Significance of suction and dual stretching on the dynamics of various hybrid nanofluids: Comparative analysis between type I and type II models

Fluid dynamics is a field of study with the objectives to not only analyse various fluid flows, but also explain the significance of some parameters to the scientists in the industry dealing with such dynamics. This has led to some reports like Qasim et al. [1], Wakif et al [2], Wakif et al [3], Wakif et al [4], Afridi et al [5], Qasim et al [6]. The significance of stretching as a sufficient tools capable to induce fluid flow has been deliberated by Gangadhar et al [7] in a study on the flow of micropolar ferrofluid, Gangadhar et al [8] in a study on the dynamics of nanofluid subject to Lorenzt force, and Gangadhar et al [9] in another study on the dynamics of Casson fluid. The mixture of one type of nanoparticles thoroughly dispersed in a base fluid otherwise known as nanofluid has attracted the attention of experts due to its unique variability of thermal property. Gangadhar et al [10] appraised the increasing effects of thermal radiation on the dynamics of titania (TiO₂) and alumina (${{Al}}_{2}{O}_{3}$) water based nanofluids while Kotha et al [11] only examined the increasing effect of thermal radiation on the dynamics of engine oil nanofluid. It was reported that the temperature distribution is an increasing property of thermal radiation. Convectively heating of the nanofluid near the wall may also affect the observed effect. Sobhana Babu et al [12] examined the dynamics of a nanofluid with a nonlinear relationship between shear stress and strain. Water conveying AA7072 and AA7075 forming hybrid nanofluid on an infinite disk subject to viscous dissipation and surface tension was modeled using II model by Ullah [13]. Table three presented in the report shows that at all the increasing values of thermal radiation, thermal slip, viscous dissipation, and exponential space heat source the Nusselt number proportional to heat transfer in the case of water conveying AA7075 is lower than that of the hybrid nanofluid.

The colloidal mixture of two different nanoparticles together with a base fluid is of importance to scientists and engineers dealing with fluid substances. For instance in solar collectors, the conversion of power energy from the Sun to electricity is termed photothermal energy. Enhancement of thermal and optical properties as in the case of photothermal energy was pointed out by Tong [14] through the addition of multi-walled carbon nanoparticles to an existing ethylene glycol mixed with ${{Fe}}_{3}{O}_{4}$ nanoparticles. The properties of insoluble titanium dioxide (titania) and it's UV resistant properties have made it a compulsory ingredient in the production of not only food coloring, but also cosmetic industry paint, and sunscreen. Some electrocatalysts can be used to reduce certain gases to organics. In this direction with major emphasis on addressing global warming, Hossain et al [15] concluded that if the homogeneous deposition-precipitation method is adopted to mix titania and copper nanoparticles with a precipitating agent like urea, then such mixture is a good substance for reduction of carbon dioxide in the vicinity to organic. Based on lattice distribution and valence of copper (Cu), Li et al [16], Assadi and Hanaor et al [17] concluded that there exists charge transfer between metal to metal in the formation of Ti-O-Cu and this process enhances photoactivity. More so, an exhibition of electron-hole separation in the mixture of ${Cu}-{{TiO}}_{2}$ (Cu doped titanium dioxide) is achievable with photoexcitation.

Viscosity, density, thermal conductivity, electrical conductivity, heat capacity, and volumetric expansion are some of the major thermo-physical properties of fluids including hybrid nanofluids. The density of hybrid nanofluid was pointed out by Babar and Ali [18] as one of the most important thermo-physical properties of hybrid nanofluid as it strongly affects stability, frictional factor, and Reynolds number. However, as in the case of GNP-Ag/water hybrid nanofluid, density negligibly augments with higher particle concentration but it reduces due to higher temperature; see Yarmand et al [19]. At different levels of volume concentrations, Suresh et al [20] noticed that the viscosity of ${{Al}}_{2}{0}_{3}-{Cu}$/water increases due to higher volume concentrations and also Newtonian (i.e. linear relationship between the shear stress and shear strain exists). The thermal conductivity of copper Cu is 401(W/mK) while the that of alumina ${{Al}}_{2}{0}_{3}$ is $40(W/{mK})$. Because of this, Suresh et al [20] noticed that the major influence in the thermal conductivity of the hybrid nanofluid can be traced to the 902.5% increase in the same thermo-physical property. The significance of increasing nanoparticles concentration and temperature on the mixture of ethylene glycol, of zinc oxide ZnO and titanium dioxide TiO₂ by Toghraie et al [21] shows that the thermal conductivity of the hybrid nanofluid κ_hnf increases with solid volume fraction and temperature for $25\,^\circ {\rm{C}}\leqslant T\leqslant 50\,^\circ {\rm{C}}$. Also, the thermal conductivity κ_hnf of the same mixture increases with temperature and solid volume fraction for 0.1% ≤ ϕ ≤ 3.5%.

The experimental research carried out by Suresh et al [22] indicates that the observed friction factor in the dynamics of ${{Al}}_{2}{O}_{3}-{Cu}$/water hybrid nanofluid is higher than that of 0.1% ${{Al}}_{2}{O}_{3}$/water nanofluid. This corroborates the observed friction factor of ${CNTs}-{{Al}}_{2}{O}_{3}$/water found to be higher due to the higher viscosity of hybrid nanofluid. In this case of hybrid nanofluid, surface adsorption is experienced and multiple types of nanoparticles in the base fluid form clustering. This leads to the higher viscosity of hybrid nanofluid; Nuim Labib et al [23]. Also remarked that the addition of CNTs to ${{Al}}_{2}{O}_{3}$/water increases the shear thinning of existing ${{Al}}_{2}{O}_{3}$/water nanofluid. In another study on ${{Al}}_{2}{O}_{3}-{Cu}$/water by Takabi and Shokouhmand [24], wall shear stress ratio and viscosity are increasing properties of volume concentration. Also, the friction factor decreases with Reynold number but found to be very large at all the values of the Reynold number for ${{Al}}_{2}{O}_{3}-{Cu}$/water. Moreover, Takabi and Shokouhmand [24] pointed out that the minimum friction factor is ascertained at all the Reynold numbers for pure water while the friction factor for ${{Al}}_{2}{O}_{3}$/water is greater than that of pure water.

