M.Bilal, Imran Khan,Taza Gul,＊,Asifa Tassaddiq, Wajdi Alghamdi, Safyan Mukhtar and Poom Kumam

1Department of Mathematics, City University of Science and Information Technology, Peshawar, 25000,Pakistan

2Department of Basic Sciences and Humanities,College of Computer and Information Sciences,Majmaah University,Al-Majmaah,11952,Saudi Arabia

3Department of Information Technology, Faculty of Computing and Information Technology, King Abdulaziz University, Jeddah,80261, Saudi Arabia

4Basic Sciences Department, Deanship of Preparatory Year, King Faisal University, Al Ahsa, 31982,Saudi Arabia

5Center of Excellence in Theoretical and Computational Science(TaCS-CoE),SCL 802 Fixed Point Laboratory,Science Laboratory Building, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, 10140,Thailand

Abstract: This article aims to investigate the Darcy Forchhemier mixed convection flow of the hybrid nanofluid through an inclined extending cylinder.Two different nanoparticles such as carbon nanotubes(CNTs)and iron oxide Fe3O4 have been added to the base fluid in order to prepare a hybrid nanofluid.Nonlinear partial differential equations for momentum, energy and convective diffusion have been changed into dimensionless ordinary differential equations after using Von Karman approach.Homotopy analysis method (HAM), a powerful analytical approach has been used to find the solution to the given problem.The effects of the physical constraints on velocity,concentration and temperature profile have been drawn as well for discussion purpose.The numerical outcomes have been carried out for the drag force, heat transfer rate and diffusion rate etc.The Biot number of heat and mass transfer affects the fluid temperature whereas the Forchhemier parameter and the inclination angle decrease the velocity of the fluid flow.The results show that hybrid nanofluid is the best source of enhancing heat transfer and can be used for cooling purposes as well.

Keywords: Mixed convection;similarity transformation;HAM;hybrid nanofluid;CNTs;Darcy Forchhemier;inclined cylinder

1 Introduction

The study of the hybrid nanofluid in the existence of mass and heat transfer has received special attention from many scientists and researchers because of its essential role in the field of science and technology[1].The convection of the hybrid nanofluid flow, together with heat and mass transfer, has several important applications in industry such as oil reservoir, suspension and colloidal solution, bioengineering, nuclear industries, polymer solution, paper production, geophysics, chemical industries and exotic lubricants etc.[2-6].The fluid like kerosene oil, water, acetone, engine oil and ethylene glycol has low thermal conductivity.In the era of modern science and technology, the extensive need for thermal energy cannot be fulfilled through commonly used fluids.However, a significant enhancement in thermal characteristics was noted when these base liquids were synthesized with the addition of small sized particles [7].Thus,this rise in the thermal properties of ordinary fluids developed the keen curiosity of scientists for further investigations.Numerous researches on nanoparticles and carbon nanotubes CNTs, both single walled carbon nanotubes and multi-walled carbon nanotubes (SWCNTs and MWCNTs) have been carried out by the researchers.CNTs are the allotropes of carbon with a nano cylindrical structure.The CNTs are frequently used in the energy sector and Nanoscience [8].The mixture of copper oxide and water was examined by Animasaun [9].The water based nanofluid fluid flow of CNTs was analyzed by Aman et al.[10].The enhancement of the heat transfer rate using the nanofluid of carbon nanotubes was examined by Raza et al.[11].The impact of the nanofluid using Arrhenius activation energy was examined by Muhammad et al.[12].The water based iron oxide nanofluid flow was studied by Qasim et al.[13].The nanofluid of magnetite-ferrium oxideFe3O4was examined by Hussanan et al.[14].The water based aluminum oxide (Al2O3) nanofluid was studied by Sheikholeslami et al.[15].The nanofluid flow through a stirring surface has been studied by Haq et al.[16].The mixed convection flow of hybrid nanofluid consisting CNTs over a stretching inclined cylinder has tremendous use in the field of mechanical engineering.The mass and heat transfer of nanofluid through an inclined surface is affected due to the buoyancy forces.This type of phenomena plays an important role in the cooling of electronic devices,automobile demister, boilers, defroster system and in solar energy system [17].The effects of slip flow over time dependent stretching sheet including mixed convection were explored by Makinde et al.[18].Rashad et al.[19] minutely examined the micropolar fluid flow using double stratified medium.Turkyilmazoglu [20] scrutinized viscoelastic fluid flow with mixed convection over a stretching porous surface.Ashraf et al.[21] investigated the 3D Maxwell fluid flow with mixed convection over an extending inclined surface.The mixed convection flow with double effect of stratification of a Jeffrey fluid past on an extending inclined cylinder has been scrutinized by Hayat et al.[22].The analytic solution of the nanofluid flow including natural convection over a linearly extending sheet has been presented by Hammad [23].A survey article has been published by Buongiorno [24] on convection transport through nanofluid.The Darcy-Forchheimer model is the most well-known extension to Darcian flow usually in resemblance with the effects of inertia.The effect of inertia is considered by the insertion of squared terms of the velocity in the momentum equation known as Forchheimer modification.This new term has been named as Forchheimer factor by Muskat [25].Mondal et al.[26,27] examined non-Darcy Forchheimer model in their research articles over a stretching surface.The Darcy Forchheimer(DF) flow over an upright surface has been studied by Anwar et al.[28].In order to understand better the problems occurred in the field of physics, it is essential to involve non-Darcy effects in convective transport analysis.The Darcy Forchheimer mixed convective flow in porous media has been examined by Seddek [29].The DF mixed convection flow of the hybrid nanofluid consisting CNTs through the impermeable inclined cylinder has been investigated minutely in this study.It will lead the researchers to new investigations.The mixed convection flow and heat transfer have several useful applications such as storage and food processing, underground disposing of nuclear wastes and geophysical system [30].This versatility makes several applications to be studied through the fluid flow over an inclined cylinder for the enhancement of heat and mass transfer.Keep in view the importance and applications of this work,we have extended the idea of[17]and revealed this problem.The results have been achieved through HAM.

