Mathematical model for graphene nanofluid flow over a stretching surface with velocity slips thermal convective and mass flux conditions

Abstract

Nowadays graphene is emerging as one of the most exciting nanomaterial due to its continuous 29 electrically conducting behavior even at zero carrier concentration. With this initiation, we investigate the flow of magnetohydrodynamic (MHD) water, water−30%EG, water−50%EG, graphene nanofluid over a stretched surface with thermal convection, and zero mass flux conditions and velocity slips comprising motile microorganisms and nanoparticles. Thermal radiation and Arrhenius activation energy are also be under consideration. The governing fluid equations are solved by Homotopy analysis method (HAM) and computed numerically with shooting technique after employing appropriate transformations. The consequence of numerous physical parameters on velocity, concentration, temperature, and density of motile microorganisms graphs as well as table are used for ethylene glycol based and water-based graphene nanoparticles. Additionally, numerically analyze the designed skin friction, Nusselt number, Sherwood number, and density of motile microorganisms. It is observed that due to heat generation and temperature the improvement of the nonlinear convection variable improves the wall friction. It is also originate that increasing the volume fraction of nanoparticles effectively boosting the thermal conductivity of water−50%EG when compared with water−30%EG and water nanofluids. Ethylene glycol based graphene nanofluids take less time for process as compared to water based nanofluids.

Introduction

Bioconvection occurs when a collection of microorganisms (less dense than water) floats upward to the surface of the liquid, accumulating and concentrating in the upper region until it become dense and unbalanced. Living micro-organisms are usually responsive to the atmosphere, including chemicals, light, and heat. Bioconvection phenomena are substantially applied in biomicrosystems such as enzyme biosensors. ¹ “Khashi et al.” ² examined the flow of $\mathrm{Cu}-\mathrm{Al}2O3/\mathrm{water}$ hybrid nanofluid of motile microorganisms in the combined convection stagnation point flow toward an fixed plate. A mathematical model for examining the heat, mass, and momentum transmission in Bio-convection flow to a rotating plate/cone in a rotating fluid with chemical reaction and thermal radiation were proposed by Raju and Sandeep. ³ De ⁴ investigate the double solution of motile microorganism with water-based nanofluid convoyed by small nanoparticles flowing on non-linearly stretching/shrinking sheet with radiation effect. Biological organism’s show behavior such as oxytaxis (respond to supply of oxygen), gravitaxis (respond to gravity), and gyrotaxis (respond to gravity and viscous torques). On other hand, taxis refers to the microbes migration to a stimulant source. Gyrotactic motile microorganism like Algae, can improve uncertainty of nanofluid. ⁵ Imran et al. ⁶ surveyed the role of a bioconvection of swimming microorganisms nanofluid with cross viscosity through a cylinder. In the presence of magnetic field, the study of the motion of electrically conducting fluid is known as magnetohydrodynamics. The magneto-hydrodynamic Carreau nano-liquid flow is proposed by “Reedy et al.” ⁷ Nanofluid was first defined by “Choi” ⁸ as a mixture of scattered nanoscale-sized particles in a base fluid. Choi said that by mixing nanoparticles with base fluid can make their thermal conductivity better.

Nanofluids are used as a coolant agent and lubricants, many applications such as mobile computer processors, air conditioning, refrigeration, and microelectronics. The model proposed by “Buongiorno” ⁹ and Tiwari and Das ¹⁰ considered two forms of nanofluid models in fluid dynamic. “Upadhya et al.” ¹¹ review that water based nanofluid take more time for process than ethylene glycol based graphene nanofluid. Sreedevi et al. ¹² analyze the Unsteady MHD mass and heat transfer analysis of hybrid nanofluid flow over extending surface with slip effects, chemical reaction, thermal radiation, and suction. Nadeem et al. ¹³ the steady flow of a Jeffrey fluid model in the presence of nano particles. “Gul and Firdous” ¹⁴ investigate the experimental work to scrutinize the stable scattering of the nanoparticles graphene and to look at nanofluid GO–H₂O flow between two rotating disks. The combination of water–ethylene glycol (50:50) as a base fluid and Aluminum Oxide $(A{l}_2{O}_3)$ and Magnetic Oxide $(F{e}_3{O}_4)$ as the nanoparticles, “Iqbal et al.” ¹⁵ explored the natural convection flow around a heated vertical spinning cone under the effect of a magnetic field. “Shankar Goud et al.” ¹⁶ utilized analysis of finite element to examine unsteady MHD free convection Casson fluid flow affected by heat over a vertical oscillating porous plate. “Zainal et al.” ¹⁷ observed at the numerical analysis of the magneto-hydrodynamic $(A{l}_2{O}_3,\mathrm{Cu})/H_{2}O$ hybrid nanofluids flow toward a porous stretching/shrinking sheet when the velocity was quadratic. “Krishna et al.” ¹⁸ studied the chemical reaction effect on MHD Casson fluid flow with a permeable stretch sheet.

Further researchers, are still searching better fluids to replace nanofluid. Some types of nanofluid known as water, water−30%EG, water−50%EG based graphene nanofluid are introduced with better thermal conductivity. Transformer cooling, Nuclear system cooling, coolant in machining, refrigeration, electronic cooling, biomedical drug reduction, and other disciplines of heat transfer have found water, water−30%EG, water−50%EG based graphene nanofluid to be more efficient than nanofluid. Researchers have worked on water, water−30%EG, water−50%EG based graphene nanofluid in real world applications. “Labib et al.” ¹⁹ quantitatively studied the influence of base fluids such as water, water−30%EG, water−50%EG based graphene nanofluid in forced convective heat transfer.

“Khan et al.” ²⁰ discussed that according to the mechanism that causes their impellent motile bacteria can be divided into different groups, with chemotactic, gyrotactic, gravitaxis, and oxytactic. In response to stimuli like self-propelled, light, gravity, and chemical attraction, motile biological organism may dynamically swim in the fluid. To make a extensive range of industrial and commercial things microbes are employed, with fertilizers, waste-derived biofuel and medicine delivery. ²¹ The effect over nanofluid bio-convection of homogeneous magnetic field over a porous vertical sheet was proposed by “Mutuku and Makinde.” ²² Motile microorganisms are denser than water. Under some conditions, these features cause instability of hydrodynamic, which result bio-convection flow. ²³

The key problem of bio-convection solutions having gyrotactic microbes and small solid particles were stated by Kuznetsov and Avramenko. ²⁴ Kuznetsov and Bubnovich, ²⁵ studied the combined effect of gyrotactic microorganism on nanofluid. Biothermal convection in perforated medium was calculated by “Chakraborty et al.” ²⁶ They verified that fluid velocity decreases with increase in magnetic field. Xu and Pop, ²⁷ studied the combined convective nanofluid flow in the existence of gyrotactic microorganism and nanoparticle on a stretching surface. Zabihi et al. ²⁸ used HAM procedure on the application of variation of parameter’s method for hydrothermal analysis on MHD squeezing nanofluid flow in parallel plates.

The aim of current analysis, is to examine the MHD Arrhenius energy of water, water−30%EG, water−50%EG based graphene nanofluids flowing over stretched surface in the existence of thermal radiation and motile microorganism. The influence of zero mass flux condition, thermal convection, and velocity slip are taken into Consideration. The transformed equations are solved by HAM and are compared with shooting method. The impacts of distinct physical parameters on velocity, temperature, nanoparticle concentration, and density of motile microorganisms is designed using Mathematica by which a thorough study have been made to analyze the effects different parameter on it. Furthermore, the intentional values for skin friction, Sherwood number, density of motile microbes, and Nusselt number are scrutinized numerically. From this study it is observed that as compared to water based nanofluids, ethylene glycol-based graphene nanofluids takes less time for execution. This is because of the thermal conductivity of ethylene glycol-based graphene nanofluid is greater than water based nanofluid.

