A mathematical model of Casson nanofluid flow over a vertically stretched porous surface along with bioconvection,Joule heating and thermal Robin conditions

Abstract

The current analysis aims is to investigate the steady-state visco-plastic Casson nanofluid flow over an electrically conducting stretching porous surface. Thermophoresis and Brownian motion parameters are utilized for modeling and analysis. Also, Joule heating, thermal radiation, thermal robin condition, thermo-solutal stratifications, and viscous dissipation are considered for energy and nanoparticle concentration. Using appropriate similarity transformation, the partial differential equations (PDEs) system is altered into ordinary differential equatons (ODEs). The Homotopy analysis method (HAM) is used for solving governing equations. The effect of distinct physical parameters on the velocity, concentration, temperature, and microorganism’s density is revealed graphically and discussed tabularly. Additionally, numerical analysis is conducted on designed Nusselt number, Sherwood number, skin friction, and motile microorganism’s density. The major finding of the study is that the fluid velocity decreases as both the magnetic parameter and Casson fluid parameter increases. Instead, increasing the values of the Deborah number and activation energy results in an increase in fluid temperature. The current study is useful for coating deposition of magnetic nanomaterials at comparatively higher temperatures.

Introduction

Researchers from all over the world are interested in nanotechnology because of its applications in a variety of fields, including chemical sensors, fuel cells (nano-plates), medicine (drug delivery, clinical trials), batteries (silver-zinc, lithium-ion), food (bottles, cartons), electronics (electronic circuits, switches, silicon nano-photonic), and solar cells (graphene, nanowires). The application of Casson nanofluid flow over electrically conducting vertically stretched porous surfaces in the field of electrochemical engineering particularly in the advanced development of batteries that is, electrochemical reactors (electroplating, water electrolysis, or wastewater treatment), battery cooling systems, fuel cells and electrolytic cells etc. In today’s industrialized world, efficient heat transfer is becoming more and more important. Conventional cooling agents might not be sufficient to meet future demands. Due to their special properties, nanofluids, which are systems that contain small particles called nanoparticles typically around 100 nanometers in size are thought to be highly effective heat conductors. Fluids including water, oil, ethylene, and glycol can all include these nanoparticles. The nanoparticles such as metals, nitrides, oxides, carbon graphite, carbides, and nanotubes can be added to the fluid and subsequently lodged inside them to increase the thermal conductivity of nanomaterials. Nanofluids are the essential in a variety of engineering uses because of their transmission of heat properties. Choi and Eastman ¹ studied the effects of nanoparticles on the thermophysical features of liquid. According to Choi the nanometer sized particles added to base fluids improved its thermal conductivity. The importance of different transfer limits of MHD bioconvection nanofluid flow was discussed by Awais et al. ² Anjum et al. ³ investigate the effect of chemical reaction with dual stratification in MHD- based nanofluid flow. The effect of radiative heat in unsteady electrically conductive MHD boundary layer Casson nanofluid flow containing chemical reactive material completely examined by Sedki and Qahiti ⁴ Khan et al. ⁵ studied the two-dimensional flow of Carreau fluid. Bilal et al. ⁶ studied the analytical solution of Casson fluids on the Riga surface. The activation energy influence on stretching flow was examined by Batool et al. ⁷ The effective approach of Prandtl’s number was applied by Lee et al. ⁸ to tackle the nanofluid problem of non-Newtonian nature. Hussain et al. ⁹ used the combined Carreau-Yasuda model for nanofluids to study the numerical solutions for the stagnation point. Farooq et al. ¹⁰ develop a mathematical model to examine the effect of chemical reactions of non-Newtonian nanofluid on the bioconvection features of self-propelled microbes, examing the fluid model of Casson and inclined stretching geometry. Minea ¹¹ examined oxide nanoparticles’ ability to transport heat in single and hybrid nanofluids for use in energy applications. The effects of non-uniform heat source and heat on magnetic nanofluid flow over a cylinder which is porous were examined by Singh et al. ¹² using the Keller box approach. The effects of injection/suction and analysis of entropy on MHD convection nanofluid flow via a cone which were inverted are studied by Vedavati et al. ¹³ Seethamahalakshmi et al. ¹⁴ calculated the significance of exploring the combined effects of convective boundaries, thermal radiation, and (MHD) slip flow over a stretching porous surface. By using a stretched cylinder, Sohail and Naz ¹⁵ were able to detect the Sutter boundary layer movement caused by nano liquid. The study of nanofluids was reviewed by Ahmad et al., ¹⁶ who also discussed the four different sizes of nanofluids: macroscale, microscale, molecular scale, and megascale. These scales are related to each other. Before discussing the use of suspensions containing both motile bacteria and nanoparticles in micro-systems, it is important to examine how these suspensions behave. Hafez et al. ¹⁷ examine the Darcy-Forchheimer Casson nanofluid flow over a stretching sheet. Specifically, the research concentrations on examining the effects of Joule dissipations and viscous induced by electro osmosis forces (EOF) of the Casson nanofluid on the boundary layer. The influence of Joule heating and viscous dissipation on Brinkman-Darcy- Forchheimer of third-grade MHD flows fluid between two parallel plates were examined by Zhang et al. ¹⁸ Chakraborty et al. ¹⁹ deliberated on the expansion below the boundary layer and the magnetic fields effect on bio-convectional nano liquid flow. The relation of Buongiorno for the flow of bio-convectional nano liquid in a porous channel with increasing relaxation and contraction was considered by Beg et al. ²⁰ Abbas et al. ²¹ investigate the steady state flow features of a Williamson Casson fluid induced by a stretchable and impermeable sheet, while seeing dissipation Ohmic effects. The analysis was carried out by Khan et al. ²² though the observation of the Oldroyd-B nanofluid bioconvection flows over the extended sheet, which is believed to demonstrate oscillations. According to Sandeep and Raju ²³ research, the rotating cone offers substantially superior heat and mass transfer when it comes to non-Newtonian fluid of MHD bioconvection flow with cross diffusion past over a rotating plate. Tarakaramu ²⁴ studied the effect on the motion of a Casson liquid of variable thermal conductivity over a porous stretching sheet. Akbar et al. ²⁵ related to a latest analysis regarded viscous flow for temperature-dependent several shaped nanoparticles in a stretchy tube. The characteristics of bioconvection EMHD nanofluid flows effect through a Riga plate with the activation energy were revealed by Bhatti and Michaelides. ²⁶ Makhdoum et al. ²⁷ examine magnetohydrodynamic (MHD) flow of 50%EG-water, 30%EG-water, water, with nanoparticles graphene with zero mass flux conditions, thermal convection, and velocity slips over a stretched surface comprising nanoparticles and motile microorganisms. Bioconvection characteristics of squeezing couple-stress nanofluids flow across a horizontal channel were studied by Srinivasacharya and Sreenath. ²⁸

