Significance of Weissenberg Number, Soret Effect and Multiple Slips on the Dynamic of Biconvective Magnetohydrodynamic Carreau Nanofuid Flow

Abstract

This study focused on the analysis of two-dimensional incompressible magnetohydrodynamic Carreau nanofluid flow across a stretching cylinder containing microorganisms with the impacts of chemical reactions and multiple slip boundary conditions. Moreover, the main objective is concerned with the enhancement of thermal transportation with the effect of heat source and bioconvection. By assigning pertinent similarity transitions to the governing partial differential equations, a series of equations (ODES) is generated. An optimum computational solver, namely the bvp5c software package, is utilized for numerical estimations. The impact of distinct parameters on thermal expansion, thermophoresis, and the Nusselt number has been emphasized, employing tables, diagrams, and surface maps for both shear thinning (n < 1) and shear thickening (n > 1) instances. Motile concentration profiles decrease with $L_b$ and the motile microorganism density slip parameter. It is observed that with increasing values of $Pr$, both the boundary layer thickness and temperature declined in both cases. The Weissenberg number demonstrates a different nature depending on the type of fluid; skin friction, the velocity profile and Nusselt number drop when n < 1 and increase when n > 1. The two- and three-dimensional graphs show the simultaneous effect of involving parameters with physical quantities. The accuracy of the existing observations is evidenced by the impressive resemblance between the contemporary and preceding remedies.

1. Introduction

The mechanisms for optimizing proactive thermal performance incorporate nanoparticles into conventional fluids, generating surfaces resilient, imparting magnetism, boosting synthesized surface irregularity, attributing fins, and embedding obstructions. Over the past couple of decades, scientists have made tremendous efforts to develop innovative nanofluids with enhanced efficiency. On a magnetohydrodynamic (MHD) boundary stream flow, the two-fold dispersion impact, the effect of fluctuating thermal rheology, and bioconvection all play a role. A substantial Péclet number is appropriate for the motile organisms [1]. Chemical process factors and stratified motile concentration parameters influence microorganism proliferation. When a magnetic field is present, higher values of the electric field parameter produce the highest drag coefficient, while smaller values of the electric field parameter produce the lowest, as explained by Areekara et al. [2]. According to Saranya and Radha [3], nanofluid bioconvection has several asylum applications in the pharmaceutical sector. Liu et al. studied the two-dimensional columnar bioconvective stream to examine the mobility of an MHD bioconvective micro-rotational nanofluid that constrains a microbe [4].

The utilization of the motion equation for the bioconvective flow of nanofluid incorporating gyrotactic microbes over a perpendicularly elongating surface was studied by Tausif et al. [5] and includes multiple slip situations at the boundary surface. Habibishandiz and Habib critically reviewed heat transfer enhancement methods in the presence of porous media, nanofluids, and microorganisms [6,7]. Khan et al. [8] scrutinized the melting processes on the unsteady wedge stream of the nanofluid in terms of the availability of thermal absorption/generation. The unsteady MHD stagnation point stream of Carreau nanoliquid across a stretching cylindrical surface with the implication of nonlinear thermal radiation is scrutinized by Azam et al. [9]. Hsiao [10] explains the effectiveness of the thermal extrusion production system employing the Carreau nanofluid with a parameters control technique. Salahuddin [11] explains the impacts of magnetic fields and nanoparticles with the generalized non-Newtonian Carreau fluid flow in two dimensions across a metal cylinder that allows heat to escape in and out but may not slip; this occurs in the sense of heat fabrication, thermal polarization, and thermal radiation.

A computational representation of the MHD radiative Carreau nanoliquid is explored by Jagadha et al. [12] for laminar, steady, and incompressible flow. Under the influence of heat generation/absorption and radiation, the first-order chemical reaction of the Carreau nanoliquid via the fixed cylinder affects the power law. The study of Carreau nanoquid with different parameters and geometry explains the heat transfer [13,14]. Kumar et al. [15] describe the bioconvection flow of Casson nanofluid through a spinning disc with gyrotactic microbes, numerous slips, and thermal radiation. A two-dimensional estimation of the convective heat transition of a low-viscosity liquid passing all through an elongated cylinder has been executed utilizing the Soret and Dufour phenomena [16]. They find heat and mass transfer rates are inversely related to simultaneous fluctuations of Soret and Dufour. To investigate MHD stagnation-point flow, Ramzan [17] has used a movable, transparent, elongated cylinder incorporating Soret–Dufour causes. Analytical solution of non-Fourier heat conduction in a 3D hollow sphere under time–space varying boundary conditions is studied by Akberi et al. [18,19]. A nonlinear mathematical analysis for magnetohyperbolic-tangent liquid featuring simultaneous aspects of a magnetic field, a heat source, and thermal stratification is explained [20,21].