1.1. Type II Model for the analysis of hybrid nanofluids

1.2. Type I model for hybrid nanofluids

Xu and Sun [32] noticed that for smaller case of nanoparticle volume fraction, the significance of the retained nonlinear terms after the expansion of the models strictly for density, viscosity, volumetric expansion, and heat capacity are insignificant; hence it can be removed. These models herein referred to as type I (q = 0) in this report does not contain such nonlinear terms after expansion. In fact, for $n-$ type of different nanoparticles, the following models are recommended

$$\begin{eqnarray*}\begin{array}{rcl}{\rho }_{{hnf}} & = & 1-{\rho }_{{bf}}\displaystyle \sum _{j=1}^{n}{\phi }_{j}+\displaystyle \sum _{j=1}^{n}{\phi }_{j}{\rho }_{{sj}},\\ {\left(\rho {c}_{p}\right)}_{{hnf}} & = & 1-{\left(\rho {c}_{p}\right)}_{{bf}}\displaystyle \sum _{j=1}^{n}{\phi }_{j}+\displaystyle \sum _{j=1}^{n}{\phi }_{j}{\left(\rho {c}_{p}\right)}_{{sj}},\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl}\displaystyle \frac{{\mu }_{{hnf}}}{{\mu }_{{bf}}} & = & \displaystyle \frac{1}{{\left(1-{\displaystyle \sum }_{j=1}^{n}{\phi }_{j}\right)}^{2.5}},\\ {\beta }_{{hnf}} & = & 1-{\beta }_{{bf}}\displaystyle \sum _{j=1}^{n}{\phi }_{j}+\displaystyle \sum _{j=1}^{n}{\phi }_{j}{\beta }_{{sj}}\end{array}\end{eqnarray*}$$

Babar and Ali [18] suggested that the models mentioned above (type 1) is suitable to model ternary-nanofluids (three different types of tiny particles). Type 1 model together with an extended Maxwell model for thermal conductivity was used by Kalidasan [33] to examine the mixture of colloidal mixture of copper and titania inside a C—shaped vacuum differentially heated. Inertia forced was noticed and it increases as the hot fluid moves towards the thermally active located at the left. Type I model was employed to examine unsteady copper, alumina, and water hybrid nanofluid along a vertical surface due to buoyancy forces and internal heat source. After using integral transforms method to solve the aforementioned problem by Saqib et al [34], it was discovered that temperature distribution increases while velocity decreases with higher volume of both nanoparticles. The aim of this study is to examine the significance of suction and dual stretching on the dynamics of various hybrid nanofluids (four base fluids and fourteen nanoparticles in the order of their densities) and establish a comparative analysis between type I and type II model of thermo-physical properties.

1.3. Research questions

The dynamics of hybrid nanofluids within the region where the effect of viscosity is significant was considered using the boundary layer argument by Prandtl [35] and Sakiadis [36]. It was assumed that dual linear stretching of the horizontal surface along x–direction and y–direction are induced with the velocities u_w(x) = ax and v_w(x) = by respectively. The switching of viscous term from type I to type II hybrid nanofluids for a better comparative analysis was modeled following Abegunrin et al [37] and Koriko et al [29]. Suitable governing equation for the dynamics is of the form

$$\begin{eqnarray}&&\displaystyle \frac{\partial u}{\partial x}+\displaystyle \frac{\partial v}{\partial y}+\displaystyle \frac{\partial w}{\partial z}=0,\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\begin{array}{l}u\displaystyle \frac{\partial u}{\partial x}+v\displaystyle \frac{\partial u}{\partial y}+w\displaystyle \frac{\partial u}{\partial z}\\ \quad =\,(1-q)\displaystyle \frac{{\mu }_{{hnfI}}}{{\rho }_{{nfI}}}\displaystyle \frac{{\partial }^{2}u}{\partial {z}^{2}}+q\displaystyle \frac{{\mu }_{{hnfII}}}{{\rho }_{{hnfII}}}\displaystyle \frac{{\partial }^{2}u}{\partial {z}^{2}},\end{array}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\begin{array}{l}u\displaystyle \frac{\partial v}{\partial x}+v\displaystyle \frac{\partial v}{\partial y}+w\displaystyle \frac{\partial v}{\partial z}\\ \quad =\,(1-q)\displaystyle \frac{{\mu }_{{hnfI}}}{{\rho }_{{nfI}}}\displaystyle \frac{{\partial }^{2}v}{\partial {z}^{2}}+q\displaystyle \frac{{\mu }_{{hnfII}}}{{\rho }_{{hnfII}}}\displaystyle \frac{{\partial }^{2}v}{\partial {z}^{2}}.\end{array}\end{eqnarray} \tag{ 3 }$$

Maintenance of airway and improvement of oxygenation in patients are achievable through oral suctioning (Propp and Gillis, [38]). In fact, less invisible separation that do occur between the boundary layer can be eliminated by either non-uniform or uniform suction but non-uniform suction is more efficient; Ramsay et al [39]. In fluid dynamics, most especially analysis of boundary layer flows, this has led to the modeling of suction at the wall as v(x = 0, y) = v_w where v_w is the suction velocity. In this study, the exertion of force on the hybrid nanofluid as it flows along x–and y–directions in order to remove empty spaces at the wall was modeled. Figure 1 shows the expressions used to account for the unnoticed relationship between suction and unequal dual stretching. In the presence of suction, the boundary conditions at z = 0, y = 0, and along x–direction

$$\begin{eqnarray}\begin{array}{rcl}u & = & {u}_{w}(x)={ax},\\ w & = & {w}_{w}\left(=-\sqrt{a{\vartheta }_{{bf}}}\displaystyle \int \displaystyle \frac{v}{{ay}}d\eta +{z}_{w}\right),\\ {\rm{at}}\,\,z & = & 0,\,{and}\,\,y=0\end{array}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&u\to 0,\,\,{\rm{as}}\,\,z\to \infty \ \ \ {and}\ \ \ y=0.\end{eqnarray} \tag{ 5 }$$

Also, in the presence of suction, the boundary conditions at z = 0, x = 0, and along y–direction

$$\begin{eqnarray}\begin{array}{rcl}v & = & {v}_{w}(x)={by},\,w={w}_{w}\left(=-\sqrt{a{\vartheta }_{{bf}}}\displaystyle \int \displaystyle \frac{u}{{ax}}d\eta +{z}_{w}\right),\\ {\rm{at}}\,\,z & = & 0,\,{and}\,\,x=0\end{array}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&v\to 0,\,\,{\rm{as}}\,\,z\to \infty \ \ \ {and}\ \ \ x=0.\end{eqnarray} \tag{ 7 }$$

Using conservation of energy, the energy equation for the flow along x–direction and y–direction is of the form

$$\begin{eqnarray}&&\begin{array}{l}u\displaystyle \frac{\partial T}{\partial x}+v\displaystyle \frac{\partial T}{\partial y}+w\displaystyle \frac{\partial T}{\partial z}\\ \quad =\,(1-q)\displaystyle \frac{{\kappa }_{{hnf}}}{{\left(\rho {c}_{p}\right)}_{{hnfI}}}\displaystyle \frac{{\partial }^{2}T}{\partial {z}^{2}}+q\displaystyle \frac{{\kappa }_{{hnf}}}{{\left(\rho {c}_{p}\right)}_{{hnfII}}}\displaystyle \frac{{\partial }^{2}T}{\partial {z}^{2}}.\end{array}\end{eqnarray} \tag{ 8 }$$

