Magnetohydrodynamic Hybrid Nanofluid Flow Past an Exponentially Stretching Sheet with Slip Conditions

Abstract

This study presents the magnetized hybrid nanofluid flow with heat source/sink over an exponentially stretching/shrinking sheet. Slip conditions are implemented to analyze the hybrid nanofluid flow for both slip and no-slip conditions. Additionally, the hybrid nanofluid of alumina and copper (hybrid nanoparticles) with blood (base fluid) has been considered and discussed with both suction and injection parameters. The appropriate similarity variables are used to convert partial differential equations (PDEs) into ordinary differential equations (ODEs) and solved analytically with the help of the homotopy analysis method (HAM). The impact of different embedded parameters has been shown in the form of graphs and tables. The numerical values of skin friction and Nusselt number are presented in the form of Tables for both slip and no-slip cases. It is summarized that the upsurge of the velocity slip parameter and magnetic parameter increases the skin friction, while the rising of the thermal slip parameter and heat generation parameter decreases the Nusselt number.

1. Introduction

In order to achieve considerable cost and energy reductions, heat transfer intensification is essential. Currently, the advancement in science and technology is encouraging the mandate for extraordinarily featured compact devices with the greatest performance, precise working, and extensive lifetime. As a result, scholars and scientists gathered to work on heat and mass transmission analysis. The investigators were persuaded by the superior thermal transport characteristics of solids compared to ordinary fluids to create a new class of fluids, which they so-called “nanofluids”. The idea of dispersing high-thermal conductivity solid particles of micro-size into the conventional fluid was first pioneered by Maxwell [1] and then continued by Hamilton and Crosser [2] to intensify the fluid thermal conductivity. Nevertheless, some limitations and flaws still occurred, such as the coagulation in the fluid flow passage. Due to that limitation, a rapid investigation is being made, and nanofluid is established. Choi and Eastman [3] invented this novel heat transfer fluid by dispersing the nanometer-sized solid particles into the base fluid, and it is believed these could overcome the coagulation of the flow passage due to its unique feature.

However, along with the development of technologies, another new class of heat transfer fluid named hybrid nanofluid has been established. It is an extension of the nanofluid invented by the dispersion of dual or multiple kinds of nanosized (size less than 100 nm) solid particles with an excellent thermal conductivity into a base fluid. Jana et al. [4] study to improve the thermal conductivity of the fluid by evaluating the single and hybrid nano additives. Surprisingly, as associated with the single nanofluid, the hybrid nanofluid $\mathrm{CNT}\text{-}\mathrm{AuNP}$ and $\mathrm{CNT}\text{-}\mathrm{CuNP}$ did not enhance the thermal conductivity. Two years later, Jha et al. [5] followed with similar research. They used a combination of silver and multiwall carbon nanotubes to investigate the thermal conductivity. They have dramatically increased the thermal conductivity compared to the single nanofluids. Recently, many researchers have been interested in studying thermal conductivity and heat transfer of the hybrid nanofluids, such as Waini et al. [6,7,8,9,10].

There are numerous parameters that significantly contributed to the heat transfer improvement of the hybrid nanofluids, for example, base fluid collection, size and shape of the nanoparticles, viscidness, temperature of the fluid, steadiness, dispersibility of the nanoparticles, virtue of the nanoparticles, preparation technique, and consistency of the nanoparticles that lead to the concordant combination of the nanofluid, which can be seen in [11,12,13,14,15]. Hybrid nanofluid is claimed to have a preferable capability concerning thermal conductivity and heat transfer due to its synergistic effects, contrasted to mono-nanofluid and regular fluid. Thermal conductivity is one of the essential parameters that contribute hugely to heat transfer improvement. Different researchers exposed the importance of thermal conductivity utilizing nanofluid [16,17,18,19,20,21,22,23,24]. Recently, many researchers investigated motivating results about hybrid nanofluids, concentrating on alumina and copper as the hybrid nanoparticles, with different arrangements of the parameters have moreover been exposed by Yashkun et al. [25], Lund et al. [26,27], Aladdin et al. [28], Rosca et al. [29], Khashi’ie et al. [30,31,32,33], Roy and Pop [34], Zainal et al. [35], Anuar et al. [36], Gangadhar et al. [37] and Wahid et al. [38].

