Irreversibility analysis of cross fluid past a stretchable vertical sheet with mixture of Carboxymethyl cellulose water based hybrid nanofluid

Irreversibility analysis of cross fluid past a stretchable vertical sheet with mixture of

Carboxymethyl cellulose water based hybrid nanofluid

Item Type Article

Authors Ali, Farhan;Arun Kumar, T.;Loganathan, K.;Reddy, C. S.;Ali Pasha, Amjad;Rahman, Mustafa M.;Al-Farhany, Khaled

Citation Ali, F., Arun Kumar, T., Loganathan, K., Reddy, C. S., Ali Pasha, A., Rahman, M. M., & Al-Farhany, K. (2022). Irreversibility analysis of cross fluid past a stretchable vertical sheet with mixture of Carboxymethyl cellulose water based hybrid nanofluid. Alexandria Engineering Journal. https://doi.org/10.1016/j.aej.2022.08.037 Eprint version Publisher's Version/PDF

DOI 10.1016/j.aej.2022.08.037

Publisher Elsevier BV

Journal Alexandria Engineering Journal

Rights Archived with thanks to Alexandria Engineering Journal under a Creative Commons license, details at: http://

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Irreversibility analysis of cross fluid past a stretchable vertical sheet with mixture of Carboxymethyl cellulose water based hybrid nanofluid

Farhan Ali

^a

, T. Arun Kumar

^b,*

, K. Loganathan

^c,d,*

, C.S. Reddy

^e

, Amjad Ali Pasha

^f

, Mustafa Mutiur Rahman

^g

, Khaled Al-Farhany

^h

aDepartment of Mathematical Sciences, Federal Urdu University of Arts, Sciences & Technology, Gulshan-e-Iqbal Karachi 75300 Pakistan

bDepartment of Science and Humanities, MLR Institute of Technology, Hyderabad, Telagana, India

cDepartment of Mathematics and Statistics, Manipal University Jaipur, Jaipur 303007, Rajasthan, India

dResearch and Development Wing, Live4Research, Tiruppur 638106, Tamil Nadu, India

eDepartment of Mathematics, Government City College, Hyderabad, Telangana, India

fAerospace Engineering Department, King Abdulaziz University, Jeddah 21589, Saudi Arabia

gMechanical Engineering Program, Physical Science and Engineering Division, King Abdullah University of Science &

Technology, Thuwal 23955, Saudi Arabia

hDepartment of Mechanical Engineering, University of Al-Qadisiyah, Al-Qadisiyah 58001, Iraq

Received 14 June 2022; revised 16 August 2022; accepted 23 August 2022

KEYWORDS Hybrid fluid;

Cross fluid;

Entropy generation;

Thermal radiation;

Carboxymethyl cellulose water

Abstract This study addresses CuOTiO2/CMC-water hybridnano-liquid in the influence of mixed convection flow and thermal radiative flow past a stretchable vertical surface. Cross nano- fluid containing Titanium dioxideðTiO2Þ;and Copper OxideðCuOÞare scattered in a base fluid of kind CMC water. In addition, theirreversibility analysis is also examined in the current problem.

A suitable transformation is utilized to transmute the momentum and thermal mathematical expres- sion in non-dimensionless form. Further, the BVP utilizer is set to solve these mathematical expres- sions. The significance of leading variables on the velocity, entropy generation,temperature, and Bejan numberare displayed and elaborated through the aid ofgraphs. The outcomes demonstrate that the larger values of the Weissenberg number reduce the velocity and entropy profiles while escalatingthe temperature distribution and Bejan number. The drag friction and heat transfer rate

Abbreviations:Cu, Copper; Ti, Titanium; PDE, Partial Differential Equations; ODE, Ordinary Differential Equations; CMC, Carboxylmethyl cellulose

* Corresponding authors at: Department of Science and Humanities, MLR Institute of Technology, Hyderabad, Telagana, India. (T. Arun Kumar); Department of Mathematics and Statistics, Manipal University Jaipur, Jaipur 303007, Rajasthan, India (K. Loganathan).