The boundary condition is of the form

$$\begin{eqnarray}&&T={T}_{w},\,\,{\rm{at}}\,\,z=0\ \ {as}\ \ (x,y)\to \infty \end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&T\to {T}_{\infty },\,\,{\rm{as}}\,\,z\to \infty ,\ \ {and}\ \ (x,y)\to \infty .\end{eqnarray} \tag{ 10 }$$

The unit of suction velocity z_w is ${m}^{2}{s}^{-2}$. The densities of various nanoparticles considered are presented as table 1. Following Wakif et al [40] the viscosity of water at 25 oc is $0.0008905$ Pa-s. Bergman et al [41] noted that the viscosity of ethylene glycol is $0.0157$ Pa-s. The viscosity of Methyl Alcohol (methanol) at 20^oc according to Shell Chemicals [42] is $0.59{mPa} \mbox{-} {\rm{s}}$ = 0.00059 Pa-s. The viscosity of blood according to Jahangiri et al [43] is $0.0033$ Pa-s. The physical quantities of interest for the transport phenomenon are the local skin friction coefficient and Nusselt number defined as

$$\begin{eqnarray*}\begin{array}{rcl}{{Nu}}_{x} & = & \displaystyle \frac{-x{\kappa }_{{hnf}}}{{\kappa }_{{bf}}({T}_{w}-{T}_{\infty })}\displaystyle \frac{\partial T}{\partial z},\\ {{Nu}}_{y} & = & \displaystyle \frac{-y{\kappa }_{{hnf}}}{{\kappa }_{{bf}}({T}_{w}-{T}_{\infty })}\displaystyle \frac{\partial T}{\partial z}\end{array}\end{eqnarray*}$$

$$\begin{eqnarray}\begin{array}{rcl}{c}_{f} & = & (1-q)\displaystyle \frac{{\mu }_{{hnfI}}}{{\rho }_{{bf}}{a}^{2}{x}^{2}}\displaystyle \frac{\partial u}{\partial z}+q\displaystyle \frac{{\mu }_{{hnfII}}}{{\rho }_{{bf}}{a}^{2}{x}^{2}}\displaystyle \frac{\partial u}{\partial z},\\ {c}_{g} & = & (1-q)\displaystyle \frac{{\mu }_{{hnfI}}}{{\rho }_{{bf}}{a}^{2}{y}^{2}}\displaystyle \frac{\partial v}{\partial z}+q\displaystyle \frac{{\mu }_{{hnfII}}}{{\rho }_{{bf}}{a}^{2}{y}^{2}}\displaystyle \frac{\partial v}{\partial z}\end{array}\end{eqnarray} \tag{ 11 }$$

Viscosity of a nanofluid at different levels of nanoparticle volume fraction ϕ by Hemmat Esfe et al [44] shows that the dynamic viscosity ratio increases with ϕ for 0 ≤ ϕ ≤ 0.02 at the rate of 19.98. In the same table, it is shown that the dynamic viscosity ratio of nanofluid presented by Brinkman [45] increases with ϕ at the rate of 2.6. The viscosity of nanofluid as suggested by Brinkman [45] and used by Dogonchi et al [46] is

$$\begin{eqnarray}&&\displaystyle \frac{{\mu }_{{nf}}}{{\mu }_{{bf}}}=\displaystyle \frac{1}{{\left(1-{\phi }_{1}\right)}^{2.5}}.\end{eqnarray} \tag{ 12 }$$

Following Kalidasan et al [33], the viscosity of hybrid nanofluid without the nonlinear terms is of the form

$$\begin{eqnarray}&&\displaystyle \frac{{\mu }_{{hnfI}}}{{\mu }_{{bf}}}=\displaystyle \frac{1}{{\left(1-{\phi }_{1}-{\phi }_{2}\right)}^{2.5}}.\end{eqnarray} \tag{ 13 }$$

The viscosity of hybrid nanofluid with nonlinear terms as presented by Devi & Devi [25, 47]

$$\begin{eqnarray}&&\displaystyle \frac{{\mu }_{{hnfII}}}{{\mu }_{{bf}}}=\displaystyle \frac{1}{{\left(1-{\phi }_{1}\right)}^{2.5}{\left(1-{\phi }_{2}\right)}^{2.5}}\end{eqnarray} \tag{ 14 }$$

Following Pak and Cho [48] and Das et al [49], the density of nanofluid is of the form

$$\begin{eqnarray}&&{\rho }_{{nf}}=(1-\phi ){\rho }_{{bf}}+\phi {\rho }_{{sp}}.\end{eqnarray} \tag{ 15 }$$

Extension of the model equation (12) to that of hybrid's density was presented by Takabi and Salehi [50], Ho et al [51], and Ahammed et al [52] as

$$\begin{eqnarray}&&{\rho }_{{hnfI}}=(1-{\phi }_{1}-{\phi }_{2}){\rho }_{{bf}}+{\phi }_{1}{\rho }_{{{sp}}_{1}}+{\phi }_{2}{\rho }_{{{sp}}_{2}}\end{eqnarray} \tag{ 16 }$$

The density of the hybrid nanofluid without nonlinear terms sequel to Devi & Devi [25, 47] is

$$\begin{eqnarray}&&{\rho }_{{hnfII}}=(1-{\phi }_{2})[(1-{\phi }_{1}){\rho }_{{bf}}+{\phi }_{1}{\rho }_{{{sp}}_{1}}]+{\phi }_{2}{\rho }_{{{sp}}_{2}}.\end{eqnarray} \tag{ 17 }$$

The specific heat capacity of nanofluid as prescribed by Xuan and Roetzel [53] is

$$\begin{eqnarray}&&{\left(\rho {c}_{p}\right)}_{{nf}}=(1-\phi ){\left(\rho {c}_{p}\right)}_{{bf}}+\phi {\left(\rho {c}_{p}\right)}_{{sp}}.\end{eqnarray} \tag{ 18 }$$

Kalidasan et al [33] and and Ahammed et al [52] suggested the heat capacity of hybrid nanofluid to be of

$$\begin{eqnarray}&&\begin{array}{l}{\left(\rho {c}_{p}\right)}_{{hnfI}}=(1-{\phi }_{1}-{\phi }_{2}){\left(\rho {c}_{p}\right)}_{{bf}}\\ \quad +\,{\phi }_{1}{\left(\rho {c}_{p}\right)}_{{{sp}}_{1}}+{\phi }_{2}{\left(\rho {c}_{p}\right)}_{{{sp}}_{2}}\end{array}\end{eqnarray} \tag{ 19 }$$