It includes amplification of the immiscible flow of Maxwell nanofluids attributable to bioconvection, where motile microbes are present, along with activating energy. Brownian motion and viscous dissipation dispersion are being explored [22]. A stretched cylinder was used to evaluate the influence of heat transmission on nanofluid flow [23]. They show that with a rise in $N_r$, the Nusselt number and magnitude of the drag force coefficient are decreased with the porosity parameter. Liaqat et al. [24] described the prominence of concentration-dependent Sutterby flow of a nanofluid in computational fluid dynamics across a certain surface. They established that the Reynolds number decreases as radial velocity increases. Islam et al. [25] present a mixed convection flow of Maxwell nanofluid with thermal energy transmission through a rotating cylinder with heat production/emtion and joule thermal effect. The combined effect of heat radiation and convection movement of nanofluid in a porous medium is caused by a stretched cylinder as well as numerous dissipative and slip boundary limitations [26]. An investigation into the flow dynamics of Cu-water nanofluid toward horizontal and exponentially porosity enalogating/detreching cylinders is conducted using the following parameters: suction/injection, thermal conduction, the shape of the nanoparticle, stagnation point flow, and heat transfer coefficient [27,28]. Saha et al. [29,30] describe the MHD flow of time-fractional Casson nanofluid using generalized Fourier and Fick’s laws over an inclined channel with applications of gold nanoparticles. Analysis and numerical simulation of fractal-fractional order nonlinear couple stress nanofluid with cadmium telluride nanoparticles was completed by [31,32].

The magnetohydrodynamic Carreau nanofluid flow containing gyrotactic microorganisms over a cylinder with a heat source and multi-slip parameters has not been investigated. In the presence of nanoparticles, due to the velocity slip, thermal slip, concentration slip, and the microorganisms’ density slip, these also have been noticed at the wall. The modeled governing equations are transmuted into a system of first-order ODEs with the help of apposite similarity transformations, which are then numerically resolved using the finite-difference-based bvp5c algorithm. To increase the novelty of the current research, the impact of various parameters on heat transfer rate for pseudo-plastic and dilatant fluids is examined using 2D graphs and 3D models on velocity, temperature, concentration, motile microorganism concentration, local skin friction coefficient, local Nusselt number, Sherwood number, and the local density of the motile microorganism, which are elaborated through graphical as well as tabular representations in the absence and presence of respective slip effects. The findings of this numerical simulation have applications in food processing, water emulsions, thermal power generation systems, chemical engineering, and nuclear reactors.

2. Problem Description

Heat and mass transfer are analyzed using a 2D mathematical model in this section. Carreau nanofluid flow with microorganisms in the presence of chemically reactive species toward a cylinder with a radius R and free stream velocity $U_{\infty }$ has been considered. The impacts of chemical reactions, the heat source, and multi-slip are involved in this paper to extend the worth of the research. Let free-stream velocity $u_w=\frac{ax}{l}$ flow over the cylinder, where a is a positive constant. The motion across the cylinder displayed in Figure 1 induces the proposed flow. The x-axis along the cylinder’s surface and r in the axial direction are both taken into consideration by the coordination system. Along the r direction, a consistent magnetic intensity ($B_0$) is implemented. The temperature and concentration at the stretching cylinder are expressed by $\left(T_w\right)$ and $\left(C_w\right)$; also, the ambient stage is represented by $T_{\infty }$ and $C_{\infty }$ when R approaches infinity. Based on these assumptions, we can derive the following governing equations: [5,33,34,35,36,37].

$$\frac{\partial \left(ru\right)}{\partial x}+\frac{\partial \left(rv\right)}{\partial r}=0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}$$

(1)

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial r}={\nu }_f\frac{{\partial }^{2}u}{\partial r^2}{\left(1+{\left({\tau }_c\frac{\partial u}{\partial r}\right)}^2\right)}^{\frac{n-1}{2}}+\frac{{\nu }_f}{r}\frac{\partial u}{\partial r}{\left(1+{\left({\tau }_c\frac{\partial u}{\partial r}\right)}^2\right)}^{\frac{n-1}{2}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}$$

$$+(n-1){\nu }_f{\left({\tau }_c\frac{\partial u}{\partial r}\right)}^2\frac{{\partial }^{2}u}{\partial r^2}{\left(1+{\left({\tau }_c\frac{\partial u}{\partial r}\right)}^2\right)}^{\frac{n-3}{2}}-\frac{\sigma B_0^2}{{\rho }_f}u,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}$$