E-mail addresses:arunk_tp@mlrinstitutions.ac.in(T. Arun Kumar),loganathankaruppusamy304@gmail.com(K. Loganathan),aapasha@kau.

edu.sa(A. Ali Pasha),Mustafa.Rahman@kaust.edu.sa(M.M. Rahman),khaled.alfarhany@qu.edu.iq(K. Al-Farhany).

Peer review under responsibility of Faculty of Engineering, Alexandria University.

Alexandria Engineering Journal (2022)xxx, xxx–xxx

H O S T E D BY

Alexandria University

Alexandria Engineering Journal

www.elsevier.com/locate/aej www.sciencedirect.com

https://doi.org/10.1016/j.aej.2022.08.037

1110-0168Ó2022 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University.

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

Please cite this article in press as: F. Ali et al., Irreversibility analysis of cross fluid past a stretchable vertical sheet with mixture of Carboxymethyl cellulose water based hybrid nanofluid, Alexandria Eng. J. (2022),https://doi.org/10.1016/j.aej.2022.08.037

are enhanced by exceeding the value of the mixed convective parameter and Biot number. The motive of this manuscript is to give more interest of entropy production study with heat and fluid flow on Cross fluid with nanoparticles and base fluid to develop the system performance. The cur- rent work is existed with the previous literaure and obtain a fantastic achievement.

Ó2022 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/

licenses/by-nc-nd/4.0/).

1. Introduction

Nanoliquid is the mixture of simple liquid containing water, oil, ethylene glycol, kerosene etc and nanometer material sized particles. Such nanometer-sized particles are known as nanoparticles. Nanoparticles are involved in ceramics, metal nitrides, metal, carbon and many more. Simple fluids have been used while a cooling agent in numerous industrial and technological processes. These simple have less thermal con- ductivity as compared to solid substances. So, the suspension of nanoparticles in traditional fluids is vastly effective in its thermal conductivity. The pioneering work in this direction was initiated via Choi [1]. In the modern era, nano-liquid has produced a novel approach to augmentingthe rate of heat in frequent features containing textile industry, medical proce- dures, defence energy systems and electronics[2–6]. A lot of scholars[7–10]discussed various kinds of nanoparticles with base fluids. Currently, various scholars have started hybrid nanofluids comprising two kinds of nanoparticles dispersed in simple fluids. The features of hybrid nanoliquid such as heat radiation and exchanger, naval structure, defence making, microelectronics and vehicle brake fluids. Ashwinkumar et al. [11] discovered MHD non-linear thermal radiation of CuO=Al2O3 hybrid nanofluids and base water in convective heat transfer. Zainal et al.[12] described the Maxwell time- dependent flow of thermal radiation in a Cu=Al2O3 hybrid nanofluid past stretching/shrinking sheet. Emad et al. [13]

investigated the MHD in a heat transfer Cu=Al2O3 hybrid nanofluids due to the porous medium with heated convective.

Chu et al. [14] reported MHD unsteady hybrid nanofluid between infinite surface. Nayak et al. [15] studied Cattaneo christov theory under the effect of Oldroyd B-fluid with hyrbid nanofluid. Puneeth et al. [16] discussed the effect of

homogenous-hetrogenous reaction regarding a hybrid nano- fluid due to needle. Olatundun et al.[17]described Blasius flow of hybrid nanofluid due to convective condition. Mohsen et al.

[18] carried out nanoparticles distribution in dual phase between cubical enclosure.

