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 oxide Fe3O4 was 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.

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:

(1) ∂(ru)∂x+∂(rv)∂r=0,

(2) ρhnf[u∂(u)∂x+v∂(u)∂r]=μhnf(∂2u∂r2+1r∂u∂r)+g(ρβT)hnf(T−T∞)g(ρβc)hnf(C−C∞)cos⁡α−νhnfuk∗−Cbk∗u2,

(3) (ρCp)hnf[u∂T∂x+v∂T∂r]=khnfr(∂2T∂r2+1r∂T∂r),

(4) [∂C∂x+v∂C∂r]=Dhnf(∂2C∂r2+1r∂C∂r).

The physical conditions for the governing equations are:

(5) u(x,r)=uw(x)=u0l,v(x,r)=0,−k∂T∂r=ht(Tf−T),−D∂C∂r=hc(Cf−C)atr=R,u(x,r)→0,T→T∞,C→C∞,asr→∞.

The velocity components u and v have been taken along the axial and normal directions of the cylinder. μhnf , νhnf and ρhnf demonstrate the dynamic, kinematic and density of hybrid nanofluid. βT and βc show the thermal expansion coefficient and the concentration expansion coefficient respectively. Whereas Tf and Cf represents the convective fluid temperature and concentration respectively.

The appropriate transformations are:

u=u0xlf′(η),v=−Rru0vlf(η),ψ(η)=−Rru0vx2lf(η),

(6) Θ(η)=T−T∞Tf−T∞,Φ(η)=C−C∞Cf−C∞,η=u0vl(r2−R22R).

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

(7) (1+2λη)ff′′+2λf′′+(1−φ1)2.5(1−φ2)2.5ρhnf(ff′′−(f′)2−Fr(f′)2)+(1−φ1)2.5(1−φ2)2.5(ρβc)hnf(Γ(Θ(ρβT)hnf+ΛΦ)cos⁡α)−krf′=0,

(8) khnfkbf[((1+2λη)Θ′′+2λΘ′)]+PrfΘ′[(1−ϕ2)(1−(1−(ρCp)Ms(Cpρ)fϕ1)+(ρCp)CNT(Cpρ)fϕ2)],

(9) (1−ϕ1)(1−ϕ2)[(1+2λη)Φ′′+2λΦ′]+ScfΦ′=0,

The transformed conditions for nonlinear differential equations are as follows:

(10) f(η)=0,f′(η)=0,Θ′(0)=−Bi1(1−Θ(0)),Φ′(0)−Bi2(1−Φ(0))atη=0,

khnf is the thermal conductivity and (ρCp)hnf is the volumetric heat capacity of hybrid nanofluid as stated in [31]:

(11) μhnf=μf(1−ϕ1)5/2(1−ϕ2)5/2,

(12) gβT=(1−ϕ2){1−(1−ρMsρf)ϕ1}+ρCNTρfϕ2,

(13) (gβT)hnf=(1−ϕ2){1−(1−(ρβT)Ms(ρβT)f)ϕ1}+(ρβT)CNT(ρβT)fϕ2,

(14) (gβc)hnf=(1−ϕ2){1−(1−(ρβc)Ms(ρβc)f)ϕ1}+(ρβc)CNT(ρβc)fϕ2,

(15) (ρCp)hnf(ρCp)f=(1−ϕ2){1−(1−(ρCp)Ms(ρCp)f)ϕ1}+(ρCp)CNT(ρCp)fϕ2,

(16) khnfkbf=1−ϕ2+2ϕ2kCNT(kCNT−kbf)−lnkCNT+kbf2kbf1−ϕ2+2ϕ2kbf(kCNT−kbf)−lnkCNT+kbf2kbf,

where

(17) kbfkf=kMS+(m−1)kf−(m−1)ϕ1(kf−kMS)kMS+(m−1)kf−ϕ1(kf−kMS).