The literature mentioned above demonstrates a wide range of studies conducted on nanofluid flow. In light of the aforementioned literature, the current study aims to examine steady-state visco-plastic Casson nanofluid flow across a vertically extended porous surface that conducts electricity. The effects of Joule heating, thermal radiation, thermal Robin condition, thermo-solutal stratifications, and viscous dissipation are taken into account. HAM is used to resolve the transformed equation. The influences of distinct parameters on temperature, concentration of nanoparticle, velocity, and motile microbes density are calculated by Mathematica, though which a systematic study has been conducted to examine the influences of distinct parameters. The current work finds applicability in magnetic nanomaterials coating deposition at substantially higher temperatures; moreover, the intended values for Sherwood number, skin friction, the motile microorganism’s density, and Nusselt number are analyzed numerically.

Formulation

Consider a visco-plastic steady-state (Casson) nanofluid flow over a vertically porous stretched surface. The flow is incompressible, laminar, and two-dimensional. A stretching flow that is electrically conducting under the influence of a magnetic field with strength $B_0$ has been formulated. The system of coordinate $(x,y)$ adopted in which the $x-\mathrm{axis}$ is connected with the plane, where the $y-\mathrm{axis}$ is located vertically to it (Figure 1). The bioconvection feature, which produces liquid movement that is macroscopic under a density gradient in conjunction with swimming microorganisms that increase the density of base liquid in a definite direction, is taken into consideration. Moreover, the features of thermal-solutal Robin conditions, thermal radiation, Joule heating, thermo-solutal stratifications, and viscous dissipation are taken into account. Under these assumptions, the governing boundary layer equations for Casson nano fluids are taken by Abbas et al. ²¹