In the modern era, the non-Newtonian fluid contemporary needs a main exceptional consideration for the researchers in this vast field. So, the mixture of such fluids in polymers, clay, plastic production, coal slurries, cosmetics, melting of plastic and many more. To study the rheological properties of non- Newtonian fluids with various model has been discussed[19–

22]. These fluids can not be identified by one constitutive rela- tion. In this way, investigators developed kinds of fluids con- taining differential and rate types. To discover the significance of these non-Newtonian fluids has increasingly the characteristics for researchers due to these non- Newtonian fluids various mathematical expressions such as Jeffery, Maxwell, Carreau, Power-law, Casson model and many others. However, these models are unable to study the characteristics of fluids at less and greater shear rates. To iden- tify the characteristics at less and greater shear rates Cross fluid model was designed by Cross [23]. Cross fluid models can reduce the challenges when the shear rate is largely enhanced. Various researchers[24,25]are investigating in this direction for blood as well as its properties. The effect of ther- mal radiative on Cross nanofluid through heat sour/sink was described by Nazeer et al.[26]. The activation energy and ther- mal radiation on Cross fluid flow past an inclined magnetic field are made by Sabir et al.[27]. Yao et al.[28]carried out a Cross fluid with the effect of magnetic dipole using different properties. Khan et al.[29]studied the influence of non-linear radiation near the stagnation point on thermophoretic and activation energy for Cross fluid.

Nomenclature

a Stretching rate (s¹)

k Mixed convection parameter (Dimensionless) k_nf Effective thermal conductivityðW=m:KÞ t_f kinematic viscosity (m²s¹)

q_f Reference density of fluid (kgm¹) q_s Reference density of solid (kgm¹) Pr Prandtl number (Dimensionless) g Similarity variable (Dimensionless) f Velocity profile (Dimensionless) h Temperature profile (Dimensionless) l_f Viscosity of fluid (m²s¹)

We Weissenberg number (Dimensionless) Rd Radiation parameter (Dimensionless) Bi Biot number (Dimensionless)

n Power – law index number (Dimensionless) k Thermal conductivity of base fluid (mkgs³K¹) u;v Components of velocity along x;y direction

ðms¹Þ

x;y Cartesian coordinates (m) q_nf Density of fluid (kgm¹)

l_nf Effective viscosity of nanofluid (m²s¹) Nu Nusselt number (Dimensionless) k_f Thermal conductivity of fluidðW=m:KÞ k_s Thermal conductivity of solidðW=m:KÞ T Local fluid temperature (K)

T1 Ambient temperature (K) T_f Temperature of the hot fluid (K)

A comparatively new trend in the observation of thermal systems is the second law of thermodynamics and its idea linked with entropy production. The entropy production has been used in various industry features that contain heat pumps, power generators, refrigerators etc. Bejan et al. [30]

examine the entropy generation in heat transport phenomena.

Later, Shit et al.[31]established the two-dimensional MHD flow for the time-dependent flow in a permeable medium past an exponential surface. Shaoo et al.[32]scrutinised a mathe- matical formulation for entropy generation on Casson nano- fluid by considering viscous dissipation and Hall current effect. Entropy production and MHD flow of Cross nano- fluid for a mixed convection flow have been explored by Abbas et al. [33]. Gabriela et al.[34] reviewed hybrid nano- fluid in the entropy generation for micro-macrochannel. Sha- fee et al.[35]described water baseFe3O4hybrid nanoparticles through a porous cavity. An entropy generation analysis and Cattaneo Christov theory on theTiO_2.CuO=Eghybrid nano- fluid over a stretching sheet has been discussed by Jamshed and Aziz[36]. Sindu and Gareesha[37]the effect of slip flow in a microchannel of hybrid nanofluid with convective boundary conditions. The numerically simluation of pertur- bation solution between parallel porous plate discussed by Nazeer et al. [38,39]. Hadi et al. [39] mentioned the entropy generation with hybrid nanofluid in 3D multi floors channel.

Many researchers are discussed on entropy genertaion with different models[40–45].

Carboxymethyl cellulose(CMC) is called cellulosic deriva- tive[46]. It has a high tendency to hold the water and modify the viscosity. This multi cellulosic derivative uses in many applications such as the food industry and many more. The ability of Carboxymethyl cellulose (CMC) escalates the beha- viour of fluid gathered with nanoparticles. It is engaged to enhance the lubricating effect[47]. Zainith et al.[48]analysed the three different nanomaterialsCuO;Al2O3 and TiO2-CMC water base solution with the wt. of 0:4%: Marjan et al.[49]

conducted heat transfer in a micro-channel of aluminium oxide with non-Newtonian fluid by using the aqueous solution of CMC water base. Rahmati et al. [50] detailed the CMC- water with the 0:5% wt influence of slip velocity and non- Newtonian nanofluid. Akinpelu and co-worker. [51] discov- ered the heat transfer and physical metal properties in Car- boxymethyl Cellulose (CMC). Alwawi et al. [52] developed the heat transfer analysis for the Casson nanoliquid on MHD carboxymethyl Cellulose (CMC) over a solid sphere.