Magnetohydrodynamics (MHD) has an essential role in studying the heat transfer of hybrid nanofluid flows. It has several applications, such as the blending process of metals in the boiler. Khadija et al. [39] investigate the impact of MHD together with buoyancy force on Carreau fluid flow. Rajesh et al. [40] studied the heat transfer analysis of the hybrid nanofluid flow with MHD. Othman et al. [41] developed a mathematical model and used an exponentially shrinking sheet to study the MHD flow of the hybrid nanofluid.

Therefore, in the present study, we intended to scrutinize the slip flow and heat transfer of the hybrid nanofluid flow of alumina and copper (hybrid nanoparticles) with blood (base fluid) and heat source/sink over an exponentially stretching/shrinking sheet in the presence of suction and injection parameters. By studying the above literature review, we found that scholars and scientists have not been investigated the present hybrid nanofluid model with such configuration impact. The numerical values of Nusselt number, skin friction and the influence of various parameters are presented in graphs and discussed in detail. The analytical method (HAM) is used to solve the transformed ODEs.

2. Mathematical Formulation

Heat transfer of the hybrid nanofluid flow with heat source/sink over an exponentially stretching/shrinking sheet in the existence of suction and injection parameters is examined. The physical model of the problem is shown in Figure 1, in which the element of the velocity $u$ is in the $x$ direction and $v$ is in the $y$ directions.

The surface velocity of the fluid is assumed as $u_w\left(x\right)=c{e}^{x/L}$, and the mass transfer velocity of the wall is stated as $v_w\left(x\right)=v_0{e}^{x/2L}$. If $v_0<0$, then it represents the mass suction, and $v_0>0$ represents the mass injection, $\lambda >0$ is the stretching constant, $\lambda <0$ is the shrinking constant, $\lambda =0$ refers to a motionless sheet, $T$ represents temperature, $T_w=T_{\infty }+T_0{e}^{x/2L}$ is fluctuating temperature of the sheet with constant $T_0$ and $q=q_0{e}^{x/L}$ is constant of the heat generation rate.

Therefore, concerning the mentioned boundary layer presumptions, equations of continuity, momentum and energy are formulated as below [42].

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$

(1)

$$u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\frac{{\partial }^{2}u}{\partial y^2}-\frac{{\sigma }_{hnf}}{{\rho }_{hnf}}{B}_0^{2}u,$$

(2)

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{hnf}}{{\left(\rho C_p\right)}_{hnf}}\frac{{\partial }^{2}T}{\partial y^2}+\frac{q}{{\left(\rho C_p\right)}_{hnf}}\left(T-T_{\infty }\right).$$

(3)

Using the boundary conditions in [43].

$$\left.\begin{array}{l}u=u_w\left(x\right)\lambda +A_1\frac{{\mu }_{mnf}}{{\rho }_{mnf}}\frac{\partial u}{\partial y}, v=v_w,T=T_w\left(x\right)+B_1\frac{\partial T}{\partial y} at y=0,\\ u\to 0, T\to T_{\infty } as y\to \infty ,\end{array}\right\}$$

(4)

where $A_1$ and $B_1$ are the slip factors for velocity and thermal energy, respectively. The correlation of the physical properties (Tayebi et al. [44], Takabi et al. [45]) and the hybrid nanofluid’s thermo-physical properties (Oztop et al. [46]) is depicted in

Table 1. Hybrid nanofluid physical properties.