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

(1)

$$\begin{array}{c}u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{\mu }{{\rho }_f}(1+\frac{1}{\beta })\frac{{\partial }^{2}u}{\partial y^2}-\frac{\sigma B_0^2}{{\rho }_f}u+\frac{1}{{\rho }_f}\hfill \\ \left[\begin{array}{c}{\rho }_f\beta g(T-T_{\infty })(1-C_{\infty })-g\gamma (N-N_{\infty })({\rho }_m-{\rho }_f)\hfill \\ \multicolumn{1}{c}{-g({\rho }_p-{\rho }_f)(C-C_{\infty })}\end{array}\right]\hfill \end{array}$$

(2)

$$\begin{array}{l}u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+{\lambda }_1(u^2\frac{{\partial }^{2}T}{\partial x^2}+v^2\frac{{\partial }^{2}T}{\partial y^2}+u\frac{\partial T}{\partial x}\frac{\partial u}{\partial x}+v\frac{\partial T}{\partial x}\frac{\partial u}{\partial y}\\ +u\frac{\partial T}{\partial y}\frac{\partial v}{\partial x}+2uv\frac{{\partial }^{2}T}{\partial x\partial y}+v\frac{\partial v}{\partial y}\frac{\partial T}{\partial y})\\ =\frac{16{\sigma }^{\ast }{T}_{\infty }^3}{3k^{\ast }{\left(\rho c\right)}_f}\frac{{\partial }^{2}T}{\partial y^2}+\frac{k_1}{{\rho }_f{\left(\rho c\right)}_f}\frac{{\partial }^{2}T}{\partial y^2}+\left(\frac{\nu }{{\left(\rho c\right)}_f}\right)\left(1+\frac{1}{\beta }\right){\left(\frac{\partial u}{\partial y}\right)}^2\\ +\tau \left[D_b\frac{\partial T}{\partial y}\frac{\partial C}{\partial y}+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial y}\right)}^2\right]+u^2\frac{\sigma B_0^2}{{\left(\rho c\right)}_f}\end{array}$$

(3)

$$\begin{array}{l}u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}+{\lambda }_2(u^2\frac{{\partial }^{2}C}{\partial x^2}+u\frac{\partial C}{\partial x}\frac{\partial u}{\partial x}+v^2\frac{{\partial }^{2}C}{\partial y^2}+v\frac{\partial C}{\partial x}\frac{\partial u}{\partial y}\\ +2uv\frac{{\partial }^{2}C}{\partial x\partial y}+v\frac{\partial C}{\partial y}\frac{\partial v}{\partial y}+\frac{\partial C}{\partial y}\frac{\partial v}{\partial x}u)\\ =D_B\frac{{\partial }^{2}C}{\partial y^2}-K{r}^2(C-C_{\infty }){\left(\frac{T}{T_{\infty }}\right)}^n\exp \left(\frac{-\mathrm{Ea}}{K_{B}T}\right)+\frac{D_T}{T_{\infty }}\frac{{\partial }^{2}T}{\partial y^2}\end{array}$$

(4)

$$u\frac{\partial n}{\partial x}+v\frac{\partial n}{\partial y}=D_m\frac{{\partial }^{2}n}{\partial y^2}-\frac{b{W}_c}{(C_w-C_0)}\left[\frac{\partial }{\partial y}\left(n\frac{\partial C}{\partial y}\right)\right]$$

(5)

Figure 1.

Geometry of the problrm.