The above mentioned study discussed nanparticles with base fluid.Cross fluid are technlogical significance and very few discussion attempted, limited aspect of this analysis. The main objective of this inclusive analysis is the increment of heat transporation with the numerical study of the entropy genera- tion and thermal radiation for Cross fluid havingCuOTiO2/ CMC water-based hybid nanoparticles with mixed convection.

From the best of our information and facts on the above- mentioned study, no investigations have been observed on hybrid nanofluid with Cross fluid model in a mixed convection flow. A numerical method is used for the solution of construc- tion. In addition, comprehensive observation is made for ordi- nary fluid CMC water and hybrid nanoparticles TiO2 and CuO. However, it attempts to discover the effect of numerous variables and heat transport. The outcomes are validated with previously available literature.

2. Mathematical formulation

The Casuhy stress tensor

shnf¼ pIþlðBÞa_ 1;l¼l₁þ l₀l₁

1þCB_n ð1Þ

Where, time constant,n is the power law index, the first Rivlin-Ericksen tensor, p the pressure, I the density tensor, B_¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2trða²₁Þ q

the sheart rate, described the zero and infinite shear rate. In the current,l₁is conisdered to be zero.Eq(1) becomes.

l¼ l₀

1þCB_n ð2Þ

The shear rate in the Cross viscosity model for the present invetigation can be given as.

B_¼ 4 @u

@x þ @u

@yþ@v

@x

2

( )1=2

ð3Þ

The velocity and temperature for two dimensional flow are.

V¼fuðx;yÞ;vðx;yÞ;0g;T¼Tðx;yÞ ð4Þ The flow is considered steady, incompressible, and mixed convective flow with a Cross hybrid-nanofluid pasta stretch- ablevertical sheet.The convective boundary condition and thermal radiation impact have been also taken in the given study. Moreover, the velocity componentuandvare repres- netd in the direction of the y-axis and x-axis as seen in Fig. 1. The constitutive equations for mass, momentum, and energy through boundary layer conjecturesalong with bound- ary conditions of Cross hybrid nanoliquid can be carved as follows:

@u

@xþ@v

@y¼0; ð5Þ

u@u

@xþv@u

@y¼v_hnf @

@y

@u@y

1þC^@u_@yn

0 B@

1 CAþ g

q_hnfðqb_TÞ_hnfðTT₁Þ;

ð6Þ

u@T

@xþv@T

@y¼ khnf

qCp

hnf

@²T

@y² 1

qCp

hnf

@q_r

@y; ð7Þ

The boundary constraintsare applied as follows:

u¼uwð Þ ¼x ax;v¼0;khnf

@T

@y^_¼hfTfT

aty¼0;

u!0;T!T₁asy! 1: ð8Þ

Meanwhile,Texhibits the temperature of the fluid,Cexhi- bits the Cross time constant, n exhibits the flow behaviour index (also referred as a power-law index), exhibits the kine- matic viscosity,q_hnfexhibits the density,khnfexhibits the ther- mal conductivity, ðqb_TÞ_hnf exhibits the thermal expansion coefficient, and qC_p

hnf exhibits the heat capacitance of the hybrid nanofluid.