(2)

$$\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}\right)=\frac{{\kappa }_f}{{\left(\rho C_p\right)}_f}\left(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}\right)+\frac{q_t}{{\left(\rho C_p\right)}_f}(T-T_{\infty }),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}$$

(3)

$$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial r}+K_c(C-C_{\infty })=D_m\left(\frac{{\partial }^{2}C}{\partial r^2}+\frac{1}{r}\frac{\partial C}{\partial r}\right)+\frac{D_m{k}_t}{T_m}\left(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}\right),\hspace{1em}\hspace{1em}$$

(4)

$$u\frac{\partial N}{\partial x}+v\frac{\partial N}{\partial r}+K_c(N-N_{\infty })=-\frac{b{W}_c}{C_w-C_{\infty }}\left[\frac{\partial }{\partial r}\left(N\frac{\partial C}{\partial r}\right)\right]+D_n\left(\frac{{\partial }^{2}N}{\partial r^2}+\frac{1}{r}\frac{\partial N}{\partial r}\right),$$

(5)

According to the problem’s geometry, the boundary conditions can be divided into the preceeding classes:

$$\begin{array}{c}\hfill \left.\begin{array}{cccc}\hfill \left(a\right)& u=U_w=\frac{ax}{l}+K_1\frac{\partial u}{\partial r}{\left(1+({\tau }_c{\frac{\partial u}{\partial r})}^2\right)}^{\frac{n-1}{2}},v=0,K_2\frac{\partial T}{\partial r}=\{T-T_w\},\hfill & & \\ \hfill & C=C_w+K_3\frac{\partial C}{\partial r},N=N_w+K_4\frac{\partial N}{\partial r}atr=R,\hfill & & \\ \hfill \left(b\right)& u\to u_w=0,T\to T_{\infty },C\to C_{\infty },N\to N_{\infty }atr\to \infty ,\hfill & \end{array}\right\}\end{array}$$

(6)

A velocity component is represented by u and v along directions x and r, respectively; the other components include thermal relaxation $\left(\delta \right)$, chemical reaction parameter $\left(K_c\right)$, magnetic field intensity $\left(B_o\right)$, microorganisms’ maximum swimming speed in nanofluids $\left(W_c\right)$, and their diffusivity $\left(D_n\right)$. Except for saturation variations that generate a thermal floatation thrust, the attributes persist. In order to reduce the complexity of the presented problem, the following non-dimensional variables must be included:

$$\begin{array}{c}\left.\begin{array}{c}\hfill \mathsf{\Psi }(x,r)=R\sqrt{u_w{\nu }_{f}x}f\left(\zeta \right),\zeta =\frac{r^2-R^2}{2R}{\left(\frac{u_w}{{\nu }_{f}x}\right)}^{\frac{1}{2}},\theta \left(\zeta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\chi \left(\zeta \right)=\frac{N-N_{\infty }}{N_w-N_{\infty }},\\ \hfill \phi \left(\xi \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }},u=\frac{1}{r}\frac{\partial \mathsf{\Psi }}{\partial r},v=-\frac{1}{r}\frac{\partial \mathsf{\Psi }}{\partial x},u_w=\frac{ax}{l},\end{array}\right\}\end{array}$$

(7)

Here, the stream functions $\mathsf{\Psi }$ and $\xi$ are dimensionless. Usually, Equation (1) is satisfied by demarcating this $\mathsf{\Psi }$ stream function. Following this, the nonlinear partial differential Equations (2)–(5) are transmuted as follows using the similarity vectors (7) mentioned above:

$$(2\zeta \Gamma +1){(1+{W_e}^2{f^{″}}^2)}^{((n-1)/2)}{f}^{‴}+(n-1){W_e}^2(2\zeta \Gamma +1){(1+{W_e}^2{f^{″}}^2)}^{((n-3)/2)}{f^{″}}^2{f}^{‴}$$

(8)

$$-M{F}^{\prime }+f{f}^{″}+2\Gamma {(1+{W_e}^2{f^{″}}^2)}^{((n-1)/2)}{f}^{″}+(n-1){W_e}^2(\Gamma ){(1+{W_e}^2{f^{″}}^2)}^{((n-3)/2)}{f^{″}}^3-{f^{\prime }}^2=0,$$

$$\begin{array}{c}\left[(2\zeta \Gamma +1){\theta }^{″}+2\Gamma {\theta }^{\prime }\right]+P_r\left[f{\theta }^{\prime }+Q_t\theta \right]=0,\end{array}$$

