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 100

1

2

3

4

θ (η

)

η










Abbreviations


Data Availability







Acknowledgments


References



















































50

55

60

65

70

ћf

fPrime (0

)



ndash4

ndash2

0

2

4

6

θprime (0

)

ћθ


TetrahedronLamina


y-axis

Column

Sphere

Tetrahedron

Hexahedron

Nanofluid

x-axis

Lamina


010 012 014 016 018 02010

15

20

25

30

uN

ϕ


05 06 07 08 09 10

15

10

20

25

30

35

A

Nu


10 12 14 16 18 20

15

20

25

30

35

Rd

Nu



05 06 07 08 09 10

15

20

25

30

35

R

Nu


00 02 04 06 08 1000

02

04

06

08

10

f (η)

η


00 02 04 06 08 1000

02

04

06

08

10

f (η)

η


00 02 04 06 08 1000

02

04

06

08

10

f (η)

η


00 02 04 06 08 1000

02

04

06

08

10f (η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

f (η

)

η



00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30θ

(η)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η





5 Conclusion



00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 100

1

2

3

4

θ (η

)

η










Abbreviations


Data Availability







Acknowledgments


References


















































05 06 07 08 09 10

15

20

25

30

35

R

Nu


00 02 04 06 08 1000

02

04

06

08

10

f (η)

η


00 02 04 06 08 1000

02

04

06

08

10

f (η)

η


00 02 04 06 08 1000

02

04

06

08

10

f (η)

η


00 02 04 06 08 1000

02

04

06

08

10f (η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

f (η

)

η



00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30θ

(η)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η





5 Conclusion



00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 100

1

2

3

4

θ (η

)

η










Abbreviations


Data Availability







Acknowledgments


References


















































00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

02

04

06

08

10

12

14

fprime(η)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30θ

(η)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η





5 Conclusion



00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 100

1

2

3

4

θ (η

)

η










Abbreviations


Data Availability







Acknowledgments


References




















































5 Conclusion



00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 1000

05

10

15

20

25

30

θ (η

)

η


00 02 04 06 08 100

1

2

3

4

θ (η

)

η










Abbreviations


Data Availability







Acknowledgments


References

























































Abbreviations


Data Availability







Acknowledgments


References

























































































Documents

The Shape Effect of Gold Nanoparticles on Squeezing ...Jul 01, 2020 · ResearchArticle The Shape Effect of Gold Nanoparticles on Squeezing Nanofluid Flow and Heat Transfer between