Pramana – J. Phys. (2021) 95:203 © Indian Academy of Scienceshttps://doi.org/10.1007/s12043-021-02217-7

Evaluating the performance of new mass flux theory on Carreaunanofluid using the thermal aspects of convective heat transport

MUHAMMAD IRFAN1 ,∗, KIRAN RAFIQ2, MUHAMMAD SHOAIB ANWAR3,MASOOD KHAN2, WAQAR AZEEM KHAN4,5 and KALEEM IQBAL2

1Department of Mathematics, Faculty of Basic Sciences, University of WAH, Quaid Avenue, Wah Cantt,Rawalpindi, Punjab 47040, Pakistan2Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan3Department of Mathematics, University of Jhang, Gojra Road, 35200 Jhang, Pakistan4School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China5Department of Mathematics, Mohi Ud Din Islamic University, Nerian Sharif, Pakistan∗Corresponding author. E-mail: muhammad.irfan@uow.edu.pk; mirfan@math.qau.edu.pk

MS received 28 October 2020; revised 23 June 2021; accepted 24 June 2021

Abstract. Nanofluids can be engineered as per requirements and have applications in microelectronic,therapeutic activities, hybrid-mechanical machineries, aeronautics zones, thermonuclear storehouses, shielding ofmiscellaneous engines etc. Here, the aspects of new mass flux theory in magneto Carreau nanofluid with convectiveand variable connectivity have been studied. Additionally, nonlinear properties of mixed convection are examined.The shear thinning–thickening properties are analysed by utilising bvp4c algorithm for influential variables. The fluidtemperature increases with thermophoresis and variable conductivity parameters. The outcomes of thermophoresisand Brownian motion parameters have conflicting influences on concentration field.

Keywords. Generalised Newtonian fluid; thermophoresis and Brownian diffusion; nanofluid; new mass fluxtheory; shear thinning–thickening properties.

PACS Nos 44.40.+a; 44.10.+I; 61.46.+w

1. Introduction

In recent times, one of the main requirements of indus-trial processes is the increase in heat transfer. Withthe advancement of technologies in paper industries,transportation industries, electronics, energy supply andtextile, new challenges arise that disturb the usageand lifetime of technical equipments. Heat is pro-duced while the electrical current goes through someresistors in series. Overheating affects the equipment’sperformance, can drastically change the efficiency ofthe equipment and shortens its lifetime. Bayomy andSaghir [1] stated that the decrease in the efficiencyof electronic devices or complete failure exponentiallyincreases with the increase of temperature. This issuegained more importance because high speed comput-ers operate for long hours. Electronic manufacturersare concerned that the sensitivity of electronic com-ponents is significantly affected by the amount ofheat generated and by the surface temperature of the

electronic chips. Earlier, oil, water and ethylene glycolwith low heat transfer rates, were recognised as coolingfluids in various processes. The idea of nanofluid wasgiven in 1995 by Choi [2] of the U.S. Argonne NationalLaboratory. Nanofluids are derived by dissolving sin-gle type of nanoparticles in the aforementioned fluidsto increase their thermal conductivity. Due to increasein base fluid thermal physical property and enhancedrate of energy exchange of the medium, nanofluids havebeen used in numerous fields. Thriveni and Mahanthesh[3] studied magnetised hybrid nanofluids consideringthe aspects of radiation. They noted that on the wallof the interior annuli, the transport of heat is max-imum. Heat and mass transfer in a nanofluid usingfractional relaxation times for the transport mechanismwere presented by Anwar [4]. Chaurasia and Sarviya[5] discussed entropy generation and thermal analysison helical screw in the presence of nanofluid. Maza-heri et al [6] studied the energy efficiency principle byanalysing microchannel heat exchanger and nanofluid.

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203 Page 2 of 9 Pramana – J. Phys. (2021) 95:203

Furthermore, noteworthy studies in these fields are givenin refs [7–20].