Formulation

Let us consider two-dimensional incompressible magnetohydrodynamics Casson nanofluid flow over a stretched surface. A strength of magnetic field $B=(0,B,0)$ is considered perpendicular in the direction of flow. The thermal radiation, thermophoresis, heat source/sink, and Brownian motion are taken in heat equation. In concentration equation, Arrhenius activation energy and binary chemical reaction are considered. The motile microorganisms are considered into attention. ² The modeled equations are:

Continuity equation

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

(1)

Momentum equation

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{{\mu }_{\mathrm{nf}}}{{\mathrm{\rho }}_{\mathrm{nf}}}\frac{{\partial }^{2}u}{\partial y^2}-\frac{{\sigma }_{\mathrm{nf}}}{{\mathrm{\rho }}_{\mathrm{nf}}}{B}_0^{2}u-\frac{\nu }{K}u$$

(2)

Energy equation

$$\begin{array}{c}u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{\mathrm{nf}}}{{\left(\mathrm{\rho }{C}_p\right)}_{\mathrm{nf}}}\frac{{\partial }^{2}T}{\partial y^2}-\frac{1}{{\left(\mathrm{\rho }{C}_p\right)}_{\mathrm{nf}}}\frac{\partial q_r}{\partial y}+\frac{{\left(\mathrm{\rho }C\right)}_p}{{\left(\mathrm{\rho }C\right)}_f}\hfill \\ [D_B\frac{\partial T}{\partial y}\frac{\partial C}{\partial y}-\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial y}\right)}^2]+\frac{q^m}{\mathrm{\rho }{C}_p}\hfill \end{array}$$

(3)

Nanoparticle’s concentration equation

$$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D_B\frac{{\partial }^{2}C}{\partial y^2}+\frac{D_T}{T_{\infty }}\frac{{\partial }^{2}T}{\partial y^2}-k_c(C-C_{\infty }){\left(\frac{T}{T_{\infty }}\right)}^n{e}^{(-\frac{E_a}{\mathrm{kT}})}$$

(4)

Density of gyrotactic microorganism

$$u\frac{\partial N}{\partial x}+v\frac{\partial N}{\partial y}=D_m\frac{{\partial }^{2}N}{\partial y^2}-\frac{B_c{v}_c}{(C_w-C_{\infty })}-\frac{\partial }{\partial y}\left(N\frac{\partial C}{\partial y}\right),$$

(5)

(BCs),

$$\begin{array}{c}u=u_w=\mathrm{ax},v=0,T=T_w=T_0+b_{1}x,C=C_w=C_0\hfill \\ +d_{1}x,N=N_w=N_0+e_{1}x\mathrm{at}y=0\hfill \\ u\to 0,T=T_{\infty }=T_0+b_{2}x,C=C_{\infty }=C_0+d_{2}x,\hfill \\ N=N_{\infty }=N_0+e_{2}x\mathrm{as}y\to \infty \hfill \end{array}$$

(6)

As shown above, the quantity $U_{\mathrm{slip}}$ describes the slip velocity, and is defined as:

$$U_{\mathrm{slip}}={\mathrm{\Delta }}_1\frac{\partial u}{\partial y}+{\mathrm{\Delta }}_2\frac{{\partial }^{2}u}{\partial y^2}+{\mathrm{\Delta }}_3\frac{{\partial }^{3}u}{\partial y^3}$$

(7)

In the above equations, $u$ and $v$ are the velocities along the x-y-axes, correspondingly. $T$ represents the nanofluid temperature, $C$ represents nanoparticle concentration, and $N$ represents microorganisms’ density. $T_{\infty }$ represents nanofluid ambient temperature, $C_{\infty }$ represents nanoparticles ambient concentration, and $N_{\infty }$ is the ambient concentration of microorganisms, $T_w$ is the nanofluid wall temperature, $C_w$ is the concentration of nanoparticles of bound-surface, $N_w$ is microorganism concentration at the surface, $D_m$ microorganism Diffusivity, $D_T$ and $D_B$ denote the thermophoresis and Brownian diffusion coefficient, $k_c$ represents thermal reaction coefficient. $T_w=T_0+b_{1}x$ where $b_1$ is the characteristic nanofluid temperature with $b_1>0(T_w>T_0)$ for supporting flow and $b_1<0(T_w<T_0)$ for opposite flow, correspondingly, where $q^m$ is the interior heat absorption/generation per unit volume.

$$q^m=\frac{k_{\mathrm{nf}}{U}_w\left(x\right)}{x\nu }[A^{*}(T_w-T_{\infty })+B^{*}(T-T\infty )]$$

(8)

Where $B^{*}$ and $A^{*}$ are temperature-dependent and space-dependent heat generation/absorption parameters. Furthermore, the thermophysical properties of nanofluids, with their ${\mu }_{\mathrm{nf}}$ dynamic viscosity, ${\mathrm{\rho }}_{\mathrm{nf}}$ density, $k_{\mathrm{nf}}$ thermal conductivity, $(\mathrm{\rho }{c}_p{)}_{\mathrm{nf}}$ heat capacity, ${\sigma }_{\mathrm{nf}}$ microorganisms’ concentration different parameter.

The following represents the thermophysical properties:

$$\begin{array}{c}\multicolumn{1}{c}{\frac{{\mu }_{\mathrm{nf}}}{{\mu }_f}=\frac{1}{{(1-Ø)}^{2.5}},\frac{{\mathrm{\rho }}_{\mathrm{nf}}}{{\mathrm{\rho }}_f}=(1-Ø)+\mathrm{\Gamma }\frac{{\mathrm{\rho }}_{\mathrm{np}}}{{\mathrm{\rho }}_f},\frac{{\left(\mathrm{\rho }{C}_p\right)}_{\mathrm{nf}}}{{\left(\mathrm{\rho }{C}_p\right)}_f}}\\ =(1-Ø)+\mathrm{\Gamma }\frac{{\left(\mathrm{\rho }{C}_p\right)}_{\mathrm{np}}}{{\left(\mathrm{\rho }{C}_p\right)}_f},\hfill \\ \frac{K_{\mathrm{nf}}}{K_f}=\frac{(K_{\mathrm{np}}+2K_f)-2Ø(K_f-K_{\mathrm{np}})}{(K_{\mathrm{np}}+2K_f)+Ø(K_f-K_{\mathrm{np}})},\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}\hfill \\ =1+\frac{3(\frac{{\sigma }_{\mathrm{np}}}{{\sigma }_f}-1)Ø}{(\frac{{\sigma }_{\mathrm{np}}}{{\sigma }_f}+2)-(\frac{{\sigma }_{\mathrm{np}}}{{\sigma }_f}-1)Ø}\hfill \end{array}$$

(9)

Where $\mathrm{\Gamma }$ represent volume fraction of nanoparticles, $\mu$ shows dynamic viscosity, $\mathrm{\rho }$ represent density and $C_p$ represent specific heat, $K$ represent thermal conductivity, and $\sigma$ represent electrical conductivity. The subscripts $\mathrm{bf}$ , $\mathrm{np}$ , $\mathrm{and}$ $\mathrm{nf}$ signify the base fluid, nanoparticles, and nanofluid, correspondingly (Table 1).

Table 1.

Thermophysical characteristic of the graphene base and fluids.