With conditions

$$\begin{array}{c}\multicolumn{1}{c}{u_w\left(x\right)=u=\mathrm{ax},v=v_w,T_w=T,T_w=T_o+a_{1}x,}\\ \multicolumn{1}{c}{C_w=C,C_o+a_{2}x=C_w,N_w=N,N_w=N_o+a_{3}x\mathrm{at}y=0,}\end{array}$$

(6)

$$\begin{array}{c}\multicolumn{1}{c}{u\to 0,T\to T_{\infty }=d_{1}x+T_0,C\to C_{\infty }=d_{2}x+C_0,N\to N_{\infty }}\\ =d_{3}x+N_0\mathrm{wheny}\to \infty .\hfill \end{array}$$

(7)

In the aforementioned equations, $(u,v)$ represent the velocity components in the $(x,y)$ directions respectively, ${\beta }^{*}$ shows the volume-expansion coefficient, g represents gravity, $\mu$ represents the dynamic viscosity of nano-liquid suspension and microorganisms, $\beta$ represents the Casson fluid parameter, $\sigma$ indicates electrical conductivity, ${\beta }_0$ represents the intensity of the magnetic field, $({\rho }_f,{\rho }_p,{\rho }_m)$ represents the density of nano liquid, nanoparticles, and microorganism particles, respectively. ${\sigma }^{*}$ represent the constant of Stefan-Boltzmann, $k_1$ represents thermal conductivity, $(T,C,n)$ represents the temperature, concentration, and concentration of microorganisms, respectively. $k^{*}$ demonstrate the mean absorption coefficient, $(C_f,T_f,n_f)$ represent the surface which is heated (concentration, temperature, concentration of microorganisms), $\tau$ represents the heat capacity ratio, $(T_0,n_0,C_0)$ represent the reference (temperature, microorganisms concentration, concentration of nanoparticles), $(D_B,D_T,D_m,)$ represent the (Brownian, thermophoresis, microorganism) coefficients of diffusion, $(v_w,u_w)$ represent the (injection/suction, stretching) velocity, $W_c$ represent the maximum swimming cell speed, $(a_1,a_2,a_{3,}{d}_{1,}{d}_2,d_3)$ show dimensional constants, $\upsilon$ represents kinematic viscosity, $(h_n,h_f,h_g)$ represent the (motile microorganisms, heat, mass) transfer coefficients, $\gamma$ shows the average microorganisms volume and $C>0$ denotes the stretching rate.

The similarity alteration for the above model is:

Similarities transformation:

$$\begin{array}{c}\eta =y{\left(\frac{a}{\nu }\right)}^{\frac{1}{2}},u=\mathrm{ax}f\prime \left(\eta \right),\psi ={\left(a\nu \right)}^{\frac{1}{2}}\mathrm{xf}\left(\eta \right),v=-{\left(a\nu \right)}^{\frac{1}{2}}f\left(\eta \right),\hfill \\ \varphi \left(\eta \right)=\frac{C-C_{\infty }}{C_w-C_0},\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_0},\phi \left(\eta \right)=\frac{N-N_{\infty }}{N_w-N_0},\hfill \end{array}$$

(8)

The transform equations are following:

$$ff″-Mf\prime -f^{{\prime }^2}+(1+\frac{1}{\beta })f″\prime +\lambda (\theta -N\varphi -R_b\phi )=0,$$

(9)

$$\begin{array}{c}(1+\frac{4}{3}R)\theta ″+\mathrm{Prf}\theta \prime -\mathrm{Pr}f\prime \theta +\mathrm{PrEcM}{f}^{{\prime }^2}+\mathrm{Pr}{N}_b\varphi \prime \theta \prime \hfill \\ +\mathrm{Pr}{N}_t{\theta }^{{\prime }^2}-\mathrm{Pr}{S}_{1}f\prime +\mathrm{PrEc}(1+\frac{1}{\beta })f^{\prime {\prime }^2}\hfill \\ -\mathrm{Pr}{\beta }_1(\theta ″f^2+S_1{f}^{{\prime }^2}+f^{{\prime }^2}\theta -S_{1}ff″-ff″\theta -ff\prime \theta \prime )=0,\hfill \end{array}$$

(10)

$$\begin{array}{c}\varphi ″+\mathrm{Scf}\varphi \prime -\mathrm{Sc}f\prime \varphi -\mathrm{Sc}{S}_{2}f\prime +\frac{N_t}{N_b}\theta ″\hfill \\ -{\beta }_2(\varphi ″f^2+S_2{f}^{{\prime }^2}+f^{{\prime }^2}\varphi -S_{2}ff″-ff″\varphi -ff\prime \varphi \prime )\hfill \\ -\sigma {(1+\delta \theta )}^n\exp \left(\frac{-E}{1+\delta \theta }\right)\varphi =0,\hfill \end{array}$$