Mixed convection which is the combination of forcedand free convections happens in several manufactur-ing and technical flows and has widespread applica-tions in nature which includes the interaction of solarreceivers with wind currents, cooling of reactors dur-ing emergency, cooling of electronic devices with fans,placement of heat exchangers at low velocity, transportphenomena in engineering processes etc. These flowsare categorised by dimensionless buoyancy parameter,the flow configuration and the surface cooling con-ditions. Outside to mixed convection section, eithercomplete forced convection or complete free convec-tion mechanism can be considered to analyse completelythe flow and temperature field. For the complete under-standing of heat transfer within the convection region,nonlinear convection term is introduced in the math-ematical model of the flow problem. With this, heattransfer rate at smaller level is completely countedtowards the average heat transfer rate of the whole flowfield. Some important research can be seen in refs [21–24].

The non-Newtonian [25–30] nanofluids combinedwith nanoparticles have significant uses in fissionablevessels and fissile structures freezing for rising vitalityaptitude. The use of non-linear nanofluids in water-chilled fissile structures could result in a significantenhancement of their fiscal performance and/or safetyboundaries. In these utilisations, the main attentionhere is to scrutinise and magnetise the convective flowof Carreau nanofluid with nonlinear mixed convectionand variable conductivity. The nonlinear ODEs of thecurrent problem are solved via bvp4c algorithm. Theinfluential factors are presented and studied using graphsand tables.

2. Development of the model

Here the properties of nonlinear mixed convection andnew mass flux conditions in transient flow of 3D Car-reau nanofluid have been elaborated with convective,variable convective and magnetic phenomena. Considerthe stretching velocities in x- and y-directions

(Uw(x, t), Vw(y, t)) =(

ax

1 − βt,

by

1 − βt

),

respectively, with stretching rate quantities a and b. Par-allel to z-axis, a strong magnetic field with strength (B0)

is applied (shown in figure 1).The Carreau fluid flow equations [23,31] in these

norms are

Figure 1. Schematic diagram.

∂u

∂x+ ∂v

∂y+ ∂w

∂z= 0, (1)

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y+ w

∂u

∂z

= ν

(∂2u

∂z2

) ⎧⎪⎨⎪⎩

[1 +

(�

∂u

∂z

)2]n−1

2

+(n − 1)

(�

∂u

∂z

)2[

1 +(

�∂u

∂z

)2]n−3

2

⎫⎪⎬⎪⎭

−σ B2(t)u

ρ f+ g[βT (T − T∞) + β∗

T (T − T∞)2

+βC(C − C∞) + β∗C (C − C∞)2], (2)

∂v

∂t+ u

∂v

∂x+ v

∂v

∂y+ w

∂v

∂z

= ν

(∂2v

∂z2

) ⎧⎪⎨⎪⎩

[1 +

(�

∂v

∂z

)2]n−1

2

+(n−1)

(�

∂v

∂z

)2[

1+(

�∂v

∂z

)2]n−3

2

⎫⎪⎬⎪⎭

−σ B2(t)v

ρ f, (3)

∂T

∂t+u

∂T

∂x+v

∂T

∂y+w

∂T

∂z= 1

(ρc) f

∂

∂z

(K (T )

∂2T

∂z2

)

+τ

[(DT

T∞

) (∂T

∂z

)2

+ DB

(∂C

∂z

∂T

∂z

)], (4)

Pramana – J. Phys. (2021) 95:203 Page 3 of 9 203

∂C

∂t+u

∂C

∂x+v

∂C

∂y+w

∂C

∂z=DB

(∂2C

∂z2

)

+DT

T∞

(∂2T

∂z2

). (5)

Uw = u=(

ax

1 − βt

), Vw = v=

(by

1 − βt

), w=0,

−K (T )∂T

∂z= h f (T f −T ),

DB

(∂C

∂z

)+ DT

T∞

(∂T

∂z

)= 0 at z = 0, (6)

u → 0, v → 0, T → T∞, C → C∞ as z → ∞.

(7)

Here, velocity components are u, v, w in x, y, z-directions, respectively. Furthermore, ν is the kinematicviscosity, � is the rate material constant, n is the powerlaw exponent, σ is the electrical conductivity, ρ f is thefluid density, (T , C) are liquid temperature and concen-tration, respectively, (ρc) f is the fluid heat capacity, g isgravity, τ is the heat capacity ratio, (βT , β∗

T ) are linear–nonlinear thermal growths, respectively, (βC , β∗

C ) arelinear–nonlinear concentration growths, (DB , DT ) arethe Brownian and thermophoresis dispersion coeffi-cients, respectively, (T∞, T f ) are the ambient and fluidtemperature, respectively, C∞ is the ambient concen-tration, h f is the heat transport quantity and β is thedimensional unsteadiness factor. Additionally, the mag-netic field and variable conductivity are given as

B(T ) = B0

(1−βt)12

, K (T ) = k∞[

1+ε

(T − T∞T f − T∞

)],

(8)

where ε is the variable conductivity factor.