Thermophysical properties	$\mathrm{Pr}$	$k$	$\sigma$	$C_p$	$\mathrm{\rho }$
Water	$6.2$	$0.613$	$0.01$	$4179$	$997.1$
Water + 30%EG	$16.62$	$0.448$	$0.00276$	$3714$	$1038$
Water + 50%EG	$29.86$	$0.425$	0.00509	$3288$	$1056$
Graphene	–	2500	$1\times$ 107	2100	2250

$q_r$ represents radiative heat flux, Roseland approximation expressed by radiative heat flux

$$q_r=-\frac{4{\sigma }^{*}}{3{\alpha }^{*}}\frac{\partial T^4}{\partial y}$$

(10)

Where ${\sigma }^{*}$ shows Boltzmann constant and ${\alpha }^{*}$ shows mean absorption coefficient, correspondingly. If the differences in temperature is small, the term $T^4$ can be written as a linear function for a Taylor series around $T_{\infty }$ , neglecting high order terms, it produces

$$T^4=4T_{\infty }^{3}T-3T_{\infty }^4$$

(11)

Chemotaxis constant are represented by $B_c$ in equation (5), and $v_c$ represents cell swim speed $\frac{B_c{v}_c}{c_w-c_{\infty }}$ represents velocity to swimming cell, using below similarity transformation equations (2)–(5), are converted into a system of differential equations.

$$\left\{\begin{array}{c}u=\mathrm{ax}{f}^{\prime }\left(\xi \right),v=-\sqrt{a{\upsilon }_f}f\left(\xi \right),\theta \left(\xi \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\phi \left(\xi \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }},\hfill \\ \multicolumn{1}{c}{\chi \left(\xi \right)=\frac{N-N_{\infty }}{N_w-N_{\infty }},\xi =\sqrt{\frac{a}{{\upsilon }_f}}y}\end{array}\right\}$$

(12)

Where prime denote differentiation with respect to $\xi$ , $\mathrm{\Delta }T=T_w-T_{\infty },\mathrm{\Delta }C=C_w-C_{\infty },\mathrm{\Delta }N=N_w-N_{\infty }$ .

When the similarities variable $(12)$ is put in equations $(2-5),$ the equation of continuity $(1)$ , is satisfied automatically.

Therefore, the simplest form of the model is given

$$\begin{array}{c}\frac{{\mu }_{\mathrm{nf}}}{{\mu }_f}(1+\frac{1}{\Lambda })f″\prime \left(\xi \right)-\frac{{\mathrm{\rho }}_{\mathrm{nf}}}{{\mathrm{\rho }}_f}(M+K_1)f\prime \left(\xi \right)\hfill \\ +\frac{{\mathrm{\rho }}_{\mathrm{nf}}}{{\mathrm{\rho }}_f}(f\left(\xi \right)f″\left(\xi \right)-{(f\prime \left(\xi \right))}^2)=0\hfill \end{array}$$

(13)

$$\begin{array}{c}(\frac{k_{\mathrm{nf}}}{k_f}+\frac{4}{3}R)\theta ″\left(\xi \right)+\frac{{\left(\mathrm{\rho }{C}_p\right)}_{\mathrm{nf}}}{{\left(\mathrm{\rho }{C}_p\right)}_f}\mathrm{Pr}\hfill \\ (N_b\theta \prime \left(\xi \right)\phi \prime \left(\xi \right)+N_t{(\theta \prime \left(\xi \right))}^2+f\left(\xi \right)\theta \prime \left(\xi \right))\hfill \\ +\frac{{\left(\mathrm{\rho }{C}_p\right)}_{\mathrm{nf}}}{{\left(\mathrm{\rho }{C}_p\right)}_f}\left(\frac{k_{\mathrm{nf}}}{k_f}(A^{*}+B^{*}\theta )\right)=0\hfill \end{array}$$

(14)

$$\begin{array}{c}\phi ″\left(\xi \right)+\frac{N_t}{N_b}\theta ″\left(\xi \right)+\mathrm{scf}\left(\xi \right)\phi \prime \left(\xi \right)\hfill \\ -\mathrm{sc}{\sigma }_R{(1+\delta \theta )}^n{e}^{-\left(\frac{E}{1+\delta \theta }\right)}\phi \left(\xi \right)=0\hfill \end{array}$$

(15)

$$\chi ″\left(\xi \right)+\mathrm{Lbf}\left(\xi \right)\chi \prime \left(\xi \right)-\mathrm{pe}(\phi ″\left(\xi \right)(\chi +\mathrm{\Omega })+\chi \prime \left(\xi \right)\phi \prime \left(\xi \right))=0$$

(16)

With (BCs),

$$\begin{array}{c}[f(0)=0,f\prime (0)=1+\alpha f″(0)+\beta f″\prime (0)+\nu f^{\mathrm{iv}}(0),f^{\prime }(\hslash )=0].\hfill \\ \mathrm{where}\alpha ={\mathrm{\Delta }}_1\sqrt{\frac{a}{{\nu }_f}},\beta ={\mathrm{\Delta }}_2\frac{a}{{\nu }_f},\gamma ={\mathrm{\Delta }}_3\frac{a}{{\nu }_f}\sqrt{\frac{a}{{\nu }_f}}\hfill \\ [\frac{k_{\mathrm{nf}}}{k_f}{\theta }^{\prime }\left(0\right)=\sigma (\theta \left(0\right)-1),\theta \left(\infty \right)=0],\mathrm{where}\sigma =\frac{h_f}{k_f}\sqrt{\frac{{\nu }_f}{a}}\hfill \\ [\mathrm{Nb}{\phi }^{\prime }\left(0\right)+\mathrm{Nt}{\theta }^{\prime }(0)=0,\phi \left(\infty \right)=0],\mathrm{where}\hfill \\ \mathrm{Nb}=\frac{a{D}_{B}C\infty }{{\nu }_f{\delta }_c},\mathrm{Nt}=\frac{a{D}_T(T_w-T_{\infty })}{{\nu }_f{T}_{\infty }}\hfill \\ [\chi \left(0\right)=1,\chi \left(\infty \right)=0]\hfill \end{array}$$

(17)

In the given equations, $\mathrm{Pr}$ represents prandtl number, ²⁹ $\mathrm{Pe}$ represents bioconvection peclet number, $R$ represents thermal radiation parameter, $\mathrm{Lb}$ represents bioconvection Lewis’s parameter, $\mathrm{Nt}$ represents thermophoresis parameter, $\mathrm{Nb}$ represents Brownian motion parameter, $\mathrm{Sc}$ represents Schmidt number, $B$ represents motile density stratification, Ω represents difference parameter of microorganism concentration.

$$\begin{array}{c}\mathrm{Pr}=\frac{k_f}{{\upsilon }_f{\left(\mathrm{\rho }{C}_p\right)}_f},\mathrm{Pe}=\frac{B_c{v}_c}{D_m},\mathrm{\Omega }=\frac{N_{\infty }}{N_w-N_0},B=\frac{e_2}{e_1},\hfill \\ \mathrm{Lb}=\frac{{\upsilon }_f}{D_m},\mathrm{Nb}=\frac{{\left(\mathrm{\rho }{C}_p\right)}_p{D}_B(C_w-C_{\infty })}{{\left(\mathrm{\rho }{C}_p\right)}_f{\upsilon }_f},\hfill \\ \mathrm{Nt}=\frac{{\left(\mathrm{\rho }{C}_p\right)}_p{D}_T(T_w-T_{\infty })}{{\left(\mathrm{\rho }{C}_p\right)}_f{T}_{\infty }{\upsilon }_f},\mathrm{Sc}=\frac{{\upsilon }_f}{D_B},R=\frac{4{\sigma }^{*}{T}_{\infty }^3}{3k^{*}{k}_f},{\sigma }_R=\frac{k_c}{a},\hfill \\ \delta =\frac{T_w-T_{\infty }}{T_{\infty }},M=\frac{{\sigma }_{\mathrm{hnf}}{B}_0^2}{a{\mathrm{\rho }}_{\mathrm{hnf}}},E=\frac{E_a}{k{T}_{\infty }},K_1=\frac{\nu }{\mathrm{aK}}\hfill \end{array}$$