2 Mathematical Formulation

The mixed convection Darcy-Forchhimier fluid flow over an inclined stretching cylinder has been examined in this research.The flow has been considered steady and axisymmetric.The physical sketch of the flow has been illustrated in Fig.1.The analysis of heat and mass transfer has been considered for the hybrid nanofluid consisting CNTs and iron oxide.The analysis of the hybrid nanofluid flow past an extended cylinder has been considered.In the coordinate system,the x-axis and r-axis are considered along the axial and normal direction to the cylinder respectively.After using the boundary layer approximations, the laws have been reduced to the form as follows[17].The basic flow equations are:

Figure 1:Geometry of the problem

The physical conditions for the governing equations are:

The velocity components u and v have been taken along the axial and normal directions of the cylinder.μhnf,νhnfand ρhnfdemonstrate the dynamic,kinematic and density of hybrid nanofluid.βTand βcshow the thermal expansion coefficient and the concentration expansion coefficient respectively.WhereasTfandCfrepresents the convective fluid temperature and concentration respectively.

The appropriate transformations are:

By using Eq.(6) in Eqs.(1)-(5),we get

The transformed conditions for nonlinear differential equations are as follows:khnfis the thermal conductivity and(ρCp)hnfis the volumetric heat capacity of hybrid nanofluid as stated in [31]:

where

The volumetric concentrations ofFe3O4and CNTs have been denoted by ?1and ?2.Each and every abbreviation has been defined individually.Furthermore,kMSandkCNTimply the thermal conductivities ofFe3O4and CNTs.ρfis the density, μ is the viscosity and (ρcp)fis the specific heat of theH2O.(ρCp)MS, ρMS, and ρCNTat constant pressure indicate specific heat capacities and densities ofFe3O4and CNTs.The Deborah number, Prandtl number and Schmidt number have been denoted by β, Pr and Sc.GrandGr*denote temperature Grashof number and mass Grashof number respectively.Biot numbersBi1andBi2are for the heat and mass transfer.