2.1 Appropriate conversions

Let us define

u =(

ax

1 − βt

)f ′(η), v =

(ay

1 − βt

)g′(η),

w = −√

aν

1 − βt[ f (η) + g(η)],

θ (η) = T − T∞T f − T∞

, φ = C − C∞C∞

, η = z

√a

ν (1 − βt).

(9)

Equations (2)–(7) after using eqs (8) and (9) become

f ′′′[1 + nWe21 f

′′2][1 + We21 f

′′2]n−32 − f ′2

+ f ′′( f + g) − S

(f ′ + 1

2η f ′′

)+ M2 f ′

+λ(1 + ξT θ)θ + λN (1 + ξCφ)φ = 0 (10)

g′′′[1 + nWe22g

′′2][1 + We22g

′′2]n−32

−g′2 + g′′( f + g) − S

(g′ + 1

2ηg′′

)+ M2g′ = 0,

(11)

(1 + εθ) θ ′′ + εθ ′2 + Pr( f + g)θ ′ − Pr S(θ + η

2θ ′)

+ Pr NT θ ′2 + Pr NBθ ′φ′ = 0, (12)

φ′′+ Pr Le( f +g)φ′− Pr LeS(φ+η

2φ′)

+(NT

NB

)θ ′′=0, (13)

f (0) = 0, g(0) = 0, f ′(0) = 1, g′(0) = α,

θ ′(0) = −γ (1 − θ(0)), NBφ′(0) + NT θ ′(0) = 0, (14)

f ′ → 0, g′ → 0, θ → 0, φ → 0 as η → ∞. (15)

Here, the local Weissenberg numbers (We1, We2),unsteadiness factor S, magnetic factor M , mixed con-vection factorλ, buoyancy ratio factor N , thermo-solutalnonlinear convection factors (ξT , ξC ), stretching ratesratio factor α, thermophoretic factor NT , Brownianmotion factor NB , thermal-Biot number γ , Prandtl num-ber Pr and Lewis number Le are

(We1, We2) =⎛⎝

√�2a3x2

ν(1 − βt)2 ,

√�2a3y2

ν(1 − βt)2

⎞⎠ ,

S = β

a, M = σ B2

0

ρ f a, λ = gβT (Tw − T∞)

aUw(x, t),

N = βC(C∞)

βT (T f − T∞), ξT = β∗

T (Tw − T∞)

βT,

ξC=β∗C(C∞)

βC, α=b

a, NT=τDT (Tw − T∞)

νT∞,

NB = τDB(C∞)

ν, γ= h f

K (T )

√ν(1 − βt)

a,

Pr = ν

α1, Le = α1

DB. (16)

3. Physical quantities

3.1 Skin friction and Nusselt number

Here C f x ,C f y and Nux are

C f x = τxz12ρ f U 2

w

and C f y = τyz12ρ f U 2

w

, (17)

Nux = −x ∂T

∂z

∣∣∣z=0(

T f − T∞) . (18)

203 Page 4 of 9 Pramana – J. Phys. (2021) 95:203

In dimensionless form

1

2C f xRe

12x = [1 + We2

1 f′′2(0)]n−1

2 f ′′(0),

1

2

(Uw

Vw

)C f yRe

12x = [1 + We2

2g′′2(0)]n−1

2 g′′(0),

(19)

Re− 1

2x Nux = −θ ′ (0) . (20)

The local Reynolds number is

Rex = ax2

ν(1 − βt).