Quantities of interest

$$\begin{array}{c}C{f}_x=\frac{{\tau }_w}{{\mathrm{\rho }}_f{u}_w^2},N{u}_x=\frac{x{q}_w}{k_f(T_w-T_{\infty })},S{h}_x=\frac{x{q}_m}{D_B(C_w-C_{\infty })},\hfill \\ N{h}_x=\frac{x{q}_n}{D_m(N_w-N_{\infty })}\hfill \end{array}$$

(18)

Where

$$\begin{array}{c}{\tau }_w={\mu }_{\mathrm{nf}}(1+\frac{1}{\Lambda })\frac{\partial u}{\partial y}{|}_{y\to 0},q_w=-k_{\mathrm{nf}}(1+\frac{16}{3}\frac{{\sigma }^{*}{T}_{\infty }}{k{k}^{*}})\frac{\partial T}{\partial y}{|}_{y\to 0},\hfill \\ q_m=D_B\frac{\partial C}{\partial y}{|}_{y\to 0},q_n=D_m\frac{\partial N}{\partial y}{|}_{y\to 0}\hfill \end{array}$$

(19)

$$\begin{array}{c}\sqrt{{\mathrm{Re}}_x}C{f}_x=(1+\frac{1}{\mathrm{\Lambda }})f″\left(0\right),\frac{1}{\sqrt{{\mathrm{Re}}_x}}N{u}_x=-(k_{\mathrm{nf}}+\frac{4}{3}R)\theta \prime \left(0\right)\hfill \\ \frac{1}{\sqrt{{\mathrm{Re}}_x}}S{h}_x=-\phi \prime \left(0\right),\frac{1}{\sqrt{{\mathrm{Re}}_x}}N{h}_x=-\chi \prime \left(0\right)\hfill \end{array}$$

Solution by HAM

The HAM are used to resolve equations (12)–(15), for given (BCs) $(16)$ . The solutions is to adjust supplementary parameters and control how near the results are to each other.

The initial suppositions are nominated as follow:

$$\begin{array}{c}{f}_0\left(\xi \right)=\frac{1}{1+\alpha -\beta +\gamma }(1-e^{-\xi }),{\theta }_0\left(\xi \right)=\frac{\sigma K_f}{K_{\mathrm{nf}}+\sigma K_f}{e}^{-\xi },\hfill \\ {\varphi }_0\left(\xi \right)=\frac{\mathrm{Nt}\sigma K_f}{\mathrm{Nb}(K_{\mathrm{nf}}+\sigma K_f)}{e}^{-\xi },{\chi }_0\left(\xi \right)=e^{-\xi }\hfill \end{array}$$

(20)

Take the linear operators as $L_f,L_{\theta },L_{\varphi },\mathrm{and}{L}_{\chi }$

$$L_f\left(f\right)=f^{″\prime }-f^{\prime },L_{\theta }\left(\theta \right)={\theta }^{″}-\theta ,L_{\varphi }\left(\varphi \right)={\varphi }^{″}-\varphi ,L_{\chi }\left(\chi \right)={\chi }^{″}-\chi$$

(21)

Which have these characteristics:

$$\begin{array}{c}{L}_f(d_1+d_2{e}^{-\xi }+d_3{e}^{-\xi })=0,L_{\theta }(d_4{e}^{-\xi }+d_5{e}^{-\xi })=0\hfill \\ L_{\varphi }(d_6{e}^{-\xi }+d_7{e}^{-\xi })=0,L_{\chi }(d_8{e}^{-\xi }+d_9{e}^{-\xi })=0\hfill \end{array}$$

(22)

Where the constants in general solution are $d_j(j=1-9)$ :

The nonlinear operations that resulted $N_f,N_{\theta },N_{\phi },\mathrm{and}{N}_{\chi }$ are given as:

$$\begin{array}{c}\multicolumn{1}{c}{N_f\left[f(\xi ;\pounds )\right]=\frac{{\mu }_{\mathrm{nf}}}{{\mu }_f}(1+\frac{1}{\beta })\frac{{\partial }^{3}f(\xi ;\pounds )}{\partial {\xi }^3}-\frac{{\mathrm{\rho }}_{\mathrm{nf}}}{{\mathrm{\rho }}_f}{\left(\frac{\partial f(\xi ;\pounds )}{\partial \xi }\right)}^2}\\ +\frac{{\mathrm{\rho }}_{\mathrm{nf}}}{{\mathrm{\rho }}_f}f(\xi ;\pounds )\frac{{\partial }^{2}f(\xi ;\pounds )}{\partial {\xi }^2}-\frac{{\mathrm{\rho }}_{\mathrm{nf}}}{{\mathrm{\rho }}_f}(M+K_1)\frac{\partial f(\xi ;\pounds )}{\partial \xi }\hfill \end{array}$$

(23)

$$\begin{array}{c}{N}_{\theta }[f(\xi ;\pounds ),\theta (\xi ;\pounds ),\varphi (\xi ;\pounds )]=(\frac{K_{\mathrm{nf}}}{K_f}+\frac{4}{3}R)\frac{1}{\mathrm{Pr}}\frac{{\partial }^2\theta (\xi ;\pounds )}{\partial {\xi }^2}\hfill \\ +\frac{{\left(\mathrm{\rho }{C}_p\right)}_{\mathrm{nf}}}{{\left(\mathrm{\rho }{C}_p\right)}_f}f(\xi ;\pounds )\frac{\partial \theta (\xi ;\pounds )}{\partial \xi }+\frac{{\left(\mathrm{\rho }{C}_p\right)}_{\mathrm{nf}}}{{\left(\mathrm{\rho }{C}_p\right)}_f}{N}_b\frac{\partial \theta (\xi ;\pounds )}{\partial \xi }\frac{\partial \varphi (\xi ;\pounds )}{\partial \xi }\hfill \\ +\frac{{\left(\mathrm{\rho }{C}_p\right)}_{\mathrm{nf}}}{{\left(\mathrm{\rho }{C}_p\right)}_f}{N}_t{\left(\frac{\partial \theta (\xi ;\pounds )}{\partial \xi }\right)}^2+\frac{{\left(\mathrm{\rho }{C}_p\right)}_{\mathrm{nf}}}{{\left(\mathrm{\rho }{C}_p\right)}_f}\left(\frac{K_{\mathrm{nf}}}{K_f}(A^{*}+B^{*}\theta (\xi ;\pounds ))\right)\hfill \end{array}$$

(24)

$$\begin{array}{c}{N}_{\varphi }[f(\xi ;\pounds ),\theta (\xi ;\pounds ),\varphi (\xi ;\pounds )]=\frac{{\partial }^2\varphi (\xi ;\pounds )}{\partial {\xi }^2}+\frac{N_t}{N_b}\frac{{\partial }^2\theta (\xi ;\pounds )}{\partial {\xi }^2}\hfill \\ +\mathrm{Scf}(\xi ;\pounds )\frac{\partial \varphi (\xi ;\pounds )}{\partial \xi }-\mathrm{Sc}{\sigma }_r(\varphi (\xi ;\pounds )+\delta \theta (\xi ;\pounds )\varphi (\xi ;\pounds )-E\varphi (\xi ;\pounds ))\hfill \end{array}$$