The physical constraints have been defined in [17]:

The Sherwood numberShx, the local Nusselt numberNuxand skin friction coefficient have been expressed in dimensional form as follows:

In which surface mass flux, surface heat flux and surface shear stress have been represented byjw,qwand τw:

Local Sherwood numberShx,the local Nusselt numberNuxand Skin friction coefficientCfare:

The local Reynolds number is Rex=

3 Problem Solution

The current problem has been solved by using HAM technique that was initiated by Liao[32-34].BVPh 2.0 package[35-42]has been used for the convergence of the modeled problem.The initial approximations for velocityf0,temperature Θ0and concentration Φ0are given as:

The linear operators ?f, ?Θand ?Φpresented as:

The solved form of ?f,?Θand ?Φare:

The series introduced by Taylor’s has been used as:

Now

The equations have been further concluded in the form of a system as:

4 Results and Discussion

This study aims to use the hybrid nanofluids flow over a stretching cylinder for the rapid heating and cooling applications in the field of thermal engineering.The hybrid nanofluid contains solid particles of theFe3O4,CNTsand base liquid ofH2O.The solid particles disperse in the base liquid and as a result the hybrid nanofluid is prepared.The analytical solution of the system has been obtained through the Homotopy analysis method (HAM).The influence of the constraints has been demonstrated in Figs.2-16.Fig.2 reveals the influence of the Farchemmier’s parameterFron velocity profile.WhenFrincreases, it enhances the transfer of mass in fluid flow which results in a decrease in the velocity of fluid flow.Fig.3 intends to perceiveKreffects on the velocity field.The rising value ofKrresults in an increase in the fluid kinematic viscosity and consequently declines the velocity of the hybrid nanofluid.The kinematic viscosity ofFe3O4is greater than the CNTs.Thus, the influence ofKris comparatively large using the iron oxide.

Fig.4 has been sketched to reveal the influences of Ψ(angle of inclination)on both CNTs andFe3O4nanofluid with velocityf′(η).The increase in the value of the parameter Ψ decreases the velocity field.In fact, decreasing the effect of gravity decreases velocity profile.The decreasing effect of gravity decreases velocity profile.The essential performance of the velocity fieldf′(η) with the rising values of λ has been shown in Fig.5.As λ is the ratio of relaxation to retardation time, thus, the increase in the values of λ(i.e.,extending the relaxation time)provides some additional resistance to the flow field.That’s why fluid velocity decreases.

Figure 2:The effects of Fr on velocity profilef ′(η) when Pr =6.2,Γ=0.1,?2 =?1 =0.1,Kr=0.5,Bi1 =0.2,λ=1.2,N =0.1 and ψ=π/4

Figure 3:The effects of Kr on velocity profilef ′(η) when Pr =6.2,Fr=1,Γ=0.1,?2 =?1 =0.1,Bi1 =0.2,λ=1.2,N =0.1 and ψ=π/4

Figure 4:The effects of Ψ on velocity profilef′(η)when Pr=6.2,Fr=1,Γ=0.1,?2 =?1 =0.1,Kr=0.5,Bi1 =0.2,λ=1.2 and N =0.1

Figure 5:The effects of λ on velocity profilef′(η) when Pr =6.2,Fr=1,Γ=0.1,?2 =?1 =0.1,Kr=0.5,Bi1 =0.2,N =0.1 and ψ=π/4

The additional resistance due to relaxation time also enhances the fluid temperature presented in Fig.6.Fig.7 shows the effects ofKrparameter on temperature distribution.The fluid temperature decreases while increasingKr.In fact, the kinematic viscosity of the fluid increases which drops the temperature normally.

Figure 6:The effects of λ on temperature distributionΘ(η) when Pr =6.2,Fr=1,Γ=0.1,?2 =?1 =0.1,Kr=0.5,Bi1 =0.2,N =0.1 and ψ=π/4

Figs.8 and 9 demonstrate the variation of temperature distribution with Biot numbersBi1andBi2respectively.Both Biot numbers consist of heat and mass transfer coefficienthtandhcrespectively.Thus,the increase in the values ofBi1andBi2enriches the thermal boundary layer and concentration field.Which results in an increase in the fluid temperature.