(11)

$$\phi ″+Scf\phi \prime -Sc{S}_{3}f\prime -\mathrm{Sc}f\prime \phi +\mathrm{Pe}[\varphi ″(\phi +\Omega )+\varphi \prime \phi \prime ]=0,$$

(12)

The transform BCs are:

$$\begin{array}{cc}\theta \left(0\right)=1,f\left(0\right)=S,\hfill & f^{\prime }\left(0\right)=1\hfill \\ \multicolumn{1}{c}{\phi \left(0\right)=1,\varphi \left(0\right)=1,}& \mathrm{at}\eta =0,\hfill \end{array}\theta \left(\infty \right)\to 0,\varphi \left(\infty \right)\to 0,f^{\prime }\left(\infty \right)\to 0,\phi \left(\infty \right)\to 0,\mathrm{as}\eta \to \infty$$

(13)

In the above equation $M$ represent the magnetic field parameter, $\beta$ represents Casson fluid parameter, $\lambda$ represent the dual convection factor, $N$ represent the buoyancy ratio factor, $R_b$ denote the bioconvection Rayleigh number, $R$ represents radiation parameter, $\mathrm{Pr}$ denotes Prandtl number, ${\beta }_1^{*}$ is the Deborah number for temperature profile, $\mathrm{Ec}$ represent the Eckert number, $N_b$ represents Brownian motion parameter, $N_t$ represent the thermophoresis parameter, $(S_1,S_2,S_3)$ represent the (thermal, concentration, motile density) stratification factor, $\mathrm{Sc}$ represent the Schmidt number, ${\beta }_2^{*}$ represent the Deborah number for concentration profile, $E$ denotes activation energy, $\sigma$ represent the parameter of chemical reaction, $\delta$ is the temperature difference parameter, $Sc$ signify bioconvection Schmidt number, $\mathrm{Pe}$ denotes Peclet number, and $\Omega$ represent the difference factor of microorganisms concentration.

$$\begin{array}{c}M=\frac{\sigma B_0^2}{a{\rho }_f},\lambda =\frac{{\beta }^{*}\gamma (1-C_{\infty })(T_f-T_0)}{a{U}_0},\hfill \\ N=\frac{({\rho }_p-{\rho }_f)(C_f-C_0)}{{\beta }^{*}{\rho }_f(T_f-T_0)},R_b=\frac{\gamma (N_f-N_0)({\rho }_m-{\rho }_p)}{{\beta }^{*}{\rho }_f(1-C_{\infty })(T_f-T_0)},\hfill \\ R=\frac{4{\sigma }^{*}{T}_{\infty }^3}{KK},\mathrm{Pr}=\frac{\mu c_f}{k},{\beta }_1=a{\lambda }_1,\mathrm{Ec}=\frac{u_w^2}{c_f(T_f-T_0)},\hfill \\ \mathrm{Nb}=\frac{\tau D_B(C_f-C_0)}{a},\mathrm{Nt}=\frac{\tau D_T(T_f-T_0)}{a{T}_{\infty }},S_1=\frac{d_1}{a_1},\hfill \\ S_2=\frac{d_2}{a_2},S_3=\frac{d_3}{a_3},\mathrm{Sc}=\frac{\nu }{D_B},{\beta }_2^{*}=a{\lambda }_2,E=\frac{E_a}{k_b{T}_{\infty }},\hfill \\ \sigma =\frac{K{r}^2}{a},\delta =\frac{T_w-T_{\infty }}{T_{\infty }},S{c}^{*}=\frac{\nu }{D_m}\mathrm{Pe}=\frac{b{W}_c}{D_m},\Omega =\frac{N_{\infty }}{N_f-N_0}\hfill \end{array}$$

(14)

The coefficient of skin friction $\left(C{f}_x\right)$ , heat and mastransfer rate $(N{u}_x,S{h}_x)$ and motile microorganism’s density $\left(N{h}_x\right)$ are given as:

$$C{f}_x=\frac{{\tau }_w}{{\rho }_f{u}_w^2},{\tau }_w=\left[\mu {\left(\frac{\partial u}{\partial y}\right)}_{y=0}(1+\frac{1}{\beta })\right],$$