(25)

$$\begin{array}{c}{N}_{\chi }[f(\xi ;\pounds ),\varphi (\xi ;\pounds ),\chi (\xi ;\pounds )]=\frac{{\partial }^2\chi (\xi ;\pounds )}{\partial {\xi }^2}+\mathrm{Lbf}(\xi ;\pounds )\frac{\partial \chi (\xi ;\pounds )}{\partial \xi }\hfill \\ -\mathrm{Pe}(\frac{\partial \varphi (\xi ;\pounds )}{\partial \xi }\frac{\partial \chi (\xi ;\pounds )}{\partial \xi }+\frac{{\partial }^2\varphi (\xi ;\pounds )}{\partial {\xi }^2}\chi (\xi ;\pounds )+\mathrm{\Omega }\frac{{\partial }^2\varphi (\xi ;\pounds )}{\partial {\xi }^2})\hfill \end{array}$$

(26)

The basic application of HAM is defined in references.^30–35 From equations (12) to (15), The zero-order problems are:

$$(1-\pounds )L_f[f(\xi ;\pounds )-f_0\left(\xi \right)]=\pounds {\hslash }_f{N}_f\left[f(\xi ;\pounds )\right]$$

(27)

$$(1-\pounds )L_{\theta }[\theta (\xi ;\pounds )-{\theta }_0\left(\xi \right)]=\pounds {\hslash }_{\theta }{N}_{\theta }[f(\xi ;\pounds ),\theta (\xi ;\pounds ),\varphi (\xi ;\pounds )]$$

(28)

$$(1-\pounds )L_{\varphi }[\varphi (\xi ;\pounds )-{\varphi }_0\left(\xi \right)]=\pounds {\hslash }_{\varphi }{N}_{\varphi }[f(\xi ;\pounds ),\theta (\xi ;\pounds ),\varphi (\xi ;\pounds )]$$

(29)

$$(1-\pounds )L_{\chi }[\chi (\xi ;\pounds )-{\chi }_0\left(\xi \right)]=\pounds {\hslash }_{\chi }{N}_{\chi }[f(\xi ;\pounds ),\varphi (\xi ;\pounds ),\chi (\xi ;\pounds )]$$

(30)

The equivalent boundary conditions are:

$$\begin{array}{c}f(\xi ;\pounds ){|}_{\xi \to 0}=0,\frac{\partial f(\xi ;\pounds )}{\partial \xi }{|}_{\xi \to 0}-\alpha \frac{{\partial }^{2}f(\xi ;\pounds )}{\partial {\xi }^2}{|}_{\xi \to 0}-\beta \frac{{\partial }^{3}f(\xi ;\pounds )}{\partial {\xi }^3}{|}_{\xi \to 0}\hfill \\ -\gamma \frac{{\partial }^{4}f(\xi ;\pounds )}{\partial {\xi }^4}=1,\frac{\partial f(\xi ;\pounds )}{\partial \xi }{|}_{\xi \to \infty }=0\hfill \\ \frac{K_{\mathrm{nf}}}{K_f}\frac{\partial \theta (\xi ;\pounds )}{\partial \xi }{|}_{\xi \to 0}=\mathrm{\sigma }(\theta (\xi ;\pounds ){|}_{\xi \to 0}-1),\theta (\xi ;\pounds ){|}_{\xi \to \infty }=0\hfill \\ N_b\frac{\partial \varphi (\xi ;\pounds )}{\partial \xi }{|}_{\xi \to 0}+N_t\frac{\partial \theta (\xi ;\pounds )}{\partial \xi }{|}_{\xi \to 0}=0,\varphi (\xi ;\pounds ){|}_{\xi \to \infty }=0\hfill \\ \chi (\xi ;\pounds ){|}_{\xi \to 0}=1,\chi (\xi ;\pounds ){|}_{\xi \to \infty }=0\hfill \end{array}$$

(31)

Wherever $\pounds \in [0,1]$ represents the imbedding parameter, ${\hslash }_f,{\hslash }_{\theta },{\hslash }_{\varphi }$ , and ${\hslash }_{\chi }$ are used to control the convergence of solution. Whenever $\pounds =0$ and $\pounds =1$ we have:

$$f(\xi ;1)=f\left(\xi \right),\theta (\xi ;1)=\theta \left(\xi \right),\varphi (\xi ;1)=\varphi \left(\xi \right),\chi (\xi ;1)=\chi \left(\xi \right)$$

(32)

Expanding $f(\xi ;\pounds ),\theta (\xi ;\pounds ),\varphi (\xi ;\pounds ),\chi (\xi ;\pounds )$ in Taylor’s series about $\pounds =0$

$$\begin{array}{c}f(\xi ;\pounds )=f_0\left(\xi \right)+\sum _{n=1}^{\infty }{f}_t\left(\xi \right){\pounds }^t\hfill \\ \theta (\xi ;\pounds )={\theta }_0\left(\xi \right)+\sum _{n=1}^{\infty }{\theta }_t\left(\xi \right){\pounds }^t\hfill \\ \varphi (\xi ;\pounds )={\varphi }_0\left(\xi \right)+\sum _{n=1}^{\infty }{\varphi }_t\left(\xi \right){\pounds }^t\hfill \\ \chi (\xi ;\pounds )={\chi }_0\left(\xi \right)+\sum _{n=1}^{\infty }{\chi }_t\left(\xi \right){\pounds }^t\hfill \end{array}$$

(33)

Where

$$\begin{array}{c}{f}_t\left(\xi \right)=\frac{1}{t!}\frac{\partial f(\xi ;\pounds )}{\partial \xi }{|}_{\pounds \to 0},{\theta }_t\left(\xi \right)=\frac{1}{t!}\frac{\partial \theta (\xi ;\pounds )}{\partial \xi }{|}_{\pounds \to 0}\hfill \\ {\varphi }_t\left(\xi \right)=\frac{1}{t!}\frac{\partial \varphi (\xi ;\pounds )}{\partial \xi }{|}_{\pounds \to 0},{\chi }_t\left(\xi \right)=\frac{1}{t!}\frac{\partial \chi (\xi ;\pounds )}{\partial \xi }{|}_{\pounds \to 0}\hfill \end{array}$$

(34)

$\begin{array}{c}\mathrm{The}\mathrm{secondary}\mathrm{constraints}{\hslash }_f,{\hslash }_{\theta },{\hslash }_{\varphi }\mathrm{and}{\hslash }_{\chi }\mathrm{are}\mathrm{chosen}\hfill \\ \mathrm{so}\mathrm{that}\mathrm{the}\mathrm{series}(33)\mathrm{converges}\mathrm{to}\pounds =1.\mathrm{Substituting}\pounds =1\hfill \\ \mathrm{into}(33),\mathrm{we}\mathrm{get}:\hfill \end{array}$

$$\begin{array}{c}f\left(\xi \right)=f_0\left(\xi \right)+\sum _{n=1}^{\infty }{f}_t\left(\xi \right)\hfill \\ \theta \left(\xi \right)={\theta }_0\left(\xi \right)+\sum _{n=1}^{\infty }{\theta }_t\left(\xi \right)\hfill \\ \varphi \left(\xi \right)={\varphi }_0\left(\xi \right)+\sum _{n=1}^{\infty }{\varphi }_t\left(\xi \right)\hfill \\ \chi \left(\xi \right)={\chi }_0\left(\xi \right)+\sum _{n=1}^{\infty }{\chi }_t\left(\xi \right)\hfill \end{array}$$

(35)