Figs.10 and 11 describe ?1and ?2(volume fraction parameters) of the mentioned nanoparticles that affect the temperature distribution.The volume fraction constraints ?1and ?2boost up the temperature field.It has been noticed that the adequate amount of volume fraction can increase thermal property of the base fluid and consequently the temperature of the fluid increases.

Figure 7:The effects of Kr on temperature distribution Θ(η) when Pr =6.2,Fr=1,Γ=0.1,?2 =?1 =0.1,Bi1 =0.2,λ=1.2,N =0.1 and ψ=π/4

Figure 8:The effects of Bi1 on temperature distribution Θ(η)when Pr=6.2,Fr=1,Γ=0.1,?2 =?1 =0.1,Kr=0.5, Bi2 =0.2,λ=1.2,N =0.1 and ψ=π/4

Figure 9:The effects of Bi2 on temperature distribution Θ(η)when Pr=6.2,Fr=1,Γ=0.1,?2 =?1 =0.1,Kr=0.5, Bi1 =0.2,λ=1.2,N =0.1 and ψ=π/4

Figure 10:The effects of ?1 on temperature distribution Θ(η)when Pr=6.2,Fr=1,Γ=0.1,?2 =0.1,Kr=0.5,Bi1 =0.2,λ=1.2,N =0.1 and ψ=π/4

Fig.12 establishes the changes in the temperature field versus Pr.The increasing value of Prreduces the thermal diffusivity,consequently it drops the fluid temperature.Fig.13 illustrates the Schmidt numberScand its effects on the concentration distribution respectively.It has been observed that Schmidt number is used to increase the thickness of associated boundary layer and concentration.

Tab.1 shows the thermophysical properties of the base fluid and nanoparticles.Tab.2 shows the effect of the parameters λ andFr.It has been observed that skin friction increases due to the resistivity created by the mentioned parameters.

Tab.3 shows the effects of λ and Pr.It has been noticed that both parameters jointly increase Nusselt number due to the rise in the numerical values of these parameters during the thermal process.Moreover,SWNCTs offer excellence behavior to MWCNTs.

Tab.4 shows the wayScbehaves to Sherwood number.It is mainly found that as the Sherwood number decreases,it results in increasing in Schmidt number.The dominant impact of the SWCNTs is still observed on MWCTs.

Figure 11:The effects of ?2 on temperature distribution Θ(η)when Pr=6.2,Fr=1,Γ=0.1,?1 =0.1,Kr=0.5,Bi1 =0.2,λ=1.2,N =0.1 and ψ=π/4

Figure 12:The effects of Pr on temperature distribution Θ(η) when Fr=1,Γ=0.1,?2 =?1 =0.1,Kr=0.5,Bi1 =0.2,λ=1.2,N =0.1 and ψ=π/4

Figure 13:The effects of Sc on concentration distribution Θ(η) when Pr=6.2,Fr=1,Γ=0.1,?2 =?1 =0.1,Kr=0.5,Bi1 =0.2,λ=1.2,N =0.1 and ψ=π/4

Table 1:Thermo-physical properties of water,CNTs and Fe3O4 nanoparticles

Table 2:Presents the numerical outcomes of skin friction f ′(0)

Table 3:Exhibits the numerical outcomes of Nusselt number Θ′(0)

Table 4:Shows the numerical outcomes of Sherwood number Φ′(0)

5 Conclusion

In this work, we have addressed the Darcy Forchhemie’r hybrid nanofluid flow past a stretched and inclined cylinder.The solid nanoparticles of the CNTs and iron oxide have been used for the preparation of hybrid nanofluid.The main findings are as below:

? It has been noticed that increase in the values of the curvature parameter results in an increase in the profiles of the temperature,concentration and velocity of the hybrid nanofluid.

? The Biot numbers are used to improve the concentration and temperature transfer rates.

? In temperature distribution,the role ofCNT+Fe3O4/H2Ois more dominant thanFe3O4.

? The variation in λ (mixed convection parameter) increases the velocity profile while the rise in the Skin friction coefficient decreases the velocity field.

? The use ofCNT+Fe3O4/H2Ois more significant to increase the thermal efficiency of the base fluid as compared to the common fluid.

Funding Statement:This research has received no specific funding.

Conflict of Interest:The authors of this research have no conflict of interest.