Properties	Hybrid Nanofluid Correlations
Density	$\begin{array}{c}{\rho }_{hnf}={\rho }_{s1}{\varphi }_{s1}+{\rho }_{s2}{\varphi }_{s2}+{\rho }_f\left(1-{\varphi }_{hnf}\right)\\ \mathrm{where} {\varphi }_{hnf}={\varphi }_{s1}+{\varphi }_{s2}\end{array}$
Heat Capacity	${\left(\rho C_p\right)}_{hnf}={\left(\rho C_p\right)}_{s1}{\varphi }_{s1}+{\left(\rho C_p\right)}_{s2}{\varphi }_{s2}+{\left(\rho C_p\right)}_f\left(1-{\varphi }_{hnf}\right)$
Dynamic Viscosity	$\frac{{\mu }_{hnf}}{{\mu }_f}=1/{\left(1-{\varphi }_{hnf}\right)}^{2.5}$
Thermal Conductivity	$\frac{k_{hnf}}{k_f}=\left[\frac{2k_f+\left\{\left({\varphi }_{s1}{k}_{s1}+{\varphi }_{s2}{k}_{s2}\right)/{\varphi }_{hnf}\right\}+2\left({\varphi }_{s1}{k}_{s1}+{\varphi }_{s2}{k}_{s2}\right)-2{\varphi }_{hnf}{k}_f}{2k_f-\left({\varphi }_{s1}{k}_{s1}+{\varphi }_{s2}{k}_{s2}\right)+\left\{\left({\varphi }_{s1}{k}_{s1}+{\varphi }_{s2}{k}_{s2}\right)/{\varphi }_{hnf}\right\}+{\varphi }_{hnf}{k}_f}\right]$

and

Table 2. Thermophysical properties.

Physical Properties	Blood	$\mathbf{A}{\mathbf{l}}_2{\mathbf{O}}_3$	$\mathbf{C}\mathbf{u}$
$\rho \left(\mathrm{k}\mathrm{g}/{\mathrm{m}}^3\right)$	1053	3970	8933
$C_p\left(\mathrm{J}/\mathrm{k}\mathrm{g}\mathrm{K}\right)$	3594	765	385
$k\left(\mathrm{W}/\mathrm{m}\mathrm{K}\right)$	0.492	40	400

, respectively. Here, the $hnf$ represents the hybrid nanofluid, $f$ refers to the base fluid, $s1$ denotes the solid nanoparticle of the alumina and $s2$ refers to the solid nanoparticle of the copper. Additionally, ${\phi }_{s1}$ and ${\phi }_{s2}$ symbolize the alumina and copper volume fraction parameters, respectively.

As to reduce the above-mentioned governing Equations (2)–(4), the following similarity variables are used (Waini et al. [47]).

$$\psi =e^{x/2L}\sqrt{2v_{f}Lcf\left(\eta \right)}, u=\frac{\partial \psi }{\partial y}, \upsilon =-\frac{\partial \psi }{\partial x}, \theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }}, \eta =y{e}^{x/2L}\sqrt{\frac{c}{2v_{f}L}}.$$

(5)

These accordingly fulfill Equation (1) and transform Equations (2) and (3) into the succeeding equations.

$$\left(\frac{{\mu }_{hnf}/{\mu }_f}{{\rho }_{hnf}/{\rho }_f}\right)f^{‴}+f{f}^{″}-2{f^{\prime }}^2-\frac{{\sigma }_{hnf}/{\sigma }_f}{{\rho }_{hnf}/{\rho }_f}M=0,$$

(6)

$$\frac{1}{\mathrm{Pr}}\left(\frac{k_{hnf}}{k_f}\right){\theta }^{″}+\frac{{\left(\rho C_p\right)}_{hnf}}{{\left(\rho C_p\right)}_f}\left(f{\theta }^{\prime }-f^{\prime }\theta \right)+\beta \theta =0.$$

(7)

Due to the transmuted boundary conditions,

$$f\left(0\right)=S, f^{\prime }\left(0\right)=\lambda +A{f}^{″}\left(0\right),\hspace{0.17em}\hspace{0.17em}{f}^{\prime }\left(\infty \right)=0, \theta \left(0\right)=1+B{\theta }^{\prime }\left(0\right), \theta \left(\infty \right)=0,$$