In addition, the thermophysical characteristics of the Cross hybrid nanoliquid are demarcated as:

lhnf lf ¼ _1u¹

1u₂ ð Þ^2:5;

q_hnf

qf ¼ð1u₂Þ ð1u₁Þ þ^u1_q^q1s

f

h i

þ^u2_q^q2s

f ;

qb_T ð Þ_hnf

qbT

ð Þ_f ¼ð1u₂Þ ð1u₁Þ þ^u1ð_qb^qbTÞ1s

ð TÞ_f

h i

þ^u2ð_qb^qbTÞ2s

ð TÞ_f ;

khnf

kf ¼^k_k^{2sþ2kf2u2ðkfk2sÞ}

2sþ2kfþu2ðkfk2sÞ ^k_k^{1sþ2kf2u1ðkfk1sÞ}

1sþ2kfþu1ðkfk1sÞ;

qcp

ð Þhnf qcp

ð Þ_f ¼ð1u₂Þ ð1u₁Þ þ^u1ð Þ^qcp _1s

qcp

ð Þ_f

þ^u2ð Þ^qcp _2s

qcp

ð Þ_f :

ð9Þ

In this equation(9), the mathematical symbolsu₁ andu₂ exhibit the solid nanoparticles volume fraction of the titanium dioxide (TiO2) and copper oxideðCuOÞnanoparticles, respec- tively. However, the subscriptsf;hnf;1sand 2sdemonstrate the base fluid, hybrid nanoliquid and the two distinct solid nanoparticles, respectively. The physical properties of these hybrid nanoparticles along with the requisite base fluid are seen inTable 1.

It is pertinent to employ the following proper transformation.

g¼y ffiffiffiffia v_f r

;u¼axf⁰ðgÞ;v¼ ffiffiffiffiffiffi av_f

p fðgÞ;hðgÞ ¼ TT1

T_fT₁: ð10Þ By applying the similaritytransmission described in Eq.

(10), Eqs.(6) and (7)are transformed into the following ODEs which depend on a single independent variablegas follows:

P1 1þ ð1nÞWef^}n f^{00 0}

h i

þP2 ff^} f^{0 2}

1þWef^}n

n o2

þP3khn1þWef^}no2

¼0 ð11Þ

h^}ðP4þ4

3RdÞ þPrP5fh⁰¼0: ð12Þ

The distortedboundary conditions are stated as:

fð Þ ¼0 0;f⁰ð Þ ¼0 1;P4h⁰ð Þ ¼ Bi0 ð1hð Þ0Þ;

f⁰ð1Þ !0;hð1Þ !0: ð13Þ

In which:

Dynamic viscosity¼P1¼l_hnf

l_f ¼ 1

1/₁/₂ ð Þ^2:5; Density¼P2¼^q_q^hnf

f ¼ð1/₂Þ ð1/₁Þ þ^/1_q^q1s

f

h i

þ^/2_q^q2s

f ;

Thermal expansion¼P3¼ðqb_TÞ_hnf qb_T ð Þ_f

¼ð1/₂Þ ð1/₁Þ þ/1ðqbTÞ_1s qb_T ð Þ_f

" #

þ/2ðqbTÞ_2s qb_T ð Þ_f

Table 1 Thermophysical properties of the hybrid nanofluid.

Physical properties C_pðJ=kgKÞ qkg=m³ k Wð =mKÞ b_Tð1=KÞ

CMC-water 4179 997.1 0.613 21

TiO2 686.2 4350 8.95 0.72

CuO 531.8 6320 76.5 1.80

Fig. 1 The problems’ physical model.

Thermalconductivity¼P4¼khnf

kf

¼k2sþ2kf2/₂ kfk2s

k_2sþ2k_fþ/₂k_fk_2s knf

whereknf¼^k1sþ2kf2/1ð^kfk1sÞ

k1sþ2kfþ/1ðkfk1sÞ;

Heat capacity¼P5¼qc_p

hnf

qc_p

f

¼ð1/₂Þ ð1/₁Þ þ/₁ qcp

1s

qcp

f

" #

þ/₂ qcp

2s

qcp

f

For the similarity solution of equation(10), we here define b_T

ð Þ_f¼xð Þb_Tf. Despite this, the set of the similarity equations comprised distinct dimensionless influential parameters which are expressed as the Weissenberg number We ¼Cxa ffiffi_a

v

p

, the mixed convection or buoyancy parameter k¼^Gr_Re^x2

x¼^{gð ÞbTf}ð^TfT1Þ

a² ; the Prandtl number Pr¼^lfð Þ^cp _x

kf , the Radiation parameter Rd¼^4rT_kk³¹

f , and the Biot number, Bi¼hf

ffiffiffitf a

q _k¹

f.