Specific heat capacity by Devi & Devi [25, 47] for type II hybrid nanofluid is

$$\begin{eqnarray}&&\begin{array}{l}{\left(\rho {c}_{p}\right)}_{{hnfII}}=(1-{\phi }_{2})[(1-{\phi }_{1}){\left(\rho {c}_{p}\right)}_{{bf}}\\ \,+\,{\phi }_{1}{\left(\rho {c}_{p}\right)}_{{{sp}}_{1}}]+{\phi }_{2}{\left(\rho {c}_{p}\right)}_{{{sp}}_{2}}\end{array}\end{eqnarray} \tag{ 20 }$$

The benchmark released by Buongiorno et al [54] reveals that the thermal conductivity of nanofluids varies with particle aspect ratio, particle loading, and particle concentration. In the same direction, the model proposed by Maxwell [55], Nadeem et al [56], Hayat et al [57], Seyyed et al [58], and Devi & Devi [25, 47] was adopted to incorporate the enhancement in the thermal conductivity of Cu-Water nanofluid with volume fraction for type I and II hybrid nanofluids as

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\kappa }_{{hnf}}}{{\kappa }_{{bf}}} & = & \displaystyle \frac{{\kappa }_{{{sp}}_{2}}+2{\kappa }_{{nf}}-2{\phi }_{2}({\kappa }_{{nf}}-{\kappa }_{{{sp}}_{2}})}{{\kappa }_{{{sp}}_{2}}+2{\kappa }_{{nf}}+{\phi }_{2}({\kappa }_{{nf}}-{\kappa }_{{{sp}}_{2}})},\\ {where}\ {\kappa }_{{nf}} & = & {\kappa }_{f}\left[\displaystyle \frac{{\kappa }_{{{sp}}_{1}}+2{\kappa }_{f}-2{\phi }_{1}({\kappa }_{f}-{\kappa }_{{{sp}}_{1}})}{{\kappa }_{{{sp}}_{1}}+2{\kappa }_{f}+{\phi }_{1}({\kappa }_{f}-{\kappa }_{{{sp}}_{1}})}\right].\end{array}\end{eqnarray} \tag{ 21 }$$

Dimensionless governing equation for the similar flow was adopted using the variables

$$\begin{eqnarray*}\begin{array}{rcl}u & = & {ax}\displaystyle \frac{{df}}{d\eta },\,v={by}\displaystyle \frac{{dg}}{d\eta },\,w=-{\left(a{\vartheta }_{{bf}}\right)}^{\tfrac{1}{2}}[f(\eta )+{cg}(\eta )],\\ \eta & = & z{\left(\displaystyle \frac{a}{{\vartheta }_{{bf}}}\right)}^{\tfrac{1}{2}},\,c=\displaystyle \frac{b}{a},\ \ \ {f}_{w}=\displaystyle \frac{-{v}_{w}}{\sqrt{a{\vartheta }_{{bf}}}},\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl}\theta & = & \displaystyle \frac{T-{T}_{\infty }}{{T}_{w}-{T}_{\infty }},\ \ \ {A}_{1}=1-{\phi }_{1}-{\phi }_{2}+{\phi }_{1}\displaystyle \frac{{\rho }_{{{sp}}_{1}}}{{\rho }_{{bf}}}+{\phi }_{2}\displaystyle \frac{{\rho }_{{{sp}}_{2}}}{{\rho }_{{bf}}},\\ {A}_{2} & = & {\left(1-{\phi }_{1}-{\phi }_{2}\right)}^{-2.5}\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl}{A}_{3} & = & (1-{\phi }_{2})\left[1-{\phi }_{1}+{\phi }_{1}\displaystyle \frac{{\rho }_{{{sp}}_{1}}}{{\rho }_{{bf}}}\right]+{\phi }_{2}\displaystyle \frac{{\rho }_{{{sp}}_{2}}}{{\rho }_{{bf}}},\\ {A}_{4} & = & {\left(1-{\phi }_{1}\right)}^{-2.5}{\left(1-{\phi }_{2}\right)}^{-2.5}\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl}{A}_{5} & = & \displaystyle \frac{{\kappa }_{{{sp}}_{2}}+2{\kappa }_{{nf}}-2{\phi }_{2}({\kappa }_{{nf}}-{\kappa }_{{{sp}}_{2}})}{{\kappa }_{{{sp}}_{2}}+2{\kappa }_{{nf}}+{\phi }_{2}({\kappa }_{{nf}}-{\kappa }_{{{sp}}_{2}})},\\ {where}\ \ {\kappa }_{{nf}} & = & {\kappa }_{f}\left[\displaystyle \frac{{\kappa }_{{{sp}}_{1}}+2{\kappa }_{f}-2{\phi }_{1}({\kappa }_{f}-{\kappa }_{{{sp}}_{1}})}{{\kappa }_{{{sp}}_{1}}+2{\kappa }_{f}+{\phi }_{1}({\kappa }_{f}-{\kappa }_{{{sp}}_{1}})}\right],\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}&&{A}_{7}=(1-{\phi }_{2})\left[(1-{\phi }_{1})+{\phi }_{1}\displaystyle \frac{{\left(\rho {c}_{p}\right)}_{{{sp}}_{1}}}{{\left(\rho {c}_{p}\right)}_{{bf}}}\right]+{\phi }_{2}\displaystyle \frac{{\left(\rho {c}_{p}\right)}_{{{sp}}_{2}}}{{\left(\rho {c}_{p}\right)}_{{bf}}}.\end{eqnarray*}$$

$$\begin{eqnarray}&&{A}_{8}=1-{\phi }_{1}-{\phi }_{2}+{\phi }_{1}\displaystyle \frac{{\left(\rho {c}_{p}\right)}_{{sp}1}}{{\left(\rho {c}_{p}\right)}_{{bf}}}+{\phi }_{2}\displaystyle \frac{{\left(\rho {c}_{p}\right)}_{{sp}2}}{{\left(\rho {c}_{p}\right)}_{{bf}}}.\end{eqnarray} \tag{ 22 }$$

Where c is the stretching ratio and f_w suction parameter. Substituting equation (22) into equations (1)–(10). Equation (1) is satisfied and others dimensionless equations are

$$\begin{eqnarray}&&(1-q)\displaystyle \frac{{A}_{2}}{{A}_{1}}\displaystyle \frac{{d}^{3}f}{d{\eta }^{3}}+q\displaystyle \frac{{A}_{4}}{{A}_{3}}\displaystyle \frac{{d}^{3}f}{d{\eta }^{3}}-\displaystyle \frac{{df}}{d\eta }\displaystyle \frac{{df}}{d\eta }+[f+{cg}]\displaystyle \frac{{d}^{2}f}{d{\eta }^{2}}=0,\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&(1-q)\displaystyle \frac{{A}_{2}}{{A}_{1}}\displaystyle \frac{{d}^{3}g}{d{\eta }^{3}}+q\displaystyle \frac{{A}_{4}}{{A}_{3}}\displaystyle \frac{{d}^{3}g}{d{\eta }^{3}}-c\displaystyle \frac{{dg}}{d\eta }\displaystyle \frac{{dg}}{d\eta }+[f+{cg}]\displaystyle \frac{{d}^{2}g}{d{\eta }^{2}}=0,\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&(1-q)\displaystyle \frac{{A}_{5}}{{A}_{8}}\displaystyle \frac{{d}^{2}\theta }{d{\eta }^{2}}+q\displaystyle \frac{{A}_{5}}{{A}_{7}}\displaystyle \frac{{d}^{2}\theta }{d{\eta }^{2}}+{P}_{r}f\displaystyle \frac{d\theta }{d\eta }+{{cP}}_{r}g\displaystyle \frac{d\theta }{d\eta }=0.\end{eqnarray} \tag{ 25 }$$