(8)

such that $Pr={(\mu Cp)}_f/kf$ represents the Prandtl number, $\beta =2q_{0}L/c{\left(\rho C_p\right)}_f$ depicts the heat generation parameter, $A=A_1\left({\mu }_{hnf}/{\rho }_{hnf}\right)e^{x/2L}\sqrt{c/2{\nu }_{f}L}$ shows the velocity slip parameter, $B=B_1{e}^{x/2L}\sqrt{c/2{\nu }_{f}L}$ denotes the thermal slip parameter, and $S=-{\nu }_0/\sqrt{{\nu }_{f}c/2L}$ refers to the mass transfer parameter of the wall, in which $S>0 \left({\nu }_0<0\right)$ is associated with mass suction and $S<0 \left({\nu }_0>0\right)$ is associated with mass injection.

The main physical quantities of attention (local skin friction coefficient $C_f$ and the local Nusselt number $N{u}_x$) are mathematically calculated as in [47].

$$C_f=\frac{{\mu }_{hnf}}{\rho f}\frac{1}{u_w^2}{\left(\frac{\partial u}{\partial y}\right)}_{y=0},{ \mathrm{Nu}}_x=\frac{k_{hnf}}{k_f}\frac{-2L}{\left(T_w-T_{\infty }\right)}{\left(\frac{\partial T}{\partial y}\right)}_{y=0},$$

(9)

by presenting the transformation, we have the following equations,

$${\mathrm{Re}}_x{}^{1/2}{C}_f=\left(\frac{{\mu }_{hnf}}{{\mu }_f}\right)f^{″}\left(0\right),{ \mathrm{Re}}_{\mathrm{x}}{}^{-1/2}N{u}_x=-\left(\frac{{\mu }_{hnf}}{{\mu }_f}\right){\theta }^{\prime }\left(0\right),$$

(10)

where $R{e}_x=2L{u}_w/{\nu }_f$ represents the local Reynolds number.

3. Solution of the Problem Using HAM

To solve the above-mentioned Equations (6) and (7) using the boundary conditions shown in Equation (8), we utilized the HAM technique in the following way [48,49].

The initial guesses are chosen as below:

$$f_0(\eta )=\frac{1}{2}\left(2s+\lambda -\lambda e^{-\eta }\right), {\theta }_0(\eta )=\frac{1}{1+B}{e}^{-\eta }.$$

(11)

The linear operators $L_f,{ \mathrm{L}}_{\theta }$ are chosen as below:

$$L_f(f)=f^{‴}-f^{\prime },{ \mathrm{L}}_{\theta }(\theta )={\theta }^{″}-\theta ,$$

(12)

which shows the succeeding characteristics.

$$L_f(c_1+c_2{e}^{-\eta }+c_3{e}^{\eta })=0,{ \mathrm{L}}_{\theta }(c_4{e}^{-\eta }+c_5{e}^{\eta })=0,$$

(13)

Here, in a general solution, $c_i(i=1-5)$ are constants.

The subsequent non-linear operators $N_f,N_{\theta }$ as follows:

$$N_f\left[f(\eta ;p)\right]=\frac{\left(\raisebox{1ex}{${\mu }_{hnf}$}\!\left/ \!\raisebox{-1ex}{${\mu }_f$}\right.\right)}{\left(\raisebox{1ex}{${\rho }_{hnf}$}\!\left/ \!\raisebox{-1ex}{${\rho }_f$}\right.\right)}\frac{{\partial }^{3}f(\eta ;p)}{\partial {\eta }^3}+f(\eta ;p)\frac{{\partial }^{2}f(\eta ;p)}{\partial {\eta }^2}-2{\left(\frac{\partial f(\eta ;p)}{\partial \eta }\right)}^2-\frac{\left(\raisebox{1ex}{${\sigma }_{hnf}$}\!\left/ \!\raisebox{-1ex}{${\sigma }_f$}\right.\right)}{\left(\raisebox{1ex}{${\rho }_{hnf}$}\!\left/ \!\raisebox{-1ex}{${\rho }_f$}\right.\right)}M.$$