The gradients are the local skin friction coefficientCf and the local Nusselt numberNuxwhich are presented by:

Fig. 2 (a-d). The impact ofWeon..f⁰ðgÞ;hðgÞ;NsðgÞ;BeðgÞ Table 2 Validation of current outcomes for Table 2. Valida-

tion of current for theh⁰ð0Þagainst.Pr:

Pr Ref[53] Current Analysis

0.7 1.0824024 1.0824

1.0 1.3336876 1.3336

10.0 4.7968167 4.796

C_f¼ s_w q_fu²_w

;Nu¼ x

kfðTwT1Þq_wð Þjq_{r y¼0}

; ð14Þ

The heat flux and wall shear stress are.

q_w¼ khnf

@T

@y _y¼0;s_w¼l_hnf

@u@y

1þC^@u_@yn

0 B@

1 CA

y¼0

; ð15Þ

In view of Eq.(10).

Re¹⁼²Nu¼ P4þ4 3Rd

hð0Þ;

Re_x¹⁼²C_f¼P1f}ð0Þð1þðWef}ð0ÞÞⁿÞ¹: ð16Þ WhereRe_x¼^uw_t^x

f called the local Reynolds number.

2.1. Entropy generation

The appearance of entropy production for cross hybrid nano- liquid over stretching sheet under energy dissipation gives.

EG¼khmf

T²₁ 1þ16rT²₁ 3k

@T

@y

" 2#

þl_hnf 1 T₁

@u

@y

2 1

1þC ^@u_@y ⁿ 0

B@

1

CA ð17Þ

First term represent in Eq.(17)heat transfer irreversibility and second term represents fluid friction irreversibility. The characteristics entropy generation is described below.

E⁰⁰⁰₀ ¼khmfðTwT1Þ

xT²₁ ð18Þ

Fig. 3 (a-b). The impact of/₁;/₂onf⁰ðgÞ;hðgÞ.

Fig. 4 (a-b). The impact ofkandBionf⁰ðgÞ;hðgÞ.

The dimensionless form of entropy generation is.

Ns¼EG

E^{0 00}₀ ð19Þ

N_G¼N_hþN_v ð20Þ

Temperature ratio and fluid friction is given by.

N_h¼P5½1þRdð Þh^{0 2};N_v¼BrP4 1þ 1 Wef}

ð Þⁿ

f⁰⁰² ð21Þ

The dimensionless variables described in Eq.(10) and Eq.

(17), Eq.(18)becomes the following constructive form.

NG¼P5½1þRdð Þh^{0 2}þBrP4 1þ 1 Wef} ð Þⁿ

f⁰⁰² ð22Þ

The Bejan number is defined by.

Be¼ P5½1þRdð Þh^{0 2} P5½1þRdð Þh^{0 2}þ þBrP4 1þ_ðWef}¹ _Þn

h i

f^}2

ð23Þ

Fig. 5 (a-c). The impact ofRdonhðgÞ;NsðgÞ;BeðgÞ.

2.2. Solution and methodology

The set of altered system of highly non-linear ODE’s Eqs.(11) and (12)with subject to the boundary conditions(13)has been numerically computed with aid of BVP4c method. For this purpose, we converted the boundary value of problem into first order by assigning a new variable by fallowing procedure.

f¼K1;f⁰¼K2;f}¼K3;f⁰⁰⁰¼K⁰₃;h¼K4;h⁰¼K5;h⁰⁰¼K⁰₅ ð24Þ

K⁰3¼P2K1K3dK2þ^g₂K3

ð ÞK2²

h i

1þðWeK3Þⁿ

f g²P3kK4f1þðWeK3Þⁿg² P1½ð1þ ð1nÞðWeK3ÞⁿÞ

ð25Þ

h^}ð1þP4

4 3RdÞ g

2dP3

P4

Prh⁰þPrP5fh⁰¼0 ð26Þ

K⁰₅¼PrP5r1r5þ^g₂d^P_P³

4PrK4

ð1þP44

3RdÞ ð27Þ

Converted boundary condition.