4. Solution procedure

Here bvp4c approach is used to discretise and revise eqs(10)–(15) into first-order differential form for solutionprocess, i.e.,

x1 = f, x2 = f ′, x3 = f ′′, x ′3 = f ′′′, (21)

x4 = g, x5 = g′, x6 = g′′, x ′6 = g′′′, (22)

x7 = θ, x8 = θ ′, x ′8 = θ ′′, (23)

x9 = φ, x10 = φ′, x ′10 = φ′′, (24)

x ′3 = −(x1 + x4)x3 + x2

2 − S(x2 + η

2 x3) − M2x2 − λ(1 + ξT x7)x7 − λN (1 + ξCx9)x9

�1;

�1 = (1 + nWe2

1x23) ∗ (1 + We2

1x23

)n−32 . (25)

x ′6 = −(x1 + x4)x6 + x2

5 − S(x5 + η

2 x6) − M2x5

�2;

�2 = (1 + nWe2

2x26) ∗ (1 + We2

2x26

)n−32 . (26)

x ′8 = − Pr(x1 + x4)x8 − εx2

8 + Pr S (x7 + (η/2)x8) − Pr NBx8x10 − Pr NT x28

�3; �3 = (1 + εx7). (27)

x ′10 = − Pr Le(x1 + x4)x10 + Pr LeS (x9 + (η/2)x10) − NT

NBx ′

8, (28)

x1 (0) = 0, x2 (0) = 1, x2 (∞) = 0, (29)

x4 (0) = 0, x5 (0) = α, x5 (∞) = 0, (30)

x8(0) = −γ (1 − x7 (∞)) , x7 (∞) = 0, (31)

NBx10 + NT x8 = 0, x9 (∞) = 0. (32)

5. Analysis of the results

Here, the phenomena of shear thinning/thickening fluidare graphically illustrated for n < 1 and n > 1, respec-tively in Carreau nanofluid considering the aspectsof variable connectivity and new mass flux condition

numerically. Furthermore, the following fixed values areconsidered in graphs, i.e., S = 0.1, M = γ = Nt = 0.2,

N = 0.3, λ = NB = 0.4, α = ε = ξT = ξC = 0.5,

We1 = We2 = Le = 1.0 and Pr = 1.3.

5.1 Graphs for θ (η)

The plots of thermal conductivity (ε) and thermophore-sis parameter (NT ) for n = 0.6 and 1.6 on temperaturefield are plotted in figures 2a, 2b, 3a and 3b. The tem-perature of Carreau nanofluid enhances for ε and NTand similar impact is noted for n < 1 and n > 1. Thethermal conductivity of Carreau liquid increases whenε increases which influences the intensification of tem-perature field. Moreover, the variation between wall andreference temperature is enhanced for higher NT whichrises the temperature field.

Figures 4a, 4b, 5a and 5b show the performance ofthermal Biot (γ ) and Prandtl number (Pr) on temper-ature scattering. Conflicting portrayal is noted for γ

and Pr when these factors are enhanced. For larger γ

the transport of heat accelerates because γ depends onheat transport which increases θ (η). Furthermore, the

Pramana – J. Phys. (2021) 95:203 Page 5 of 9 203

(a) (b)

Figure 2. Plot of η vs. θ(η) for various values of ε when (a) n = 0.6 and (b) n = 1.6.

(a) (b)

Figure 3. Plot of η vs. θ(η) for various values of NT when (a) n = 0.6 and (b) n = 1.6.

small thermal diffusivity causes larger Pr and smallerheat transport from penetrating surface to cold adjacentfluid. Hence, higher Pr decreases θ(η) for n = 0.6 and1.6.

5.2 Graphs for φ (η)

The concentration field for the values of thermophore-sis parameter (NT ) and Brownian motion (NB) for

shear thinning (n = 0.6) and shear thickening (n = 1.6)

instances are depicted in figures 6a, 6b, 7a and 7b. Thelarger NT enhances, while higher NB decreases φ (η).The thermophoretic force increases when NT increasesand transport of nanoparticles occurred to lesser tem-perature from the area of higher temperature whichreports higher concentration field. Additionally, the par-ticle collision of fluid increases for advanced NB whichdecreases φ (η).

203 Page 6 of 9 Pramana – J. Phys. (2021) 95:203

(a) (b)

Figure 4. Plot of η vs. θ(η) for various values of γ when (a) n = 0.6 and (b) n = 1.6.