Subject to the dimensionless boundary conditions

$$\begin{eqnarray}&&\displaystyle \frac{{df}}{d\eta }=1,\,\,f={f}_{w},\,\,\displaystyle \frac{{dg}}{d\eta }=c,\,\,g=\displaystyle \frac{{f}_{w}}{c},\,\,\theta =1\,\,{\rm{at}}\,\,\eta =0,\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{df}}{d\eta }\to 0,\,\,\displaystyle \frac{{dg}}{d\eta }\to 0,\,\,\theta \to 0\,\,{\rm{as}}\,\,\eta \to \infty .\end{eqnarray} \tag{ 27 }$$

Using the transformation variable in equation (22), $\sqrt{{{Re}}_{x}}=\tfrac{{a}^{1/2}x}{{\vartheta }_{{bf}}^{1/2}}$ and $\sqrt{{{Re}}_{y}}=\tfrac{{a}^{1/2}y}{{\vartheta }_{{bf}}^{1/2}}$ to obtain the following dimensionless physical quantities for type I and type II hybrid nanofluid

$$\begin{eqnarray*}&&\displaystyle \frac{{{Nu}}_{x}}{\sqrt{{{Re}}_{x}}}=\displaystyle \frac{{{Nu}}_{y}}{\sqrt{{{Re}}_{y}}}=-\theta ^{\prime} (0),\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl}f^{\prime\prime} (0) & = & (1-q){\left(1-{\phi }_{1}-{\phi }_{2}\right)}^{2.5}{c}_{f}\sqrt{{{Re}}_{x}}\\ & & +q{\left(1-{\phi }_{1}\right)}^{2.5}{\left(1-{\phi }_{2}\right)}^{2.5}{c}_{f}\sqrt{{{Re}}_{x}},\end{array}\end{eqnarray*}$$

$$\begin{eqnarray}\begin{array}{rcl}g^{\prime\prime} (0) & = & (1-q){\left(1-{\phi }_{1}-{\phi }_{2}\right)}^{2.5}\displaystyle \frac{{c}_{g}\sqrt{{{Re}}_{y}}}{c}\\ & & +q{\left(1-{\phi }_{1}\right)}^{2.5}{\left(1-{\phi }_{2}\right)}^{2.5}\displaystyle \frac{{c}_{g}\sqrt{{{Re}}_{y}}}{c}.\end{array}\end{eqnarray} \tag{ 28 }$$

The numerical solutions of the dimensionless governing equation equations (23)–(27) were obtained using bvp4c package in MATLAB and shooting techniques along with 4th order integration scheme ShT4RK.

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Table 1.  Density, thermal conductivity, and specific heat capacity of some base fluids and nanoparticles

Source	Base fluid	ρ (kg/m−3)	κ (W/mK)	cp (J/kgK)	${P}_{r}=\tfrac{\mu {c}_{p}}{\kappa }$
Tlili et al [26]	Methanol	792	0.2035	2,545	7.3786
Animasaun et al [59]	Water ${H}_{2}O$	997.1	0.613	4,180	6.1723
Koriko et al [60]	Blood	1,050	0.52	3,617	22.9540
Nayak et al [61]	Ethylene Glycol ${C}_{2}{H}_{6}{O}_{2}$	1,114.4	0.252	2,415	150.46

Source	Nomenclature of Nano-particles	ρ (kg/m−3)	κ (W/mK)	cp (J/kgK)
Abbas et al [28]	Multiple wall CNT	1,600	3000	796
Selimefendigil et al [62]	Silicon dioxide SiO2	2,200	1.2	703
Mahanthesh et al [63]	Single wall CNT SWCNT	2,600	6600	425
Nayak et al [61]	Aluminium Al	2,702	237	903
Tlili et al [26]	Magnesium oxide MgO	3,580	48.4	960
Ref. [64] & Ref. [65]	Aluminium Oxide ${{Al}}_{2}{O}_{3}$	$3,970$	40	765
Acharya et al [66]	Titanium dioxide TiO2	4,250	8.954	686.2
Sandeep et al [67]	Iron(iii)oxide (magnetite ${{Fe}}_{3}{O}_{4}$)	5,180	9.7	670
Nayak [61]	Zinc oxide ZnO	5,600	13	495.2
Acharya et al [66]	Copper(II)oxide CuO	6,320	76.5	531.80
Nayak et al [61]	Zinc Zn	7,140	116	389
Xu and Chen [68]	Copper Cu	8,933	400	385
Hayat et al [57]	Silver Ag	10,500	429	235
Koriko et al [60]	Gold Au	19,300	318	129

In order to show that the new results are reliable, the solutions of equations (23)–(27) were obtained for a limiting case as shown in tables 2 and 3. It is worth remarking that the solutions of both methods are accurate as it yields almost equal results. Based on the level of the numerical accuracy, it is valid to conclude that the results are reliable and valid. The obtained values for type I model (q = 0) and type II model (q = 1) are approximately equivalent for ϕ₁ + ϕ₂ = 0.02. The insignificant difference in the velocities using both models can be associated to the fact that an increases in $\tfrac{{\mu }_{{hnfI}}}{{\mu }_{{bf}}}$ with (ϕ₁, ϕ₂) in equation (13) and $\tfrac{{\mu }_{{hnfII}}}{{\mu }_{{bf}}}$ with (ϕ₁, ϕ₂) in equation (14) for 0 ≤ (ϕ₁, ϕ₂) ≤ 0.02 are equal for ϕ₁ = ϕ₂ = 0.01; see table 4. Based on the analysis presented in table 4, it is seen that the percentage difference between $\tfrac{{\mu }_{{hnfI}}}{{\mu }_{{bf}}}$ equation (13) and $\tfrac{{\mu }_{{hnfII}}}{{\mu }_{{bf}}}$ equation (14) at ϕ₁ = ϕ₂ = 0.3 was estimated as −39.79122679%. This percentage decreases is too large. In fact, the outcome of the experimental research on Ag − MgO/water mixture by Hemmat Esfe et al [44] also suggests that the nanoparticle volume fraction of viscosity model related to that of Brinkman model should be strictly satisfies ϕ ≤ 0.02. Considering the fact that changes in the viscosity of hybrid nanofluid is strongly influenced by both volume fraction and temperature, the relative viscosity model of hybrid nanofluid suggested by Amini et al [69] is recommended. There are different mode of stretching. A few are linear stretching, non-linear stretching, and dual stretching. Dual stretching is a case in which the stretching at the wall is in two different directions. As presented in equation (4)–(7), the first stretching is parallel to x–direction while the second stretching is parallel to y–direction. Table 3 shows that as the ratio of stretching rate c increases, the obtained results are reliable due to closeness in the solution of bvp4c and ShT4RK.