(15)

$$\frac{x{q}_w}{k(T_f-T_0)}=N{u}_x,q_w=-(\frac{16T_{\infty }^3{\sigma }^{*}}{3k^{*}}+k){\left(\frac{\partial T}{\partial y}\right)}_{y=0},$$

(16)

$$q_m={(-D_B\frac{\partial \varphi }{\partial y})}_{y=0},S{h}_x=\frac{x{q}_m}{(C_f-C_0)D_B},$$

(17)

$$N{h}_x=\frac{x{q}_n}{D_m(n_f-n_0)},q_n={(-D_m\frac{\partial \phi }{\partial y})}_{y=0}.$$

(18)

After simplification, equation (15)–(18) we acquire

$$C{f}_x{\mathrm{Re}}_x^{1/2}f″\left(0\right)(1+\frac{1}{\beta }),$$

(19)

$${\mathrm{Re}}_x^{-1/2}N{u}_x=-(1+\frac{4}{3}R)\theta \prime \left(0\right),$$

(20)

$$S{h}_x{\mathrm{Re}}_x^{-1/2}-\varphi \prime \left(0\right),$$

(21)

$$N{h}_x{\mathrm{Re}}_x^{-1/2}-\phi \prime \left(0\right),$$

(22)

in which ${\mathrm{Re}}_x\frac{x{u}_w}{\upsilon }$ illustrates Reynolds number.

HAM solution

In order to solve equations (9)–(12) under the BCs (13), the HAM is used with a subsequent technique. The solutions exhibit and control the solutions convergence through the incorporation of auxiliary parameters $\hslash$ .

The following are initial guesses chosen as

$$f_0\left(\eta \right)=1-e^{-\eta },{\theta }_0\left(\eta \right)=e^{-\eta },{\varphi }_0\left(\eta \right)=e^{-\eta },{\phi }_0\left(\eta \right)=e^{-\eta }$$

(23)

$L_f,L_{\theta },L_{\varphi },\mathrm{and}{L}_{\phi }$ represents linear operators:

$$\begin{array}{c}\multicolumn{1}{c}{L_{\theta }\left(\theta \right)=\theta ″-\theta ,L_{\varphi }\left(\varphi \right)=\varphi ″-\varphi ,L_f\left(f\right)=f″\prime }\\ -f\prime ,L_{\phi }\left(\phi \right)=\phi ″-\phi \hfill \end{array}$$

(24)

This contains the subsequent properties:

$$\begin{array}{c}{L}_f(s_1+s_2{e}^{-\eta }+s_3{e}^{\eta })=0,L_{\theta }(s_4{e}^{\eta }+s_5{e}^{-\eta })\hfill \\ =0,L_{\varphi }(s_6{e}^{\eta }+s_7{e}^{-\eta })=0,L_{\phi }(s_8{e}^{-\eta }+s_9{e}^{\eta })=0,\hfill \end{array}$$

(25)

Where the general solution constants are $s_i(i=1-9)$ :

$N_f,N_{\theta },N_{\varphi },N_{\phi }$ are resultant non-linear operatives:

$$\begin{array}{c}{N}_f[f(\eta ;J),\phi (\eta ;J),\theta (\eta ;J),\varphi (\eta ;J)]\hfill \\ =(1+\frac{1}{\beta })\frac{{\partial }^{3}f(\eta ;J)}{\partial {\eta }^3}-M\frac{\partial f(\eta ;J)}{\partial \eta }-{\left(\frac{\partial f(\eta ;J)}{\partial \eta }\right)}^2\hfill \\ +\lambda (-R_b\phi (\eta ;J)-N\varphi (\eta ;J)+\theta (\eta ;J))+\frac{{\partial }^{2}f(\eta ;J)}{\partial {\eta }^2}f(\eta ;J)\hfill \end{array}$$