(14)

$$\left.\begin{array}{cc}{N}_{\theta }\left[f(\eta ;p),\theta (\eta ;p)\right]=\frac{1}{\mathrm{Pr}}\left(\frac{k_{hnf}}{k_f}\right)\frac{{\partial }^2\theta (\eta ;p)}{\partial {\eta }^2}+& \\ & \frac{{\left(\rho C_p\right)}_{hnf}}{{\left(\rho C_p\right)}_f}\left(f(\eta ;p)\frac{\partial \theta (\eta ;p)}{\partial \eta }-\theta (\eta ;p)\frac{\partial f(\eta ;p)}{\partial \eta }\right)+\beta \theta (\eta ;p)\hfill \end{array}\right\}.$$

(15)

The 0th-order problems from Equations (6) and (7) are:

$$(1-p)L_f\left[f(\eta ;p)-f_0(\eta )\right]=p{ħ}_f{N}_f\left[f(\eta ;p)\right].$$

(16)

$$(1-p)L_{\theta }\left[\theta (\eta ;p)-{\theta }_0(\eta )\right]=p{ħ}_{\theta }{N}_{\theta }\left[f(\eta ;p),\theta (\eta ;p)\right].$$

(17)

The equivalent boundary constraints are:

$$\left.\begin{array}{l}{\left.f(\eta ;p)\right|}_{\eta =0}=S, {\left.\frac{\partial f(\eta ;p)}{\partial \eta }\right|}_{\eta =0}=\lambda +{\left.\frac{{\partial }^{2}f(\eta ;p)}{\partial {\eta }^2}\right|}_{\eta =0},{\left.\frac{\partial f(\eta ;p)}{\partial \eta }\right|}_{\eta \to \infty }=0\\ {\left.\theta (\eta ;p)\right|}_{\eta =0}=1+B{\left.\frac{\partial \theta (\eta ;p)}{\partial \eta }\right|}_{\eta =0}=0, {\left.\theta (\eta ;p)\right|}_{\eta \to \infty }=0,\end{array}\right\}$$

(18)

Here the implanting parameters ${ħ}_f, {ħ}_{\theta }$ for $p\in [0,1]$ are used to control the convergence of the solution. By putting $p=0 \mathrm{and} \mathrm{p}=1$, we have:

$$f(\eta ;1)=f(\eta ), \theta (\eta ;1)=\theta (\eta ),$$

(19)

Expanding $f(\eta ;p), \theta (\eta ;p)$ using Taylor’s series about $p=0$.

$$\left.\begin{array}{l}f(\eta ;p)=f_0(\eta )+{\displaystyle \sum _{m=1}^{\infty }{f}_m(\eta )p^m},\\ \theta (\eta ;p)={\theta }_0(\eta )+{\displaystyle \sum _{m=1}^{\infty }{\theta }_m(\eta )p^m},\end{array}\right\}$$

(20)

where

$$f_m(\eta )={\left.\frac{1}{m!}\frac{\partial f(\eta ;p)}{\partial \eta }\right|}_{p=0}, {\theta }_m(\eta )={\left.\frac{1}{m!}\frac{\partial \theta (\eta ;p)}{\partial \eta }\right|}_{p=0}$$

(21)

The secondary parameters ${ħ}_f, {ħ}_{\theta }$ are taken for the purpose that the Series (20) converges at $p=1$, and by putting $p=1$ in Equation (20), we obtain the following:

$$\left.\begin{array}{l}f(\eta )=f_0(\eta )+{\displaystyle \sum _{m=1}^{\infty }{f}_m(\eta )},\\ \theta (\eta )={\theta }_0(\eta )+{\displaystyle \sum _{m=1}^{\infty }{\theta }_m(\eta ).}\end{array}\right\}$$

(22)

The $m^{th}$-order deformation problem as follows:

$$\left.\begin{array}{l}{L}_f\left[f_m(\eta )-{\chi }_m{f}_{m-1}(\eta )\right]={ħ}_f{R}_m^f(\eta ),\\ L_{\theta }\left[{\theta }_m(\eta )-{\chi }_m{\theta }_{m-1}(\eta )\right]={ħ}_{\theta }{R}_m^{\theta }(\eta ).\end{array}\right\}$$