Table 2.  Validation and Reliability of Results for ϕ₁ = ϕ₂ = 0, P_r = 6.1723, ρ_bf = 997.1, ρ_sp1 = 1,600, ρ_sp2 = 2,200, κ_bf = 0.613, κ_sp1 = 3,000, κ_sp2 = 1.2, cp_bf = 4,180, cp_sp1 = 796, cp_sp2 = 703, f_w = 0.5, c = 0.5, and ${\eta }_{\infty }=100.$

	bvp4c	ShT4RK	bvp4c	ShT4RK
	$f^{\prime\prime} (0)$	$f^{\prime\prime} (0)$	$-\theta ^{\prime} (0)$	$-\theta ^{\prime} (0)$
q = 0	−1.67304252334	−1.67304960218	7.039 992 370 31	7.039 994 715 08
q = 1	−1.67304252334	−1.67304021793	7.039 992 370 31	7.039 999 628 37

Table 3.  Validation and Reliability of Results for ρ_sp1 = 1,600, ρ_sp2 = 2,200, ϕ₁ = ϕ₂ = 0.01, q = 1, P_r = 6.1723, ρ_bf = 997.1, ρ_sp1 = 1,600, ρ_sp2 = 2,200, κ_bf = 0.613, κ_sp1 = 3,000, κ_sp2 = 1.2, cp_bf = 4,180, cp_sp1 = 796, cp_sp2 = 703, f_w = 0.5, and ${\eta }_{\infty }=100$.

	bvp4c	ShT4RK	bvp4c	ShT4RK
c	$f^{\prime\prime} (0)$	$f^{\prime\prime} (0)$	$-\theta ^{\prime} (0)$	$-\theta ^{\prime} (0)$
0.1	−1.58987854780	−1.5898701479	6.900 643 063 5	6.900 696 031 2
0.5	−1.63542932561	−1.635429963201	7.043 168 497 99	7.043 169 614 75
1	−1.76141307300	−1.76141960108	7.440 041 259 83	7.440 049 630 18
1.5	−1.93858007280	−1.93858874931	7.999 807 937 29	7.999 809 687 41

Table 4.  Comparative analysis of the viscosity models between type I and type II.

ϕ1	ϕ2	$\tfrac{{\mu }_{{nf}}}{{\mu }_{{bf}}}$ equation (12)	$\tfrac{{\mu }_{{hnfI}}}{{\mu }_{{bf}}}$ equation (13)	$\tfrac{{\mu }_{{hnfII}}}{{\mu }_{{bf}}}$ equation (14)
0	0	1	1	1
0.005	0.005	1.012 610 201	1.025 444 154	1.025 379 419
0.01	0.01	1.025 444 154	1.051803982	1.051535713
0.015	0.015	1.038 506 989	1.079 122 293	1.078 496 767
0.02	0.02	1.051 803 982	1.107 444 364	1.106 291 617
0.025	0.025	1.065 340 557	1.136 818 119	1.134 950 503
0.03	0.03	1.079 122 293	1.167 294 303	1.164 504 922
0.035	0.035	1.093 154 927	1.198 926 691	1.194 987 695
0.04	0.04	1.107 444 364	1.231 772 295	1.226 433 02
0.1	0.1	1.301 348 831	1.746 928 107	1.693 508 781
0.2	0.2	1.746 928 107	3.586 095 691	3.051 757 813
0.3	0.3	2.439 242 06	9.882 117 688	5.949 901 827

When ϕ₁ = ϕ₂ = 0.01, with the aid of type II hybrid nanofluid models, an attempt had been made to determine the variations in the local skin friction coefficients and temperature a few distances away from the wall in the dynamics of SiO₂ and multi-walled carbon nanoparticles by (i) methanol, (ii) water, (iii) blood, and (iv) ethylene glycol. As seen in tables 5–8, local skin friction coefficient that is directly proportional to friction decreases as stretching related parameter c increases at different rates. The rate of decrease in ${(1-{\phi }_{1})}^{2.5}{\left(1-{\phi }_{2}\right)}^{2.5}{c}_{f}\sqrt{{{Re}}_{y}}$ with c is higher in all the four cases of hybrid nanofluids compared to the observed changes in ${(1-{\phi }_{1})}^{2.5}{\left(1-{\phi }_{2}\right)}^{2.5}{c}_{f}\sqrt{{{Re}}_{x}}$ with c. In all the four dynamics of hybrid nanofluids, it is worth concluding that the temperature distribution decreases in each transport phenomenon with higher c. This is not true in the case of ethylene glycol-based nanofluid where its density is $1,114.4\,{\mathrm{kgm}}^{-3}$ and P_r = 150.46. With a change in the base fluid from methanol to water, the density increases but the Prandtl number decreases. These changes are not capable to cause a significant variation in the local skin friction coefficients as shown in table 6. In the dynamics of blood conveying SiO₂ and multi-walled carbon nanoparticles, temperature distribution a few distances away from the wall is seen to depreciate with c at the rate of −0.00261707; see table 7. In the case of highly dense ethylene glycol-based hybrid nanofluid, the kinematic viscosity is approximately zero thus affected the velocity and temperature distribution as shown in table 8. Figure 2 shows that the horizontal velocity for flow along x–direction decreases with a higher stretching ratio c. However, the horizontal velocity for the dynamics along y–direction increases near the wall due to a higher stretching ratio c as shown in figure 3. Physically, the stretching rate 'a' which is a variable in the similarity variable $\eta =z{\left(\tfrac{a}{{\vartheta }_{{bf}}}\right)}^{\tfrac{1}{2}}$ is also a major variable in the stretching velocity u_w(x) along x–direction. A higher magnitude of stretching ratio c corresponds to a lower stretching rate a. This is the major reason why the horizontal velocity decreases with a higher stretching ratio in the flow along x–direction but increases in the flow along y–direction. Figure 4 reveals that the temperature distribution in the dynamics of hybrid nanofluid decreases with growth in stretching ratio c. In heat transfer, when velocity is increasing, higher heat energy is transported away from the domain. The observed results in figure 4 is justifiable because the observed increase in the flow along y–direction is not only substantial but also dominates the decrease in the flow along x–direction. This is also responsible for the variation in the temperature gradient proportional to the heat transfer rate as shown in figure 5. At different levels of stretching ratio, figures 6 and 7 present the variation in the vertical velocity of the flow along x–direction and y–direction. As stretching ratio increases within the domain 0.5 ≤ c ≤ 4.5, a turning point in the vertical velocity of the flow along x–direction is ascertained a few distances away from η = 0.6. In this case, as shown in figure 6, the maximum vertical velocity is observable at the extreme free stream when the stretching ratio is small in magnitude. The reverse is the pattern of changes in the vertical velocity of the flow along y–direction as stretching ratio strongly affects the suction in this case as shown in figure 7. At the wall η = 0, it is worthy that f = f_w and $g=\tfrac{{f}_{w}}{c}$. As methanol flows in x–direction and y–direction conveying multiple wall CNT and Silicon dioxide nanoparticles, it is shown in figures 8 and 9 that the vertical velocities (f(η) and g(η)) increase with suction f_w at different rates. The observed curve in the vertical velocity of the flows in y–direction is more pronounced. It is important to remark that the corresponding force due to suction is parallel to the vertical velocity. Each time the suction works, the pressure is lower and higher. Thus, it pushes the flow in an upward direction and reduces the flow in a forward direction as shown in figures 10 and 11. The observed decrease in the horizontal flow along x–direction and y–direction is associated with the fact that the corresponding force due to suction is perpendicular to the vertical velocity. The suction introduced in equations (4) and equation (6) above actually reduces pressure. Due to the nature of dual stretching in both horizontal directions ($x-,\,y-$), vertical velocity of the flow along ($x-,\,y-$) direction increases with suction but the horizontal velocity of the flow along ($x-,\,y-$) direction decreases with suction due to the inherent relationship between stretching ratio and suction. In a study by Stevens et al [70] it was remarked that a significant reduction in the flow of blood is achievable through higher suction. It is discovered that the local skin friction coefficients decrease with suction. For brevity, this is not shown as it can be easily deduced from tables 9–15. The temperature distribution is a decreasing property of suction due to enhancement in the Nusselt number due to higher suction as shown in from tables 9–15. However, the rate of decrease in $f^{\prime\prime} (0)$ and $g^{\prime\prime} (0)$ is optimal in the case of methanol conveying highly dense nanoparticles like silver and gold. Using the slope of linear regression through the data point suggested by Shah et al [71], Animasaun et al [72, 73], and Wakif et al [74].