(9)

$$(1+2\zeta \Gamma ){\phi }^{″}+(Lef+2\Gamma ){\phi }^{\prime }+Sr\left[(1+2\zeta \Gamma ){\theta }^{″}+2\Gamma {\theta }^{\prime }\right]-Le{C}_h\phi =0,$$

(10)

$$(1+2\zeta \Gamma ){\chi }^{″}+(Lbf+2\Gamma ){\chi }^{\prime }$$

$$\begin{array}{c}-Pe[(1+2\zeta \Gamma )\chi {\phi }^{″}+\Gamma (\chi +\omega ){\phi }^{\prime }+\omega (1+2\zeta \Gamma ){\phi }^{″}+(1+2\xi \Gamma ){\phi }^{\prime }{\chi }^{\prime }]-Lb{C}_h\chi =0,\end{array}$$

(11)

Moreover, the modified boundary limitations are expressed as;

$$\begin{array}{c}\hfill \left.\begin{array}{cccc}& f\left(\zeta \right)=0,f^{\prime }\left(\zeta \right)=1+{\lambda }_1{f}^{″}{(1+{W_e}^2{f^{″}}^2)}^{((n-1)/2)},\theta \left(\zeta \right)=1+{\lambda }_2{\theta }^{\prime },\hfill & & \\ & \chi \left(\zeta \right)=1+{\lambda }_3{\phi }^{\prime },\chi \left(\zeta \right)=1+{\lambda }_4{\chi }^{\prime }at\zeta =0,\hfill & & \\ & f^{\prime }\left(\zeta \right)\to 1,\theta \left(\zeta \right)\to 0,\phi \left(\zeta \right)\to 0,\chi \left(\zeta \right)\to 0at\zeta \to \infty ,\hfill & \end{array}\right\}\end{array}$$

(12)

3. Physical Quantities

The various involving factors in Equations (8) to (12) are also described, as are the numerous participating factors: Prandtl number $\left(P_r\right)$, magnetic parameter $\left(M\right)$, traditional Lewis number $\left(L_e\right)$, chemical reaction parameter $\left(C_h\right)$, velocity slip parameter $\left({\lambda }_1\right)$, Péclet number $\left(P_e\right)$, Lewis number $\left(L_b\right)$, heat source $\left(Q_t\right)$, Soret number $\left(S_r\right)$, microorganisms difference parameter $\left(w\right)$, thermal slip factor $\left({\lambda }_2\right)$, concentration slip factor $\left({\lambda }_3\right)$, motile microbes density slip factor $\left({\lambda }_4\right)$, Weissenberg number $\left(We\right)$ and the curvature parameter $(\Gamma )$.

$$\begin{array}{c}\left.\begin{array}{c}\hfill M=\frac{{\sigma }_f{B^2}_{0}l}{{\rho }_{f}a},Q_t=\frac{q_T}{{\left(\rho C_p\right)}_f},Sr=\frac{k_t(T_w-T_{\infty })}{T_m(C_w-C_{\infty })},C_h=\frac{K_{c}l}{a},\\ \hfill {\left(W_e\right)}^2=\frac{{\tau }_c{a}^3{x}^2{r}^2}{{\nu }_f{l}^3{R}^2}Le=\frac{{\mu }_f}{{\rho }_f{D}_m},Pr=\frac{{\mu }_f}{{\rho }_f{\alpha }_f},\\ \hfill \Gamma =\frac{1}{R}\sqrt{\frac{{\mu }_{f}l}{{\rho }_{f}a}},Pe=\frac{b{W}_c}{D_m},\omega =\frac{N_{\infty }}{N_w-N_{\infty }},Lb=\frac{{\mu }_f}{{\rho }_f{D}_m},{\alpha }_f=\frac{{\kappa }_f}{{\left(\rho c\right)}_f},\end{array}\right\}\end{array}$$

4. Engineering Quantities

The obligatory facets of engineering concern the skin friction factor $\{C{f}_{x}R{e}_x^{1/2}\}$, local Sherwood number $\{S{h}_{x}R{e}_x^{-1/2}\}$, local Nusselt number $\{N{u}_{x}R{e}_x^{-1/2}\}$, and local density of the motile microorganism $\{N_{nx}R{e}_x^{-1/2}\}$. The following specific interests include:

• $C_{fx}=\frac{2{\mu }_f}{{\rho }_f{U_w}^2}\left({\tau }_{rx}\right),N{u}_x=-\frac{x}{T_w-T_{\infty }}{\left(\frac{\partial T}{\partial r}\right)}_{r=R}$
• $S{h}_x=\frac{-x}{C_w-C_{\infty }}{\left(\frac{\partial C}{\partial r}\right)}_{r=R},N_{nx}=\frac{-x{D}_n}{D_n(N_w-N_{\infty })}{\left(\frac{\partial N}{\partial r}\right)}_{r=R}$

where

• ${\tau }_{rx}={\left({\mu }_f\frac{\partial u}{\partial r}{\left(1+({\tau }_c{\frac{\partial u}{\partial r})}^2\right)}^{\frac{n-1}{2}}\right)}_R$

By utilizing the above similarity transformations, (7) can be calculated as:

• ${Cf}_x=\sqrt{R{e}_x}{Cf}_x=f^{″}\left(0\right){\left(1+{W_e}^2{\left(f^{″}\right(0\left)\right)}^2\right)}^{(n-1)/2},\frac{N{u}_x}{\sqrt{R{e}_x}}=\left\{{\theta }^{\prime }\left(0\right)\right\},$
• $\frac{S{h}_x}{\sqrt{R{e}_x}}=-{\phi }^{\prime }\left(0\right),\frac{N_{nx}}{\sqrt{R{e}_x}}={\chi }^{\prime }\left(0\right),$

Here, $R{e}_x=x{u}_w/{\nu }_f$ is the local Reynolds number.

5. Implementation of Method

The finite-difference-based bvp5c algorithm is adopted to numerically rectify the impaired cluster of ODEs. The results are observed by initiating with a technical guess. An effective first estimation must be selected based on the values of the defined parameters. The obtained consequences are used as an initial point for the problem’s solution with only a small change in the settings. This allows us to tweak the parameters periodically until they are as close as possible to the optimal values. This approach is discussed by Shampine et al. [38]. The first step is to transform the partial governing equations into a group of ordinary first-order differential equations. In the end, the collection of Equations (8)–(12) appears as follows, and firstly assumed that:

• $f={\mathsf{\Pi }}_1,f^{\prime }={\mathsf{\Pi }}_2,f^{″}={\mathsf{\Pi }}_3$,
• $\theta ={\mathsf{\Pi }}_4,{\theta }^{\prime }={\mathsf{\Pi }}_5,\phi ={\mathsf{\Pi }}_6,{\phi }^{\prime }={\mathsf{\Pi }}_7,$
• $\chi ={\mathsf{\Pi }}_8,{\chi }^{\prime }={\mathsf{\Pi }}_9$,
• ${\mathsf{\Pi }}_1^{\prime }={\mathsf{\Pi }}_2$,
• ${\mathsf{\Pi }}_2^{\prime }={\mathsf{\Pi }}_3$,
• ${\mathsf{\Pi }}_3^{\prime }=\frac{(-1)\left(-M{\mathsf{\Pi }}_2+2\Gamma {(1+{W_e}^2{{\mathsf{\Pi }}_3}^2)}^{((n-1)/2)}{\mathsf{\Pi }}_3+{\mathsf{\Pi }}_1{\mathsf{\Pi }}_3+(n-1){W_e}^2(\Gamma ){(1+{W_e}^2{{\mathsf{\Pi }}_3}^2)}^{((n-3)/2)}{{\mathsf{\Pi }}_3}^3-{{\mathsf{\Pi }}_2}^2\right)}{\left((2\zeta \Gamma +1){(1+{W_e}^2{{\mathsf{\Pi }}_3}^2)}^{((n-1)/2)}+(n-1){W_e}^2(2\zeta \Gamma +1){(1+{W_e}^2{{\mathsf{\Pi }}_3}^2)}^{((n-3)/2)}{{\mathsf{\Pi }}_3}^2\right)},$
• ${\mathsf{\Pi }}_4^{\prime }={\mathsf{\Pi }}_5$,
• ${\mathsf{\Pi }}_5^{\prime }=\frac{-1}{\left(\right(1+2\Gamma \xi \left)\right)}[\left(P_r\right)\left({\mathsf{\Pi }}_1{\mathsf{\Pi }}_5\right)-P_r{Q}_t{\mathsf{\Pi }}_5+2\Gamma {\mathsf{\Pi }}_5]$,
• ${\mathsf{\Pi }}_6^{\prime }={\mathsf{\Pi }}_7$,
• ${\mathsf{\Pi }}_7^{\prime }=\frac{-1}{(1+2\Gamma \xi )}[(2\Gamma +L_e{\mathsf{\Pi }}_1){\mathsf{\Pi }}_7+S_r(1+2\Gamma \xi )\left({\mathsf{\Pi }}_5^{\prime }\right)+2\Gamma {\mathsf{\Pi }}_5)-L_e{C}_h{\mathsf{\Pi }}_6]$,
• ${\mathsf{\Pi }}_8^{\prime }={\mathsf{\Pi }}_9$,
• ${\mathsf{\Pi }}_9^{\prime }=\frac{-1}{(1+2\Gamma \xi )}[(2\Gamma +L_b{\mathsf{\Pi }}_1){\mathsf{\Pi }}_9-Pe(\Gamma ({\mathsf{\Pi }}_8+w){\mathsf{\Pi }}_7+((1+2\Gamma \xi ){\mathsf{\Pi }}_8+w(1+2\Gamma \xi )){\mathsf{\Pi }}_7^{\prime }+(1+2\Gamma \xi ){\mathsf{\Pi }}_7{\mathsf{\Pi }}_9)-L_{b}Ch{\mathsf{\Pi }}_8]$,