(23)

The consistent boundary constraints are given below:

$$\left.\begin{array}{l}{f}_m(0)={f^{\prime }}_m(0)-{f^{″}}_m(0)={\theta }_m(0)-B{{\theta }^{\prime }}_m(0)=0\\ {f^{\prime }}_m(\infty )={\theta }_m(\infty )=0,\end{array}\right\}$$

(24)

Here,

$$R_m^f(\eta )=\frac{\left(\raisebox{1ex}{${\mu }_{hnf}$}\!\left/ \!\raisebox{-1ex}{${\mu }_f$}\right.\right)}{\left(\raisebox{1ex}{${\rho }_{hnf}$}\!\left/ \!\raisebox{-1ex}{${\rho }_f$}\right.\right)}{f}_{m-1}^{‴}+{\displaystyle \sum _{k=0}^{m-1}{f}_{m-1-k}}{f}^{″}-2{\displaystyle \sum _{k=0}^{m-1}{f}_{m-1-k}^{\prime }}{f}_k^{\prime }-\frac{\left(\raisebox{1ex}{${\sigma }_{hnf}$}\!\left/ \!\raisebox{-1ex}{${\sigma }_f$}\right.\right)}{\left(\raisebox{1ex}{${\rho }_{hnf}$}\!\left/ \!\raisebox{-1ex}{${\rho }_f$}\right.\right)}M,$$

(25)

$$R_m^{\theta }(\eta )=\frac{1}{\mathrm{Pr}}\left(\frac{k_{hnf}}{k_f}\right){\theta }_{m-1}^{″}+\frac{{\left(\rho C_p\right)}_{hnf}}{{\left(\rho C_p\right)}_f}\left({\displaystyle \sum _{k=0}^{m-1}{f}_{m-1-k}}{\theta }_k^{\prime }-{\displaystyle \sum _{k=0}^{m-1}{f}_{m-1-k}^{\prime }}{\theta }_k\right)+\beta {\theta }_{m-1},$$

(26)

where

$${\chi }_m=\left\{\begin{array}{l}0, \mathrm{if} \mathrm{p}\le 1\\ 1, \mathrm{if} \mathrm{p}>1\end{array}\right.$$

4. Results and Discussion

Solutions of the Equations (6) and (7) with the boundary conditions shown in Equation (8) are analytically computed using the HAM method in Mathematica software. The appropriate initial guess, boundary conditions and different values of parameters need to be chosen and well-adjusted in the coding of the HAM method to calculate the foremost exact outcomes. We considered the stretching ($\lambda >0$) and shrinking ($\lambda <0$) sheets. The Prandtl number for alumina is fixed in the present study, which is $\mathrm{Pr}=6.2$, while the other parameters, such as velocity $A$ and thermal $B$ slips, heat generation $\beta$ and suction $S$, are set to be vary with the purpose to study their influence on the fluid flow and heat transmission.

For validation, we compared our result with the numerical method (the Shooting method). We find that the current and the numerical results are rationally interrelated. We thus guarantee that the methodology and the results of this study are valid.

Figure 2 shows the variations in the skin friction ${\mathrm{Re}}_x{}^{1/2}{C}_f$, and it is noticed that skin friction has a decreasing trend as the velocity slip parameter $A$ and magnetic parameter $M$ escalate.

Figure 3 shows the variations in the local Nusselt number ${\mathrm{Re}}_{\mathrm{x}}{}^{-1/2}N{u}_x$, and it is noticed that the enhancement in the value of $B$ and $\mathrm{Pr}$ promotes the increment in the Nusselt number ${\mathrm{Re}}_{\mathrm{x}}{}^{-1/2}N{u}_x$.

Figure 4 and Figure 5 show the impact of the velocity slip parameter $A$ on $f^{\prime }\left(\eta \right)$ and temperature profiles $\theta \left(\eta \right)$, respectively. It shows that the slip parameter has an important impact on velocity and temperature distributions. This illustrates that by increasing the values of the velocity slip parameter $A$, there is a gradual decline in the field of velocity and temperature profiles.