The nth-order problem has the following characteristics:

$$\begin{array}{c}{L}_f[f_t\left(\xi \right)-{\chi }_t{f}_{t-1}\left(\xi \right)]={\hslash }_f{R}_t^f\left(\xi \right)\hfill \\ L_{\theta }[{\theta }_t\left(\xi \right)-{\chi }_t{\theta }_{t-1}\left(\xi \right)]={\hslash }_{\theta }{R}_t^{\theta }\left(\xi \right)\hfill \\ L_{\varphi }[{\varphi }_t\left(\xi \right)-{\chi }_t{\varphi }_{t-1}\left(\xi \right)]={\hslash }_{\varphi }{R}_t^{\varphi }\left(\xi \right)\hfill \\ L_{\chi }[{\chi }_t\left(\xi \right)-{\chi }_t{\chi }_{t-1}\left(\xi \right)]={\hslash }_{\chi }{R}_t^{\chi }\left(\xi \right)\hfill \end{array}$$

(36)

The related boundary conditions are:

$$\begin{array}{c}{f}_t\left(0\right)=0,f_t^{\prime }\left(0\right)-\alpha f_t^{″}\left(0\right)-\beta f_t^{″\prime }\left(0\right)-\gamma f_t^{\mathrm{iv}}\left(0\right)=1,\hfill \\ f_t\left(\infty \right)=0\hfill \\ \frac{K_{\mathrm{nf}}}{K_f}{\theta }_t^{\prime }\left(0\right)-\sigma {\theta }_t\left(0\right)=0,{\theta }_t\left(\infty \right)=0\hfill \\ N_b{\varphi }_t^{\prime }(0)+N_t{\theta }_t^{\prime }\left(0\right)=0,{\varphi }_t\left(\infty \right)=0\hfill \\ {\chi }_t\left(0\right)=0,{\chi }_t\left(\infty \right)=0\hfill \end{array}$$

(37)

Here

$$\begin{array}{c}\multicolumn{1}{c}{R_t^f\left(\xi \right)=\frac{{\mu }_{\mathrm{nf}}}{{\mu }_f}(1+\frac{1}{\beta })f_{t-1}^{″\prime }-\frac{{\mathrm{\rho }}_{\mathrm{nf}}}{{\mathrm{\rho }}_f}\sum _{k=0}^{t-1}{f}_{t-1-k}^{\prime }{f}_k^{\prime }}\\ +\frac{{\mathrm{\rho }}_{\mathrm{nf}}}{{\mathrm{\rho }}_f}\sum _{k=0}^{t-1}{f}_{t-1-k}{f}_k^{″}-\frac{{\mathrm{\rho }}_{\mathrm{nf}}}{{\mathrm{\rho }}_f}(M+K_1)f_{t-1}^{\prime }\hfill \end{array}$$

(38)

$$\begin{array}{c}\multicolumn{1}{c}{R_t^{\theta }\left(\xi \right)=(\frac{K_{\mathrm{nf}}}{K_f}+\frac{4}{3}R)\frac{1}{\mathrm{Pr}}{\theta }_{t-1}^{″}+\frac{{\left(\mathrm{\rho }{C}_p\right)}_{\mathrm{nf}}}{{\left(\mathrm{\rho }{C}_p\right)}_f}\sum _{k=0}^{t-1}{f}_{t-1-k}{\theta }_k^{\prime }}\\ +\frac{{\left(\mathrm{\rho }{C}_p\right)}_{\mathrm{nf}}}{{\left(\mathrm{\rho }{C}_p\right)}_f}{N}_b\sum _{k=0}^{t-1}{\theta }_{t-1-k}^{\prime }{\varphi }_k^{\prime }\hfill \\ +\frac{{\left(\mathrm{\rho }{C}_p\right)}_{\mathrm{nf}}}{{\left(\mathrm{\rho }{C}_p\right)}_f}{N}_t\sum _{k=0}^{t-1}{\theta }_{t-1-k}^{\prime }{\theta }_k^{\prime }\hfill \\ +\frac{{\left(\mathrm{\rho }{C}_p\right)}_{\mathrm{nf}}}{{\left(\mathrm{\rho }{C}_p\right)}_f}\left(\frac{K_{\mathrm{nf}}}{K_f}(A^{*}+B^{*}{\theta }_{t-1})\right)\hfill \end{array}$$

(39)

$$\begin{array}{c}\multicolumn{1}{c}{R_t^{\varphi }\left(\xi \right)={\varphi }_{t-1}^{″}+\frac{N_t}{N_b}{\theta }_{t-1}^{″}+\mathrm{Sc}\sum _{k=0}^{t-1}{f}_{t-1-k}{\varphi }_k^{\prime }}\\ -\mathrm{Sc}{\sigma }_r({\varphi }_{t-1}+\delta \sum _{k=0}^{t-1}{\theta }_{t-1-k}{\varphi }_k-E{\varphi }_{t-1})\hfill \end{array}$$

(40)

$$\begin{array}{c}\multicolumn{1}{c}{R_t^{\chi }\left(\xi \right)={\chi }_{t-1}^{″}+\mathrm{Lb}\sum _{k=0}^{t-1}{f}_{t-1-k}{\chi }_k^{\prime }}\\ -\mathrm{Pe}(\sum _{k=0}^{t-1}{\varphi }_{t-1-k}^{\prime }{\chi }_k^{\prime }+\sum _{k=0}^{t-1}{\varphi }_{t-1-k}^{″}{\chi }_k+\mathrm{\Omega }{\varphi }_{t-1}^{″})\hfill \end{array}$$

(41)

Where

$${\chi }_t=\left\{\begin{array}{c}0,\mathrm{if}\pounds \le 1\hfill \\ \multicolumn{1}{c}{1,\mathrm{if}\pounds >1}\end{array}\right\}$$

(41)