The volumetric concentrations of Fe3O4 and CNTs have been denoted by ϕ1 and ϕ2 . Each and every abbreviation has been defined individually. Furthermore, kMS and kCNT imply the thermal conductivities of Fe3O4 and CNTs. ρf is the density, μ is the viscosity and (ρcp)f is the specific heat of the H2O . (ρCp)MS , ρMS , and ρCNT at constant pressure indicate specific heat capacities and densities of Fe3O4 and CNTs. The Deborah number, Prandtl number and Schmidt number have been denoted by β , Pr and Sc. Gr and Gr∗ denote temperature Grashof number and mass Grashof number respectively. Biot numbers Bi1 and Bi2 are for the heat and mass transfer.

The physical constraints have been defined in [17]:

λ=vlu0R2 , Γ=GrRex2 , Λ=Gr∗Grβc(Cf−C∞)βT(Cf−C∞) , Pr=μCpk ,

(18) kr=νf2u0k∗,

Bi1=htku0vl , Bi2=hcDu0vl , Fr=CpxRk∗ , Sc=vD .

The Sherwood number Shx , the local Nusselt number Nux and skin friction coefficient have been expressed in dimensional form as follows:

(19) Cf=τw12ρuw2,,

In which surface mass flux, surface heat flux and surface shear stress have been represented by jw , qw and τw :

(20) τw=μhnf(∂u∂r)r=R.

Local Sherwood number Shx , the local Nusselt number Nux and Skin friction coefficient Cf are:

(21) Rex−0.5Shx=(1−ϕ1)(1−ϕ2)Φ′(0),Re−0.5Nux=−khnfkbfΘ′(0)12Re0.5Cfx=1(1−ϕ1)2.5(1−ϕ2)2.5f″(0).

The local Reynolds number is Rex=u0x2vl.

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 velocity f0 , temperature Θ0 and concentration Φ0 are given as:

(22) f0(η)=1−eη,Θ0(η)=Bi11+Bi1e−η,Φ0(η)=Bi21+Bi2e−η.

The linear operators Lf , LΘ and LΦ presented as:

(23) Lf(f)=f′′′−f′,LΘ(Θ)=Θ′′−Θ,andLΦ(Φ)=Φ′′−Φ.

The solved form of Lf , LΘ and LΦ are:

(24) Lf[ς1+ς2η+ς3η2+ς4η3]=0,LΘ[ς5+ς6η+ς7η2]=0,LΦ[ς8+ς9η+ς10η2]=0.

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

(25) f(η;ξ)=f0(η)+∑l=1∞fl(η)ξl,

(26) Θ(η;ξ)=Θ0(η)+∑l=1∞Θl(η)ξl,

(27) Φ(η;ξ)=Φ0(η)+∑l=1∞Φl(η)ξl,

Now

(28) Φl(η)=1l!dΦ(η;ς)dη|ς=0,

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

(29) LΦ[Φl(η)−NlΦl−1(η)]=LΦRlΦ(η).

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 the Fe3O4 , CNTs and base liquid of H2O . 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 parameter Fr on velocity profile. When Fr increases, it enhances the transfer of mass in fluid flow which results in a decrease in the velocity of fluid flow. Fig. 3 intends to perceive Kr effects on the velocity field. The rising value of Kr results in an increase in the fluid kinematic viscosity and consequently declines the velocity of the hybrid nanofluid. The kinematic viscosity of Fe3O4 is greater than the CNTs. Thus, the influence of Kr is comparatively large using the iron oxide.

Fig. 4 has been sketched to reveal the influences of Ψ (angle of inclination) on both CNTs and Fe3O4 nanofluid with velocity f′(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 field f′(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.

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

Figs. 8 and 9 demonstrate the variation of temperature distribution with Biot numbers Bi1 and Bi2 respectively. Both Biot numbers consist of heat and mass transfer coefficient ht and hc respectively. Thus, the increase in the values of Bi1 and Bi2 enriches the thermal boundary layer and concentration field. Which results in an increase in the fluid temperature.

Figs. 10 and 11 describe ϕ1 and ϕ2 (volume fraction parameters) of the mentioned nanoparticles that affect the temperature distribution. The volume fraction constraints ϕ1 and ϕ2 boost 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.

Fig. 12 establishes the changes in the temperature field versus Pr . The increasing value of Pr reduces the thermal diffusivity, consequently it drops the fluid temperature. Fig. 13 illustrates the Schmidt number Sc and 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 and Fr . 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 way Sc behaves 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.

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.