Figure 6 illustrates the influence of the magnetic parameter $M$ on the velocity field $f^{\prime }\left(\eta \right)$. It is revealed that the velocity of the fluid declined due to enhancement in magnetic parameters. The reality behind this is the application of a crosswise magnetic field, which leads to that of a drag force comparable to that of Lorentz, which shows up a resistance in the fluid flow, and as a result, it reduces the velocity of the fluid.

Figure 7 and Figure 8 show the impact of $S$ on $f^{\prime }\left(\eta \right)$ and $\theta \left(\eta \right)$, respectively. It is noted that when $S$ rises, the fluid velocity and temperature slow down. A comparison of the results shown in Figure 6 clearly indicates that the effect of the slip parameter on velocity and temperature is quite similar to the effect of the magnetic parameter. Therefore, it is clear from the above-mentioned example that the combination of partial slip, magnetic effect and mass transfer has a great influence on the fluid flow and temperature.

In Figure 9 and Figure 10, the influences of the thermal slip parameter $B$ and heat generation parameter $\beta$ over the temperature profile $\theta \left(\eta \right)$ are portrayed. Since these parameters do not affect the velocity profile $f^{\prime }\left(\eta \right)$, it is only necessary to provide the temperature profile $\theta \left(\eta \right)$ in this present study. The temperature profile $\theta \left(\eta \right)$ rises when the thermal slip parameter B decreases or the heat generation parameter β increases. The appropriate amounts of thermal slip and heat generation parameters need to be controlled to obtain the required heat transfer performance, as both parameters produce a different impact that could offset each other.

Table 3. Numerical values of the local skin friction coefficient $-C_{f}R{e}_x^{1/2}$ for various constraints when $\beta =0.3,\mathrm{Pr}=0.7,B=0.1,h=-0.3$.

$\mathit{M}$	$\mathit{\lambda }$	$\mathit{A}$	$\mathit{S}$	$-{\mathit{C}}_{\mathit{f}}\mathit{R}{\mathit{e}}_{\mathit{x}}^{1/2}$ Cylindrical Shape	$-{\mathit{C}}_{\mathit{f}}\mathit{R}{\mathit{e}}_{\mathit{x}}^{1/2}$ Spherical Shape
0.4	0.3	0.5	0.4	−0.117305 *	−0.117305
0.5				−0.113748	−0.113748
0.6				−0.110191	−0.110191
0.4	0.3			−0.117305	−0.117305
	0.4			−0.108694	−0.108694
	0.5			−0.101673	−0.101673
	0.3	0.5		−0.117305	−0.117305
		0.6		−0.110741	−0.110741
		0.7		−0.104883	−0.104883
		0.5	0.4	−0.117305	−0.117305
			0.5	−0.113921	−0.113921
			0.6	−0.110592	−0.110592

* Shadow in the table: similar values or initial values.

shows the impact of different constraints on cylindrical and spherical shapes of the skin friction coefficient. It is noticed that with the increase in magnetic number $M$, the slip parameter $A$, stretching parameter $\lambda$ and the wall mass transfer parameter $S$, both the cylindrical and spherical shapes of the skin friction coefficient increase.

Table 4. Numerical values of the local Nusselt number $N{u}_x$ for different parameters when $A=0.5,\lambda =0.3,M=0.4,s=0.4,h=-0.3$.

$\mathit{P}\mathit{r}\mathbf{}$	$\mathit{B}$	$\mathit{\beta }$	$\mathit{N}{\mathit{n}}_{\mathit{x}}\mathit{R}{\mathit{e}}_{\mathit{x}}^{-1/2}$ Cylindrical Shape	$\mathit{N}{\mathit{n}}_{\mathit{x}}\mathit{R}{\mathit{e}}_{\mathit{x}}^{-1/2}$ Spherical Shape
0.7	0.1	0.3	0.444816 *	0.501535
0.8			0.462944	0.512572
0.9			0.477043	0.521157
0.7	0.1		0.444816	0.501535
	0.2		0.420066	0.467725
	0.3		0.397374	0.437983
	0.1	0.3	0.444816	0.501535
		0.4	0.439307	0.496025
		0.5	0.433797	0.490516

* Shadow in the table: similar values or initial values.

represents the influence of various parameters on cylindrical and spherical shapes of the local Nusselt number. It shows that because of a raise in the Prandtl number $Pr$, the cylindrical and spherical shapes of the Nusselt number are enhanced. Conversely, cylindrical and spherical shapes of the Nusselt number decline due to an increase in the slip parameter $B$ and heat generation parameter $\beta$.