K1ð Þ ¼0 0;K2ð Þ ¼0 1;K5ð Þ ¼ c0 ð1K4ð Þ0Þ;

K2ð Þ ¼1 0;K4ð Þ ¼1 0: ð28Þ The iterative process has been used and the accuracy of the solution is obtained to 10⁶.

3. Result and discussion

To examine the effect of numerous variables on f⁰ðgÞ, hðgÞ,NsðgÞandBeðgÞ, the numerical outcomes have beendis- playedoppositeof various flow parameters contain We;Bi;Pr;Rd;u₁;u2,kandu₁¼TiO2;u₂¼Cuo. The numeri- cal value of flow are considered tp be contant We¼0:7; n¼1:0;Bi¼0:3;k¼1;Pr¼6:2; Rd¼0:7;/₁¼0:025;/₂¼ 0:06. Table 2 shows the verification of outcome against the hð0ÞforPrand it is obtained an outstanding agreement with the work of Wakif[53]. It is noted for the escalating value of nanoparticles volume fraction, mixed convection parameter (k), and biot number ðBiÞ exhibit a larger upsurging in the

skinfriction and the Nusselt number for hybrid nanoliquid as compared to usual nanoliquid.

Fig. 2(a-d) displayedthe impact of the Weissenberg number on the velocity profilef⁰ðgÞ, the temperature distributionhðgÞ, entropyNsðgÞ, and the Bejan numberBeðgÞ. The velocity pro- file and momentum boundary layer thickness reduce due to rises inWe. Hybrid nanofluid is largely influenced by boosting the value ofWe. Physically, the Weissenberg number is therate of shear with time that climbs the thickness of the fluid, due to this fluid the velocity depreciates with arise ofWe.Fig. 2(b) reveals the fluctuation of the We on hðgÞ. The temperature and relatedthickness of boundary layer upsurge for improving the value ofWe. Actually, theWehas a specific significance to intensify the temperature profile. Moreover, the thermal layer thickness is superior for hybrid nanoliquid compared to nano- liquid. The effect of We on the entropyNsðgÞand Bejan num- ber Beð Þg has been demonstrated in Fig. 2 (c-d). Profile of NsðgÞdecays with a larger value ofWefor nanofluid and lower entropy created in the nanoliquid as compared to the hybrid nanoliquid. A reverse trend is observed for Bejan numberBe.

Fig. 6 The impact ofPronhðgÞ.

Fig. 7 Impacts ofkonRe¹⁼²_x CfversusWe.

Fig. 8 Impacts ofRdonRe¹⁼²_x Nuversus.Bi:

Fig. 3(a-b) sees the influence ofu₁andu₂onf⁰ðgÞ, andhðgÞ.

Computations are obtained for bothu₁ andu₂. It is detected that the velocity declines with a larger value of u₁;u₂ and increases in a conventional fluid. Physically, it is shown that hybrid-nanoparticles are dense and the main reason for the depreciation of the fluid velocity.Fig. 3(b) displays the mount in the fluid temperature with escalating values of nanoparticles volume frication. The normal nano liquids thermal conductiv- ity is lower than that of hybrid nanofluid.Moreover, it is viewed that the temperature of hybridnanofluid is greater in comparison of nanofluid. Fig. 4 (a-b) represents the mixed convection kon the velocity profile and Biot number Bi on the temperature distribution. The profile off⁰ðgÞisthe reducing function ofk, and it happens due to the existence of a bouncy force. Bouncy force applies as a suitable pressure gradient thus stronger bouncy force accommodates the flow in a lower direc- tion which is the reason the velocity decreases. Fig. 4(b) divulges the temperature profile for Biot number Bi. Physi- cally, Biot number is the proportion of convection at the boundary of conduction inside the boundary. That’s the rea- son,the temperature profile hiked for the escalating value ofBi: Fig. 5 (a-c) portrays the effect of Rd on the temperature field hðgÞ, entropy generation NsðgÞ and Bejan number BeðgÞ. The temperature distribution and thermal layer thick- ness exaggerated the growing value ofRdas shown inFig. 5 (a). In fact, additional heat is achieved, causing the thermal radiation procedure due to which the radiation parameter is augmented. Fig. 5 (b-c) is viewed as similar behaviour for the entropy and Bejan number.Fig. 6shows the influence of Pr on the temperature profile.Theaugment inPr reduces the temperature profile. Physically, it is mean that the Prandtl number is the proportion of thermal diffusivity due to this temperature distribution decaying.