(a) (b)

Figure 5. Plot of η vs. θ(η) for various values of Pr when (a) n = 0.6 and (b) n = 1.6.

The dependence of Lewis number (Le) on φ (η) forn = 0.6 and 1.6 is shown in figures 8a and 8b. The analo-gous portrayal for n = 0.6 and 1.6 is acknowledged anddecreases the concentration field. When Le increases,the Brownian diffusion (DB) decreases and hence φ(η)

decreases.

5.3 Comparison tables

For the confirmation of the present outcomes a compar-ative table 1 of − f ′′(0) for S and table 2 − f ′′(0) and−g′′(0) for α are presented. An outstanding agreementis noted between the current study and earlier studies.

Pramana – J. Phys. (2021) 95:203 Page 7 of 9 203

(a) (b)

Figure 6. Plot of η vs. φ(η) for various values of NT when (a) n = 0.6 and (b) n = 1.6.

(a) (b)

Figure 7. Plot of η vs. φ(η) for various values of NB when (a) n = 0.6 and (b) n = 1.6.

Furthermore, outcomes of −θ ′(0) for n = 0.6 and 1.6are displayed in table 3 for ε, M, NB and NT . Hereε, NB and NT decrease the heat transport amount; how-ever enhances for M and γ .

6. Closing remarks

Here, the nonlinear mixed convection and convectivephenomenon in magnetised Carreau nanofluid after

203 Page 8 of 9 Pramana – J. Phys. (2021) 95:203

(a) (b)

Figure 8. Plot of η vs. φ(η) for various values of Le when (a) n =0.6 and (b) n =1.6.

Table 1. Comparative outcomes of − f ′′ (0) for Newto-nian case.

− f ′′ (0)

- S Ref. [32] Ref. [33] Current (bvp4c)

0.3 – – 1.1015960.5 – – 1.1672140.8 1.261042 1.261512 1.2610461.2 1.377722 1.378052 1.3777291.5 – – 1.4596741.8 – – 1.5375122.0 1.587362 – 1.587381

considering the theory of new mass flux have been stud-ied. Additionally, variable conductivity is investigated.The major strategic findings are given as follows:

• Larger thermal Biot causes increased performance.• The shear thinning and thickening have the same

impact on temperature field and the fluid tempera-ture of Carreau fluid for variable conductivity andthermophoretic factors.

• The Prandtl and Lewis numbers decrease the tem-perature and concentration fields, respectively.

• Conflicting portrayal of Brownian and thermophoreticfactors are reported on concentration scattering.

• The Nusselt number is increased for thermal Biotand magnetic factors.

Table 2. Comparative outcomes of − f ′′ (0) and −g′′ (0) when We1 = We2 = S = M = λ = N = ξT = ξC = 0 and n = 1.

− f ′′ (0) −g′′ (0) − f ′′ (0) −g′′ (0)

α Ref. [34] Ref. [35] Ref. [34] Ref. [35] Current (bvp4c) Current (bvp4c)

0.0 −1 −1 0 0 −1 −10.25 −1.048813 −1.048813 −0.194564 −0.194565 −1.048808 −0.19456990.50 −1.093097 −1.093096 −0.465205 −0.465206 −1.093091 −0.46521360.75 −1.134485 −1.134486 −0.794622 −0.794619 −1.134483 −0.79461631.0 −1.173720 −1.173721 −1.173720 −1.173721 −1.173721 −1.1737210

Pramana – J. Phys. (2021) 95:203 Page 9 of 9 203

Table 3. Computational outcomes of −θ ′ (0) for different parameters.

ε M NB NT γ −θ ′ (0)

n = 0.6 n = 1.6

0.5 0.2 0.4 0.2 0.2 0.1618750 0.16266680.8 0.1606176 0.16145711.2 0.1588894 0.15979411.5 0.1575573 0.15851180.5 0.5 0.1626623 0.1632708

0.7 0.1636174 0.16402700.9 0.1646705 0.16489730.2 0.1 0.1619159 0.1626943

0.2 0.1618886 0.16267600.3 0.1618795 0.16266990.4 0.4 0.1617133 0.1625082

0.5 0.1616317 0.16242800.6 0.1615495 0.16234740.2 0.3 0.2202771 0.2217516

0.4 0.2680569 0.27023680.5 0.3076041 0.3104582

References

[1] A M Bayomy and M Z Saghir, Int. J. Heat MassTransf. 107, 181 (2017)