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Table 5.  Variation in the local skin friction coefficients and temperature a few distance away from the wall in the dynamics of SiO₂ and multi walled carbon nanoparticles by methanol when f_w = 0.5 and P_r = 7.3786.

c	$f^{\prime\prime} (0)$	$g^{\prime\prime} (0)$	$\theta (0.2)$
0.5	−1.641141403270246	−0.721238901159391	0.149 442 725 680 186
1.5	−1.950050209965838	−3.271585190865442	0.097 744 783 400 392
2.5	−2.391940396308734	−7.727602127704166	0.063 814 375 318 287
Slp	−0.375399497	−3.503181613	−0.042814175

Table 6.  Variation in the local skin friction coefficients and temperature a few distance away from the wall in the dynamics of SiO₂ and multi walled carbon nanoparticles by water when f_w = 0.5 and P_r = 6.0723.

c	$f^{\prime\prime} (0)$	$g^{\prime\prime} (0)$	θ(0.2)
0.5	−1.629941698439091	−0.715958722254514	0.240 581 727 059 485
1.5	−1.937445601340791	−3.251333623983448	0.142 126 903 412 985
2.5	−2.377269739263540	−7.683598714554352	0.089 527 594 457 403
Slp	−0.37366402	−3.483819996	−0.075527066

Table 7.  Variation in the local skin friction coefficients and temperature a few distance away from the wall in the dynamics of SiO₂ and multi walled carbon nanoparticles by blood when f_w = 0.5 and P_r = 22.9540.

c	$f^{\prime\prime} (0)$	$g^{\prime\prime} (0)$	θ(0.2)
0.5	−1.633276905648857	−0.718848083862862	0.005 562 433 170 759
1.5	−1.936133294049687	−3.248644262172203	0.001 800 749 355 756
2.5	−2.374540457775160	−7.675184714776091	0.000 328 292 505 257
Slp	−0.370631776	−3.478168315	−0.00261707

Table 8.  Variation in the local skin friction coefficients and temperature a few distance away from the wall in the dynamics of SiO₂ and multi walled carbon nanoparticles by ethylene glycol when f_w = 0.5 and P_r = 150.46.

c	$f^{\prime\prime} (0)$	$g^{\prime\prime} (0)$	θ(0.2)
0.5	−1.6309318322029	−0.71787509361440	0.0000
1.5	−1.9335665611968	−3.24443539626813	0.0000
2.5	−2.3714283543743	−7.6658377187914	0.0000
Slp	−0.370248261	−3.473981313	0

Table 9.  Base fluid-Water, n_p 1—Multiple wall CNT, n_p 2-Silicon dioxide: Local skin friction coefficients and Nusselt number of various hybrid nanofluids for ϕ₁ = ϕ₂ = 0.01, q = 1, P_r = 6.1723 and ${\eta }_{\infty }=100$.

		fw = 0.5			fw = 3
c	$f^{\prime\prime} (0)$	$g^{\prime\prime} (0)$	$-\theta ^{\prime} (0)$	$f^{\prime\prime} (0)$	$g^{\prime\prime} (0)$	$-\theta ^{\prime} (0)$
1	−1.7704046	−1.7704046	7.438 845 6	−6.1041835	−6.1041835	37.484 387 5
2	−2.1609375	−5.2546902	8.657 769 4	−6.3186562	−13.0725849	37.891 629 7
3	−2.6364227	−10.6929008	10.144 171 6	−6.6357148	−21.4790001	38.531 592 6
Slp	−0.43300905	−4.4612481	1.352 663	−0.26576565	−7.6874083	0.523 602 55

Table 10.  Base fluid -Water, n_p 1-Single wall CNT SWCNT, n_p 2-Aluminium: Local skin friction coefficients and Nusselt number of various hybrid nanofluids for ϕ₁ = ϕ₂ = 0.01, q = 1, P_r = 6.1723 and ${\eta }_{\infty }=100$.

		fw = 0.5			fw = 3
c	$f^{\prime\prime} (0)$	$g^{\prime\prime} (0)$	$-\theta ^{\prime} (0)$	$f^{\prime\prime} (0)$	$g^{\prime\prime} (0)$	$-\theta ^{\prime} (0)$
1	−1.7929205	−1.7929205	7.436 777 5	−6.2118529	−6.2118529	37.484 012 9
2	−2.1868946	−5.3138042	8.654 063 9	−6.4268365	−13.2898686	37.890 341 9
3	−2.6666298	−10.8061843	10.138 576 1	−6.7450933	−21.8123067	38.528 978 7
Slp	−0.43685465	−4.5066319	1.350 899 3	−0.2666202	−7.8002269	0.522 482 9