Results and discussions

The geometry of the problem displays in Figure 1. For the purpose of authentication, the HAM and numerical results are compared as the limiting case. We find that numerical imitation and analytical solution is in outstanding agreement, showed graphically in Figure 2(a–d) and presented in Tables 2 to 5. Lastly, Figure 3 shown the entire residual error. A decreasing performance of residual error is detected for higher-order errors. Figure 4 displays that an increase in the value of $M$ a decay in velocity and the boundary layer thickness. This confirms the general behavior of the magnetic effect. The drag force rises with the continuously growing values of $M$ causing the velocity field to decline. Figure 5 shows the effect of Darcy number $K_1$ the velocity profile. An rise in $K_1$ values lead the decay to velocity profiles. The presence of porous space boosting the fluid stream safety by dropping fluid velocity and the energy layer related with it. Physically, the occurrence of porosity causes internal friction to fluid motion, resulting in a decrease in fluid velocity. Figure 6 shows that for the increasing value of Casson parameter $\beta$ velocity profile decreases. Because when the Casson parameter increases the fluid viscosity also increases and hence the velocity of fluid decreases. Figure 7 shows the changes of thermophoresis parameter $\mathrm{Nt}$ with θ(ξ). The hot particles are drawn to the cold region and out of the hot region, increasing the fluid temperature. Figure 8 shows the key features of the volume concentration of nanoparticles for $\mathrm{Nt}$ . When $\mathrm{Nt}$ increases, the concentration profile enhances slowly. According to Brownian motion theory, the speed of nanoparticles is directly related to their temperature. As temperature increases, the kinetic energy of nanoparticles increases, causing them to transfer faster as shown in Figure 9, as the amount of $\mathrm{Nb}$ increases, both the boundary layer thickness and temperature profile. Figure 10 shows the influences of Brownian motion parameter $\mathrm{Nb}$ on the concentration profile. It is originated that the concentration profile decreases for growing the value of $\mathrm{Nb}$ . In Figure 11, as the value of $R$ increases, the temperature profile $\theta (\xi )$ rises. Substantially, it emphasizes the certainty that the process of radiation produces more heat. Figure 12 displays the consequence of Schmidt number $\mathrm{Sc}$ on concentration profile $\phi \left(\xi \right)$ . For the increasing value of $\mathrm{Sc}$ , $\phi \left(\xi \right)$ decreasing. This is because the Schmidt number and mass diffusivity has an inverse relation therefore for high value of $\mathrm{Sc}$ brings weaker mass distribution as a result concentration of nanoparticles fall. Figure 13 displays the effect of activation energy parameter $E$ on the concentration field. The animation shows that as the value of $E$ increases, the nanofluid concentration also increases. Figure 14 shows the influences of microorganism concentration difference parameter $\mathrm{\Omega }$ on the density of microorganism. It is shown that for increasing the value of $\mathrm{\Omega }$ the density of microorganisms decreases. Figure 15 shows the peclet number $\mathrm{Pe}$ effect on the motile microorganisms density $\chi (\xi )$ . The graph displays that as the value of $\mathrm{Pe}$ is increased, the motile characteristic of fluid declines. It can be determined that the number of motile microbes and number of peclet are proportional inversely. The $\mathrm{pe}$ is the relationship among the orientation and velocity of microorganisms, as a higher $\mathrm{pe}$ results in greater orientation of the microorganisms, which in turn decreases the motile microbe profile. Figure 16 shows the consequence of the bioconvection Lewis number $\mathrm{Lb}$ , on motile microorganisms $\chi (\xi )$ . It must be noted that as the bioconvection Lewis number increases, the fluid motility decreases. Convection is produced by moving motile particles in the fluid which is driving the rate of heat transfer. Figure 17 displays the effect of magnetic field parameter $M$ on skin friction. It is detected that when the value of $M$ is boosting, the skin friction rises close to the wall and decreases as it moves from the wall. Figure 18 demonstrates the consequence of radiation parameter $R$ on the Nusselt number. When the value of radiation parameter $R$ is high, the value of the Nusselt number is high. Figure 19 displays that the value of the Sherwood number increases as the value of the $\mathrm{Nb}$ increases. Figure 20 shows that as the bioconvection Lewis number increases the density of microorganisms also increases.

Figure 1.

Flow geometry.

Figure 2.

(a) Comparison of HAM and Numerical solutions for $f\prime (\xi )$ , (b) comparison of HAM and Numerical solutions for $\theta (\xi )$ , (c) comparison of HAM and numerical solutions for $\phi (\xi )$ , and (d) comparison of HAM and numerical solutions for $\chi (\xi )$ .

Table 2.

Comparison of numerical solution and analytic solution for velocity.

$\xi$	$\mathrm{HAM}\mathrm{solution}$	$\mathrm{Numerical}\mathrm{soltuion}$	$\mathrm{Absolute}\mathrm{Error}$
0	$0.881874$	$0.877359$	$0.004516$
0.5	$0.681842$	$0.680107$	$0.001735$
1	$0.481919$	$0.481537$	$0.000382$
1.5	$0.320478$	$0.320556$	$0.000078$
2	$0.205401$	$0.205576$	$0.000176$
2.5	$0.128779$	$0.128936$	$0.000157$
3	$0.079684$	$0.079798$	$0.000114$
3.5	$0.048918$	$0.048994$	$0.000076$
4	$0.029887$	$0.029936$	$0.000049$
4.5	$0.018208$	$0.018239$	$0.000031$
5	$0.006125$	$0.011092$	$0.000019$

Table 3.

Comparison of numerical solution and analytic solution for temperature profile.

$\xi$	$\mathrm{HAM}\mathrm{solution}$	$\mathrm{Numerical}\mathrm{soltuion}$	$\mathrm{Absolute}\mathrm{Error}$
$0$	$4.086860$	$4.087770$	$0.000712$
$0.5$	$2.231080$	$2.231430$	$0.000331$
$1$	$1.277290$	$1.277370$	$0.000080$
$1.5$	$0.5890240$	$0.590224$	$0.000016$
$2$	$0.384508$	$0.384469$	$0.000037$
$2.5$	$0.229390$	$0.229357$	$0.000034$
$3$	$0.136696$	$0.136671$	$0.000024$
$3.5$	$0.075511$	$0.075495$	$0.000016$
$4$	$0.044875$	$0.044864$	$0.000011$
$4.5$	$0.027994$	$0.027987$	$5.895660\times {10}^{-6}$
$4.5$	$0.015266$	$0.015263$	$3.268950\times {10}^{-6}$

Table 4.

Comparison of numerical solution and analytic solution for concentration profile.

$\xi$	$\mathrm{HAM}\mathrm{solution}$	$\mathrm{Numerical}\mathrm{soltuion}$	$\mathrm{Absolute}\mathrm{Error}$
$0$	$-0.059214$	$-0.059265$	$0.000052$
$0.5$	$-0.031844$	$-0.031889$	$0.000043$
$1$	$-0.024329$	$-0.024361$	$0.000030$
$1.5$	$-0.014335$	$-0.014366$	$0.000026$
$2$	$-0.008301$	$-0.008314$	$0.000020$
$2.5$	$-0.005638$	$-0.005647$	$7.119920\times {10}^{-6}$
$3$	$-0.004318$	$-0.004324$	$4.008250\times {10}^{-6}$
$3.5$	$-0.002272$	$-0.002277$	$2.068300\times {10}^{-6}$
$4$	$-0.001267$	$-0.001269$	$1.872180\times {10}^{-6}$
$4.5$	$-0.000663$	$-0.000664$	$1.138650\times {10}^{-6}$
$4.5$	$-0.000363$	$-0.000363$	$6.928180\times {10}^{-7}$

Table 5.

Comparison of numerical solution and analytic solution for density of microorganisms.

$\xi$	$\mathrm{HAM}\mathrm{solution}$	$\mathrm{Numerical}\mathrm{soltuion}$	$\mathrm{Absolute}\mathrm{Error}$
0	$1.000000$	$1.000000$	$2.220450\times {10}^{-16}$
0.5	$0.784823$	$0.784969$	$0.000146$
1	$0.567011$	$0.567196$	$0.000185$
1.5	$0.383945$	$0.384104$	$0.000159$
2	$0.249188$	$0.249305$	$0.000116$
2.5	$0.157512$	$0.157591$	$0.000078$
3	$0.097965$	$0.098016$	$0.000051$
3.5	$0.060332$	$0.060364$	$0.000032$
4	$0.036933$	$0.036953$	$0.000020$
4.5	$0.022527$	$0.022539$	$0.000012$
5	$0.013710$	$0.013717$	$7.425570\times {10}^{-6}$

Figure 3.

Total residue versus order of approximations.

Figure 4.

Effect of $M$ on $f\prime (\xi )$ .

Figure 5.

Effect of $K_1$ on $f\prime (\xi )$ .

Figure 6.

Effect of $\beta$ on $f\prime (\xi )$ .

Figure 7.

Effect of $\mathrm{Nt}$ on $\theta (\xi )$ .

Figure 8.

Effect of $\mathrm{Nt}$ on $\phi (\xi )$ .

Figure 9.

Effect of $\mathrm{Nb}$ on $\theta (\xi )$ .

Figure 10.

Effect of $\mathrm{Nb}$ on $\phi (\xi )$ .

Figure 11.

Effect of $R$ on $\theta (\xi )$ .

Figure 12.

Effect of $\mathrm{Sc}$ on $\phi (\xi )$ .

Figure 13.

Effect of $E$ on $\phi (\xi )$ .

Figure 14.

Effect of $\mathrm{\Omega }$ on $\chi (\xi )$ .