with

• ${\mathsf{\Pi }}_1\left(0\right)=0,\mathsf{\Pi }2\left(0\right)=1+{\lambda }_1{\mathsf{\Pi }}_3{(1+{W_e}^2{{\mathsf{\Pi }}_3}^2)}^{((n-1)/2)},{\mathsf{\Pi }}_4\left(0\right)=1+{\lambda }_2{\mathsf{\Pi }}_5,{\mathsf{\Pi }}_6\left(0\right)=1+{\lambda }_3{\mathsf{\Pi }}_7,{\mathsf{\Pi }}_8\left(0\right)=1+{\lambda }_4{\mathsf{\Pi }}_9,$
• ${\mathsf{\Pi }}_2=0,{\mathsf{\Pi }}_4=0,{\mathsf{\Pi }}_6=0,{\mathsf{\Pi }}_8=0$,

The infinity condition has been rescaled to $\zeta$ = 10. The accuracy of the numerical procedure is validated through a restrictive comparison with the authors work shown in

Table 1. Assessment of $-f^{″}\left(0\right)$ for diverse values of n, $W_e$, $\Gamma$ and remaining parameters are held constant.

n	${\mathit{W}}_{\mathit{e}}$	$\mathbf{\Gamma }$	BVP5C [39]	RKF45 [39]	RK4 [39]	Current Result
0.5	1	0.2	0.989702	0.989709	0.989709	0.989701
	2	0.4	0.901331	0.901343	0.901343	0.901342
	3	0.6	0.830948	0.830972	0.830972	0.830949
1.5	1	0.2	1.145708	1.145708	1.145708	1.145704
	2	0.4	1.367059	1.367061	1.367061	1.367058
	3	0.6	1.611580	1.611585	1.611585	1.611580

and

Table 2. Assessment of $-{\theta }^{\prime }\left(0\right)$ for numerous values of $Pr$ and remaining factors are held at zero.

$\mathit{Pr}$	Pop [40]	Sabu [39]	Current Results
0.7	0.4539	0.453932	0.453933
2.0	0.9113	0.911359	0.911359
7.0	1.8954	1.895428	1.895422
20	3.3539	3.354174	3.354172

.

6. Results and Discussion

Several variables have an impact on the momentum profile. $f^{\prime }\left(\zeta \right)$, thermal distribution $\theta \left(\zeta \right)$, nanoparticle concentration profile $\phi \left(\xi \right)$, motile concentration profile $\chi \left(\zeta \right)$ with $\{N{u}_{x}R{e}_x^{-1/2}\}$, $\{C{f}_{x}R{e}_x^{1/2}\}$ and Sherwood number $\{S{h}_{x}R{e}_x^{-1/2}\}$ are clarified using graphs and tables. To accomplish this purpose, figures and tables are presented. The numerical results obtained using bvp5c are contrasted with those obtained using bvp5c and RKF45, which are shown in Table 1 and Table 2.

Table 3. The variation of the Nusselt number $\{N{u}_{x}R{e}_x^{-1/2}\}$, skin friction coefficient $\{C{f}_{x}R{e}_x^{1/2}\}$ and Sherwood number $\{S{h}_{x}R{e}_x^{-1/2}\}$ for numerous values of $M,\Gamma ,W_e,{\lambda }_1$ while remaining parameters be fixed.