[2] S U S Choi and J A Eastman, ASME Int. Mech. Eng.Congr. Expo. 99, (1995)

[3] K Thriveni and B Mahantesh, Heat Transf. 49, 4759(2020)

[4] M S Anwar, Phys. Scr. 95, 035211 (2020)[5] S R Chaurasia and R M Sarviya, Int. Commun. Heat

Mass Transf. 122, 105138 (2021)[6] N Mazaheri, M Bahiraei and S Razi, Powder Technol.

383, 484 (2021)[7] J Philip and P D Shima, Adv. Colloid Interface Sci. 183–

184, 30 (2012)[8] R U Haq, Z H Khan and W A Khan, Physica E 63, 215

(2014)[9] P V S Narayana, J. Nanofluids 6, 1181 (2017)

[10] T Hayat, M Waqas, M I Khan and A Alsaedi, J. Mol.Liq. 225, 302 (2017)

[11] A Rasheed and M S Anwar, Comput. Math. Appl. 74,2485 (2017)

[12] M Khan, M Irfan and W A Khan, Int. J. Mech. Sci. 130375 (2017)

[13] M Waqas, M I Khan, T Hayat and A Alsaedi, Comput.Meth. Appl. Mech. Eng. 324, 640 (2017)

[14] B Mahanthesh, I L Animasaun, M Rahimi-Gorji and IM Alarifi, Physica A 535, 122471 (2019)

[15] K Thriveni and B Mahanthesh, Eur. Phys. J. Plus 135,459 (2020)

[16] D Mahanty, R Babu and B Mahanthesh, Multidisc.Model. Mater. Struct. 16, 915 (2020)

[17] K Rafiq, M Irfan, M Khan, M S Anwar and W A Khan,Phys. Scr. 96, 045002 (2020)

[18] B Mahantesh and J Mackolil, Int. Commun. Heat MassTransf. 120, 105040 (2021)

[19] M Irfan, R Aftab and M Khan, Chin. J. Phys. 71, 444(2021)

[20] K Thriveni and B Mahanthesh, J. Therm. Anal. Calor.143, 2729 (2021)

[21] W Ibrahim and C Zemedu, Math. Prob. Eng. 2020,3596368 (2020)

[22] M Waqas, M I Khan, Z Asghar, S Kadry, Y M Chu andW A Khan, J. Mater. Res. Technol. 9, 11080 (2020)

[23] M Irfan, Surfaces Interfaces 23, 100926 (2021)[24] P M Patil and H F Shankar,Alex. Eng. J. 60, 1043 (2021)[25] T Hayat, M Rashid, A Alsaedi and S Asghar, J. Braz.

Soc. Mech. Sci. Eng. 41, 86 (2019)[26] B C Prasannakumara, M G Reddy, G T Thammanna and

B J Gireesha, Nonlinear Eng. 7, 197 (2018)[27] M Khan, M Irfan, W A Khan and M Ayaz, Pramana –

J. Phys. 91: 14 (2018)[28] M Khan, M Irfan and W A Khan, Pramana – J. Phys.

92: 17 (2019)[29] M Madhu, B Mahanthesh, N S Shashikumar, S A She-

hzad, S U Khan and B J Gireesha, Int. Commun. HeatMass Transf, 117, 104761 (2020)

[30] P V S Narayana, N Tarakaramu, G Sarojamma and I LAnimasaun, Therm. Sci. Eng. Appl. 13, 021028 (2021)

[31] M Irfan, W A Khan, M Khan and M Mudassar Gulzar,J. Phys. Chem. Solids 125, 141 (2018)

[32] S Sharidan, T Mahmood and I Pop, Int. J. Appl. Mech.Eng. 11, 647 (2006)

[33] A J Chamkha, A M Aly and M A Mansour, Chem. Eng.Commun. 197, 846 (2010)

[34] C Y Wang, Phys. Fluids 27, 1915 (1984)[35] I C Liu and H I Anderson, Int. J. Heat Mass Transf. 51,

4018 (2008)