Figs. 7 and 8 display to notice the effect of k, and Rd againstWe,Biover the shear wall stressand the rate of heat transfer. From the figure,Cf is boosting the function ofkand surface drag inverse relation withk. Whereas the Nusselt num- ber is enhanced for Biand Rd:Moreover, the rate of heat transport is an increase forRd:

Numeric outcomes of the drag friction and the Nusselt numberfor the numerous values ofWe;n;k;/₁;/₂;BiandRd in the case of nano liquid and hybrid nano liquid have been evaluated inTable 3. It is shown that the causal nano liquid and hybrid nano liquid are upsurged inCfandNu.

Figs. 9 and 10display the bar graph comparative between normal nanoliquid and hybrid nanoliquid for the magnitude ofWeandBi. It is clearly shown the impact of hybrid nano liq- uid is superior in contrast to normal nanoliquid. Finally, Table 3 Values of the skin friction and the Nusselt number.

C_fRe¹⁼²_x NuRe¹⁼²_x

We n k Bi Rd CuO TiO2 CuOþTiO2 CuO TiO2 CuOþTiO2

1.0 1. 1.0 0.2 0.5 1.3015 1.4279 2.7294 0.1236 0.1352 0.2588

1.1 0.3 1.6204 1.8078 3.4282 0.2334 0.2542 0.4876

1.2 0.4 2.1375 2.4515 4.5890 0.3316 0.3597 0.6913

1.0 1.0 1.3 0.2 0.7 1.2886 1.4137 2.7023 0.1397 0.1632 0.3029

1.1 0.3 1.5994 1.7840 3.3834 0.2630 0.3051 0.5681

1.2 0.4 2.0994 2.4055 4.5049 0.3727 0.4289 0.8016

1.0 1.0 1.5 0.2 0.9 1.2601 1.4044 2.6645 0.1557 0.1818 0.3375

1.1 0.3 1.5656 1.7682 3.3338 0.2924 0.3389 0.6313

1.2 0.4 2.0745 2.3745 4.44900 0.4136 0.4762 0.8898

Fig. 9 Bar graph for the skin friction versusWe.

Fig. 10 Bar graph for the Nusselt number versusBi.

Figs. 11 and 12describe the streamlined flow for the different values of nanoparticles. The streamlines patterns are simple curves and closed to the surface.

4. Conclusion

In this study, a (CuOTiO2/CMC-water) hybrid nanofluid using a Crossflow model with thermal radiativepast a stretch- able vertical sheet incorporated by the addition of entropy

generation has been studied. The model is computed via Mat- lab BVP4c method. The important out come sare given below.

The velocity curve depreciates as the Weissenberg number (We) esclates. Whereas the opposite effect is examined in the power index number. Hybrid nanofluid is more preju- diced than casual nanofluid.

The temperature profile is enhanced for bothnandWe:

The trends of entropy generation and Bejan number are quite opposite forWeandn:

The temperature field, entropy minimization, and Bejan number are enhanced asRd increases.

Biot number and mixed convective variables are improved for the thermal distribution and the velocity field.

TheRe¹⁼²_x Cf andRe¹⁼²_x Nuare increasing trend on nanopar- ticles and hybrid nanoparticles.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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