Table 11.  Base fluid -Water, n_p 1-Magnesium oxide, n_p 2-Aluminium Oxide: Local skin friction coefficients and Nusselt number of various hybrid nanofluids for ϕ₁ = ϕ₂ = 0.01, q = 1, P_r = 6.1723 and ${\eta }_{\infty }=100$.

		fw = 0.5			fw = 3
c	$f^{\prime\prime} (0)$	$g^{\prime\prime} (0)$	$-\theta ^{\prime} (0)$	$f^{\prime\prime} (0)$	$g^{\prime\prime} (0)$	$-\theta ^{\prime} (0)$
1	−1.8265552	−1.8265552	7.432 859 1	−6.3733607	−6.3733607	37.478 714 8
2	−2.2256234	−5.4018988	8.647 644 5	−6.5890835	−13.6156985	37.883 672 2
3	−2.7116561	−10.9747892	10.129 258 8	−6.9090833	−22.3117962	38.520 313 9
Slp	−0.44255045	−4.574117	1.348 199 85	−0.2678613	−7.96921775	0.520 799 55

Table 12.  Base fluid -Water, n_p 1-Titanium dioxide, n_p 2-Iron(iii)oxide: Local skin friction coefficients and Nusselt number of various hybrid nanofluids for ϕ₁ = ϕ₂ = 0.01, q = 1, P_r = 6.1723 and ${\eta }_{\infty }=100$.

		fw = 0.5			fw = 3
c	$f^{\prime\prime} (0)$	$g^{\prime\prime} (0)$	$-\theta ^{\prime} (0)$	$f^{\prime\prime} (0)$	$g^{\prime\prime} (0)$	$-\theta ^{\prime} (0)$
1	−1.8545799	−1.8545799	7.430 398 6	−6.5085280	−6.5085280	37.478 818 5
2	−2.2578510	−5.4751105	8.643 176 5	−6.7248458	−13.8882984	37.882 643 6
3	−2.7490849	−11.1147152	10.122 480 8	−7.0462564	−22.7294002	38.517 642 6
Slp	−0.4472525	−4.63006765	1.346 041 1	−0.2688642	−8.1104361	0.519 412 05

Table 13.  Base fluid -Water, n_p 1-Zinc oxide, n_p 2-Copper(II)oxide: Local skin friction coefficients and Nusselt number of various hybrid nanofluids for ϕ₁ = ϕ₂ = 0.01, q = 1, P_r = 6.1723 and ${\eta }_{\infty }=100$.

		fw = 0.5			fw = 3
c	$f^{\prime\prime} (0)$	$g^{\prime\prime} (0)$	$-\theta ^{\prime} (0)$	$f^{\prime\prime} (0)$	$g^{\prime\prime} (0)$	$-\theta ^{\prime} (0)$
1	−1.8914593	−1.8914593	7.426 995 9	−6.6872068	−6.6872068	37.477 968 0
2	−2.3002045	−5.5711970	8.637 133 6	−6.9042799	−14.2485322	37.880 303 9
3	−2.7982204	−11.2980959	10.113 400 7	−7.2274920	−23.2808748	38.513 141 6
Slp	−0.45338055	−4.7033183	1.343 202 4	−0.2701426	−8.296834	0.517 586 8

Table 14.  Base fluid -Water, n_p 1-Zinc, n_p 2-Copper: Local skin friction coefficients and Nusselt number of various hybrid nanofluids for ϕ₁ = ϕ₂ = 0.01, q = 1, P_r = 6.1723 and ${\eta }_{\infty }=100$.

		fw = 0.5			fw = 3
c	$f^{\prime\prime} (0)$	$g^{\prime\prime} (0)$	$-\theta ^{\prime} (0)$	$f^{\prime\prime} (0)$	$g^{\prime\prime} (0)$	$-\theta ^{\prime} (0)$
1	−1.9526683	−1.9526683	7.421 295 8	−6.9857050	−6.9857050	37.476 124 4
2	−2.3703598	−5.7300434	8.627 084 1	−7.2039667	−14.8500554	37.875 995 2
3	−2.8794785	−11.6006046	10.098 355 2	−7.5300380	−24.2008349	38.505 255 9
Slp	−0.4634051	−4.82396815	1.338 529 7	−0.2721665	−8.60756495	0.514 565 75

Table 15.  Base fluid -Water, n_p 1-Silver, n_p 2-Gold: Local skin friction coefficients and Nusselt number of various hybrid nanofluids for ϕ₁ = ϕ₂ = 0.01, q = 1, P_r = 6.1723 and ${\eta }_{\infty }=100$.

		fw = 0.5			fw = 3
c	$f^{\prime\prime} (0)$	$g^{\prime\prime} (0)$	$-\theta ^{\prime} (0)$	$f^{\prime\prime} (0)$	$g^{\prime\prime} (0)$	$-\theta ^{\prime} (0)$
1	−2.1520237	−2.1520237	7.403 053 0	−7.9733768	−7.9733768	37.470 773 9
2	−2.5977109	−6.2422743	8.594 998 2	−8.1950224	−16.8382930	37.862 692 7
3	−3.1417367	−12.5707751	10.050 447 4	−8.5294019	−27.2346992	38.480 418 5
Slp	−0.4948565	−5.2093757	1.323 697 2	−0.27801255	−9.6306612	0.504 822 3

In this report, the significance of suction and dual stretching on the dynamics of various hybrid nanofluids (four base fluids and fourteen nanoparticles) had been investigated using type I and type II models of thermophysical properties. Based on the observed analysis of results, it is worth concluding that

• 1.  
  type I and type II models of viscosity should not be considered for the modeling of hybrid nanofluid for volume fraction ϕ₁ + ϕ₂ > 0.02 for there exists a significant difference between $\tfrac{{\mu }_{{hnfI}}}{{\mu }_{{bf}}}$ in equation (13) and $\tfrac{{\mu }_{{hnfII}}}{{\mu }_{{bf}}}$ equation (14) as presented in table 4.
• 2.  
  combine nature of dual stretching and suction is responsible for the major reason why the vertical velocity of the flow along ($x-,\,y-$) direction increases with suction but the horizontal velocity of the flow along ($x-,\,y-$) direction decreases with suction.
• 3.  
  local skin friction coefficients and temperature distribution are decreasing properties of suction. However, optimal Nusselt number is ascertained at a larger value of stretching ratio and suction in the dynamics of water conveying (less dense nanoparticles) Multiple wall CNT and silicon dioxide.