Figure 15.

Effect of Pe on $\chi (\xi )$ .

Figure 16.

Effect of Lb on $\chi (\xi )$ .

Figure 17.

M versus local skin friction.

Figure 18.

R versus local Nusselt number.

Figure 19.

Nb versus local Sherwood number.

Figure 20.

Lb versus density of microorganism.

The changes of $f^{\prime }\prime \left(0\right)$ for growing parameters $M,\Lambda ,\mathrm{and}K1$ are in Table 6. It is determined that for growing the parameters value $M,\Lambda ,\mathrm{and}K1$ decreases the variance of $f^{″}\left(0\right)$ . Table 7 displays the influence of distinct parameter on local Nusselt number that is when the thermophoresis parameter and Brownian motion parameter goes up the Nusselt number declines and when the radiation parameter increases the value of Nusselt number also increases. Table 8 shows various important factors affect local Sherwood number that is, when the thermophoresis parameter increases, the Sherwood number also rises. Whereas a rise in $\mathrm{Nb}$ will cause the value of Sherwood number to decrease. Table 9 displays the value of distinct parameters on microorganisms’ density. It displays that for the increase in the value of bioconvection peclet number $(\mathrm{Pe})$ and Lewis number $(\mathrm{Lb})$ the density of microorganisms will increase.

Table 6.

$\mathrm{Local}\mathrm{skin}\mathrm{friction}\mathrm{coefficients}\mathrm{for}\mathrm{distinct}\mathrm{parameter}$ $M,\Lambda ,\mathrm{and}K1$ .

M	Λ	K1	$\begin{array}{l}C{f}_x\mathrm{R}{e}_x^{-\frac{1}{2}}\\ (H_2\mathrm{o})\end{array}$	$\begin{array}{l}C{f}_x\mathrm{R}{e}_x^{-\frac{1}{2}}\\ (30\%H_2\mathrm{o})\end{array}$	$\begin{array}{l}C{f}_x\mathrm{R}{e}_x^{-\frac{1}{2}}\\ (50\%H_2\mathrm{o})\end{array}$
$0.1$	$0.2$	$0.1$	$1.504541$	$1.50940$	$1.51142$
0.2			$1.45431$	$1.45948$	$1.46163$
0.25			$1.42951$	$1.43483$	$1.43705$
0.1	0.21		$1.27901$	$1.28325$	$1.28501$
	0.23		$0.95022$	$0.95343$	$0.95477$
	0.24		$0.82995$	$0.83274$	$0.83390$
	0.2	0.2	$1.45431$	$1.45948$	$1.46163$
		0.25	$1.42951$	$1.43483$	$1.43705$
		0.3	$1.40493$	$1.41040$	$1.41267$

Table 7.

Local Nusselt number for distinct parameters $\mathrm{Nt},\mathrm{Nb},\mathrm{and}R$ .

Nt	Nb	R	$\begin{array}{l}N{u}_x\mathrm{R}{e}_x^{-\frac{1}{2}}\\ (H_2\mathrm{o})\end{array}$	$\begin{array}{l}N{u}_x\mathrm{R}{e}_x^{-\frac{1}{2}}\\ (30\%H_2\mathrm{o})\end{array}$	$\begin{array}{l}N{u}_x\mathrm{R}{e}_x^{-\frac{1}{2}}\\ (50\%H_2\mathrm{o})\end{array}$
$0.5$	$0.5$	0.1	$1.10048$	$1.11703$	$1.12128$
1.0			$1.08827$	$1.10471$	$1.10880$
1.5			$1.07606$	$1.09239$	$1.09632$
0.5	0.8		$1.10048$	$1.11703$	$1.12128$
	1.2		$1.10038$	$1.11503$	$1.12028$
	1.6		$1.10018$	$1.11312$	$1.11006$
	0.5	0.2	$1.19792$	$1.21761$	$1.22268$
		0.3	$1.29491$	$1.31803$	$1.32400$
		0.4	$1.39147$	$1.41828$	$1.42522$

Table 8.

Local Sherwood number for distinct parameters $\mathrm{Nt},\mathrm{Nb}$ .

Nt	Nb	$\begin{array}{l}S{h}_x\mathrm{R}{e}_x^{-\frac{1}{2}}\\ (H_2\mathrm{o})\end{array}$	$\begin{array}{l}S{h}_x\mathrm{R}{e}_x^{-\frac{1}{2}}\\ (30\%H_2\mathrm{o})\end{array}$	$\begin{array}{l}S{h}_x\mathrm{R}{e}_x^{-\frac{1}{2}}\\ (50\%H_2\mathrm{o})\end{array}$
$0.5$	$0.5$	$0.07978$	$0.07977$	$0.07977$
1.0		$0.13856$	$0.13855$	$0.13855$
1.5		$0.20834$	$0.20833$	$0.20833$
$0.5$	$0.8$	$0.04361$	$0.04362$	$0.04363$
	1.2	$0.02807$	$0.02707$	$0.02767$
	1.6	$0.02280$	$0.02281$	$0.02283$

Table 9.

Motile microorganism number for distinct parameters $\mathrm{Lb},\mathrm{Pe}$ .

$\mathrm{Lb}$	$\mathrm{Pe}$	$\begin{array}{l}\mathrm{N}{h}_x\mathrm{R}{e}_x^{-\frac{1}{2}}\\ (H_2\mathrm{o})\end{array}$	$\begin{array}{l}\mathrm{N}{h}_x\mathrm{R}{e}_x^{-\frac{1}{2}}\\ (30\%H_2\mathrm{o})\end{array}$	$\begin{array}{l}\mathrm{N}{h}_x\mathrm{R}{e}_x^{-\frac{1}{2}}\\ (50\%H_2\mathrm{o})\end{array}$
$1$	1	$1.36431$	$1.3643$	$1.3643$
$1.5$		$1.37983$	$1.37982$	$1.37982$
$2$		$1.40558$	$1.40557$	$1.40557$
$1$	$1.2$	$1.36869$	$1.36868$	$1.36867$
	$1.7$	$1.38963$	$1.38962$	$1.38962$
	$2.2$	$1.40058$	$1.40057$	$1.40057$

Conclusion

In this study the consequence of MHD Arrhenius activation energy of water, water−30%EG, water−50%EG based graphene nanofluid flowing over a stretched surface in the occurrence of motile micro-organism and thermal-radiation is examined. Some key findings of the study are:

1. The ethylene glycol-based graphene nanofluids takes less time for process as compared to water-based nanofluids. This is because of the thermal conductivity of ethylene glycol-based graphene nanofluid is higher than that of water-based nanofluids.

2. As the magnetic field increases the velocity of fluid decreases, as opposed to this, the temperature and concentration of the fluid increases.

3. The fluid velocity declines with growing value of both Darcy number and Casson fluid parameter $\beta$ .

4. As the activation energy parameter $E$ enhanced, the nanoparticle concentration increased slightly.

5. Growing the thermophoresis parameter $\mathrm{Nt}$ rises the nanoparticle temperature and concentration profile.

6. Increasing the value of Brownian motion parameter $\mathrm{Nb}$ rises fluid temperature and decreasing the nanoparticle concentration.

7. Increasing the value of Schmidt number $\mathrm{Sc}$ increases the nanoparticle concentration profile.

8. Microorganism density profiles are minimized to optimized the material parameters. The bioconvection Lewis parameter $\mathrm{Lb}$ , microorganism concentration difference parameter $\mathrm{\Omega }$ , and peclet number $\mathrm{Pe}$ .

Future work

We will extend our work for unsteady flow nanofluid and Hybrid nanofluid, as well as for polar and cylindrical coordinates.