Table 5. Results of calculation for error analysis.

Numerical Iteration (m)	$\mathit{f}(\mathit{\eta })$ Velocity Profile	$\mathit{\theta }(\mathit{\eta })$ Temperature Profile
4	$9.84368\times {10}^{-6}$	0.18939
8	$5.90287\times {10}^{-7}$	0.0025323
12	$6.67716\times {10}^{-8}$	$3.65053\times {10}^{-5}$
16	$1.09998\times {10}^{-8}$	$5.41439\times {10}^{-7}$
20	$2.6821\times {10}^{-9}$	$7.9137\times {10}^{-9}$

and Figure 11 present the error analysis. The absolute error is decreasing with the increasing number of iterations.

Table 6. Comparison of HAM and the numerical solution for $f^{\prime }\left(\eta \right)$.

$\mathit{\eta }$	HAM Solution	Numerical Solution (Shooting Method)	Absolute Error
0.0	0.658510	0.659357	0.000847
0.5	0.490030	0.495299	0.005269
1.0	0.372866	0.382860	0.009995
1.5	0.283323	0.297442	0.014119
2.0	0.210825	0.227093	0.016268
2.5	0.151427	0.166819	0.015392
3.0	0.103891	0.115291	0.011399
3.5	0.067335	0.072721	0.005385
4.0	0.040156	0.039517	0.000639
4.5	0.019973	0.015563	0.004410
5.0	0.004288	$-3.564410\times {10}^{-8}$	0.004288

(Figure 12) and

Table 7. Comparison of HAM and the numerical solution for $\theta \left(\eta \right)$.

$\mathit{\eta }$	HAM Solution	Numerical Solution (Shooting Method)	Absolute Error
0.0	0.942573	0.943156	0.000583
0.5	0.881382	0.882394	0.001008
1.0	0.812507	0.813847	0.001340
1.5	0.736136	0.737613	0.001477
2.0	0.652458	0.653804	0.001347
2.5	0.561613	0.562549	0.000936
3.0	0.463738	0.463989	0.000251
3.5	0.358965	0.358280	0.000685
4.0	0.247422	0.245590	0.001832
4.5	0.129223	0.126099	0.003123
5.0	0.004492	$-9.900090\times {10}^{-10}$	0.004492

(Figure 13) show the comparison of HAM and numerical solution for velocity and temperature profiles. An excellent agreement has been seen between analytical and numerical methods. The validity can be seen very clearly from these tables.

5. Conclusions

Heat transfer of the magnetohydrodynamic hybrid nanofluid flow past an exponentially stretching surface with slip conditions in the presence of suction and injection has been investigated. The hybrid nanofluid of alumina and copper (hybrid nanoparticles) with blood (base fluid) has been considered. The homotopy analysis method (HAM) has been used to solve the stated problem analytically. The key points of the present article are given below.

• The rise in velocity slip and the magnetic parameters increases the skin friction. Consequently, it indicates a decline in the velocity of the fluid.
• The rise in the stretching/shrinking parameters decreases the skin friction. Therefore, it shows an increment in the velocity of the fluid.
• The Nusselt number is diminished with the increase in the thermal slip parameter and heat generation parameter.
• The temperature distribution rises with the enhancement of the heat generation parameter, while it decreases when the thermal slip parameter and mass transfer parameter rise.
• It is concluded from Table 4 that the heat transfer rate increases with raising $\mathrm{Pr}$, while it decreases with the enhancement of $B \mathrm{and} \beta$.