n	M	$\mathbf{\Gamma }$	${\mathit{W}}_{\mathit{e}}$	${\mathit{\lambda }}_1$	${\mathit{Nu}}_{\mathit{x}}{\mathit{Re}}_{\mathit{x}}^{-1/2}$	${\mathit{Cf}}_{\mathit{x}}{\mathit{Re}}_{\mathit{x}}^{1/2}$	${\mathit{Sh}}_{\mathit{x}}{\mathit{Re}}_{\mathit{x}}^{-1/2}$
0.5	0.5	-	-	-	1.038195	0.845774	−0.244909
-	1.5	-	-	-	0.875339	1.006156	−0.135125
-	2.5	-	-	-	0.619765	1.119215	0.051506
-	-	0.1	-	-	0.952643	0.921192	−0.275204
-	-	0.4	-	-	0.964925	0.960081	−0.040858
-	-	0.7	-	-	0.977579	0.996118	0.157372
-	-	-	2	-	0.964925	0.960081	−0.040858
-	-	-	3	-	0.917302	0.872254	−0.006914
-	-	-	4	-	0.870742	0.808093	0.026700
-	-	-	-	0.2	0.964925	0.960081	−0.040858
-	-	-	-	0.4	0.845168	0.816136	0.047052
-	-	-	-	0.6	0.718793	0.707215	0.140396
1.5	0.5	-	-	-	1.088769	1.082961	−0.275481
-	1.5	-	-	-	0.982632	1.342749	−0.208953
-	2.5	-	-	-	0.879563	1.531233	−0.138016
-	-	0.1	-	-	1.028744	1.187827	−0.323431
-	-	0.4	-	-	1.042463	1.295725	−0.093998
-	-	0.7	-	-	1.052298	1.392372	0.104310
-	-	-	2	-	1.042463	1.295725	−0.093998
-	-	-	3	-	1.047745	1.357226	−0.096997
-	-	-	4	-	1.050866	1.406824	−0.098531
-	-	-	-	0.2	1.042463	1.295725	−0.093998
-	-	-	-	0.4	0.888681	0.987318	0.016635
-	-	-	-	0.6	0.745529	0.804817	0.121383

shows Nusselt number $\{N{u}_{x}R{e}_x^{-1/2}\}$, $\{C{f}_x\sqrt{R{e}_x}\}$ and $\{S{h}_x/\sqrt{R{e}_x}\}$ for numerous values of $M,\Gamma ,W_e,{\lambda }_1$ for scenarios of shearing thinning $(n<1)$ and thickening $(n>1)$. The local Nusselt number decreases with M and ${\lambda }_1$ but increases with the curvature parameter for $(n<1)$ and $(n>1)$. The skin friction and Nusselt number show a dual nature with $W_e$ as both decrease for $(n<1)$, whereas they increase for $(n>1)$. Sherwood’s number decreases with M for dilatant fluid and pseudoplastic fluid. The impact of the magnetic parameter M with the simultaneous effect of velocity slip ${\lambda }_1$ on the velocity field for shearing thinning $(n<1)$ and shear thickening $(n>1)$ situations is shown in Figure 2a,b. We can see from this diagram that the dimensionless velocity falls as the magnetic effect increases in both cases.

The presence of a transverse magnetic intensity in an electrically steering liquid causes the Lorentz force by slowing the flow of the liquid inside the boundary stream area. Physically, the magnetic parameter is connected to the Lorentz force, which is a resistive force for fluid flow. As the magnetic parameter increases, more resistive force is provided, which causes the velocity to decrease. Furthermore, the slip parameter describes the interaction between a fluid and a rigid body. When ${\lambda }_1$ is enhanced, it creates a large gap between the solid and the fluid, thus reducing the fluid’s motion. Hence, the velocity profile decreases with the velocity slip parameter. Figure 3a,b describes the fluctuation of velocity $f\left(\zeta \right)$ with various local Weissenberg number values for dilatant and quasi elastic liquids with the simultaneous effect of velocity slip. When $n<1$, it should be observed that for increasing local Weissenberg number values, the fluid velocity drops but increases in the shear thickening case. The Weissenberg number increases fluid thickness, which reduces velocity, because it measures the fluid’s relaxation time in relation to a particular procedure time. The same trend is shown with the Weissenberg number on the velocity profile for dilatant and quasi-elastic liquids by Sabu et al. [39].

Figure 4a,b expresses the effects of magnetic intensity M on the temperature profile in the presence of thermal slip ${\lambda }_2$ at the boundary in both shearing dilution $(n<1)$ and shear condensation $(n>1)$ scenarios. A transverse magnetic field reduces the fluid’s velocity and greatly slows down the rate of transmission. As M rises, the Lorentz force grows, creating increasing flow resistance. Thus, the thermal boundary layer’s viscosity becomes thicker in both situations. The susceptibility of the fluid flow within the boundary layer is also decreased by raising the thermal slip parameter ${\lambda }_2$, which minimizes the quantity of heat produced and thus shrinks the thermal boundary layer. Figure 5a,b designates the inspiration of the Prandtl number $P_r$ with the simultaneous effect of thermal slip ${\lambda }_2$ on the Carreau fluid temperature profile for situations of shearing dilution $(n<1)$ and shear condensing $(n>1)$. The given result is a good fit with Khan et al. [41].

The Prandtl number is a non-dimensional quantity, which is described as the ratio of two quantities: momentum and thermal diffusivity. Thermal diffusivity diminishes when $P_r$ rises, which further causes a drop in the boundary layer thickness and temperature in both cases. An illustration of the impact of the chemical reaction parameter $C_h$ and effect of concentration slip ${\lambda }_3$ on the concentration profile for both dilatant and pseudoplastic fluids can be seen in Figure 6a,b. A decrease in concentration profile can be observed with increased chemical reaction parameter values. In the case of high $C_h$ values, the reaction rate and molecular diffusivity increase, resulting in a decline in nanoparticle concentration profiles. A decrease in concentration profile can be observed with increased chemical reaction parameter values for $n<1$ and $n>1$. Moreover, because of the sheet’s deleterious nanoparticle intensity elevation, the nanoparticle volume fraction layer lowers as the nanoparticle concentration slip parameter increases. Figure 7a,b illustrates the impact of the bioconvective Lewis number $\left(Lb\right)$ on the concreteness field of gyrotactic motile microorganisms with the simultaneous effect of the microorganism slip parameter ${\lambda }_4$ in both shearing retreating $(n<1)$ and shear condensing $(n>1)$ instances. For each case, the fluid’s motility is decreased when the bioconvection Lewis number is raised, as can be shown.

Digestion behavior is typically instigated by motile microbes that move unceasingly through the liquid while the heat transfer ratio is being executed. Bio-nanoparticles that move to carry out heat transfer are the motile microbes in this case of bioconvection according to Lewis’s equation. The diffusion of microorganisms from the cylinder surface to the main body of the nanofluid is delayed as a result of coupling with the microorganism density gradient. This delay prevents microorganisms from moving from the cylinder surface to the free stream and manifests as a reduction in the thickness of the microorganism species boundary layer with ${\lambda }_4$. Therefore, incorporating microorganism slip at the cylinder surface yields results that are inferior to those noted in no-slip models. Figure 8a,b explores the impact of $W_e$ on skin friction and thermal slip ${\lambda }_2$ on the Nusselt number for scenarios of shearing diminishing $(n<1)$ and shearing condensing $(n>1)$. The mass transfer escalates, but the Nusselt number declines with the effect of the respective parameters. Figure 9a,b show the simultaneous effect of $P_e$ and ${\lambda }_3$ parameters on $\{N{n}_{x}R{e}_x^{-1/2}\}$, which is shown by surface plot. The common effect of these parameters is observed as decreasing the valves of parameters intensifies the $\{N{n}_{x}R{e}_x^{-1/2}\}$; similarly, the impact of ${\lambda }_3$ and M on $\{S{h}_{x}R{e}_x^{-1/2}\}$ is described for the shear thickening $(n>1)$ case.

7. Concluding Remarks

The current research focused on the analysis of two-dimensional incompressible MHD Carreau nanofluid flow across a stretching cylinder containing microorganisms in the presence of chemically reactive agents with impacts of chemical reaction, the heat source, and multiple slip effects. Through similarity transformations, the governing PDEs are reduced to ODEs. With shooting proficiency, the numerical results are determined using the bvp5c approach. We keep providing the quantitative solution for the engineering quantities as various aspects are adapted to the key parameters. Efficacious parameters’ contemporaneous implications on corporeal quantities are portrayed in three-dimensional diagrams.

• The temperature distribution is declining with Prandtl number and thermal slip parameter.
• Nusselt number and local skin friction are intensified with the curvature parameter $\Gamma$ and diminished with the velocity slip parameter for both $n<1$ and $n>1$.
• The concentration profile and motile concentration profile diminish with the chemical reaction factor and $L_b$, respectively.
• Thermal diffusivity declines as the values of $Pr$ increase, creating a drop in boundary layer thickness and temperature in both cases.
• Weissenberg number demonstrates a different nature on the velocity profile, skin friction, and heat transfer; these are reduced for shear diminishing $(n<1)$ and increased when shear condensing $(n>1)$.
• The magnetic parameter M with the simultaneous effect of velocity slip ${\lambda }_1$ creates a decreasing impact on the velocity field for shear thinning and shear thickening.

The parametric implications on the dynamics of the incompressible magnetohydrodynamic Carreau nanofluid flow with multiple slip boundary conditions have been satisfactorily clarified by this successful numerical computational effort. Maxwell dusty nanofluid, Casson nanofluid, and hybrid nanofluid may all be included in this study’s expansion.