(26)

$$\begin{array}{l}{N}_{\theta }\left[\theta \left(\eta ;J\right),\varphi \left(\eta ;J\right),f\left(\eta ;J\right)\right]=\frac{{\partial }^2\theta \left(\eta ;J\right)}{\partial {\eta }^2}\left(1+\frac{4}{3}R\right)\\ +Prf\left(\eta ;J\right)\frac{\partial \theta \left(\eta ;J\right)}{\partial \eta }-Pr\frac{\partial f\left(\eta ;J\right)}{\partial \eta }\theta \left(\eta ;J\right)+PrEc\left(1+\frac{1}{\beta }\right)\\ {\left(\frac{{\partial }^{2}f\left(\eta ;J\right)}{\partial {\eta }^2}\right)}^2+PrEcM{\left(\frac{\partial f\left(\eta ;J\right)}{\partial \eta }\right)}^2+Pr{N}_b\frac{\partial \varphi \left(\eta ;J\right)}{\partial \eta }\frac{\partial \theta \left(\eta ;J\right)}{\partial \eta }\\ +Pr{N}_t{\left(\frac{\partial \theta \left(\eta ;J\right)}{\partial \eta }\right)}^2-Pr{S}_1\frac{\partial f\left(\eta ;J\right)}{\partial \eta }-Pr{\beta }_1^{\ast }\\ \left(\begin{array}{c}{S}_1{\left(\frac{\partial f\left(\eta ;J\right)}{\partial \eta }\right)}^2+\frac{{\partial }^2\theta \left(\eta ;J\right)}{\partial {\eta }^2}{\left(f\left(\eta ;J\right)\right)}^2+\theta \left(\eta ;J\right){\left(\frac{\partial f\left(\eta ;J\right)}{\partial \eta }\right)}^2\\ -f\left(\eta ;J\right)\frac{{\partial }^{2}f\left(\eta ;J\right)}{\partial {\eta }^2}\theta \left(\eta ;J\right)-S_{1}f\left(\eta ;J\right)\frac{{\partial }^{2}f\left(\eta ;J\right)}{\partial {\eta }^2}\\ -f\left(\eta ;J\right)\frac{\partial f\left(\eta ;J\right)}{\partial \eta }\frac{\partial \theta \left(\eta ;J\right)}{\partial \eta }\end{array}\right)\end{array}$$

(27)

$$\begin{array}{l}{N}_{\varphi }\left[\theta \left(\eta ;J\right),\varphi \left(\eta ;J\right),f\left(\eta ;J\right)\right]=\frac{{\partial }^2\varphi \left(\eta ;J\right)}{\partial {\eta }^2}+Sc\frac{\partial \varphi \left(\eta ;J\right)}{\partial \eta }f\left(\eta ;J\right)\\ -Sc\frac{\partial f\left(\eta ;J\right)}{\partial \eta }\varphi \left(\eta ;J\right)-Sc{S}_2\frac{\partial f\left(\eta ;J\right)}{\partial \eta }+\frac{N_t}{N_b}\frac{{\partial }^2\theta \left(\eta ;J\right)}{\partial {\eta }^2}-{\beta }_2^{\ast }\\ \left(\begin{array}{c}{\left(f\left(\eta ;J\right)\right)}^2\frac{{\partial }^2\varphi \left(\eta ;J\right)}{\partial {\eta }^2}+S_2{\left(\frac{\partial f\left(\eta ;J\right)}{\partial \eta }\right)}^2+{\left(\frac{\partial f\left(\eta ;J\right)}{\partial \eta }\right)}^2\\ \varphi \left(\eta ;J\right)-S_2\frac{{\partial }^{2}f\left(\eta ;J\right)}{\partial {\eta }^2}f\left(\eta ;J\right)-f\left(\eta ;J\right)\frac{{\partial }^{2}f\left(\eta ;J\right)}{\partial {\eta }^2}\varphi \left(\eta ;J\right)\\ -\frac{\partial f\left(\eta ;J\right)}{\partial \eta }\frac{\partial \varphi \left(\eta ;J\right)}{\partial \eta }f\left(\eta ;J\right)\end{array}\right)\\ -\sigma \left(\varphi \left(\eta ;J\right)+\delta \varphi \left(\eta ;J\right)\theta \left(\eta ;J\right)-E\varphi \left(\eta ;J\right)\right)\end{array}$$

(28)

$$\begin{array}{c}{N}_{\phi }[\varphi (\eta ;J),f(\eta ;J),\phi (\eta ;J)]=\frac{{\partial }^2\phi (\eta ;J)}{\partial {\eta }^2}\hfill \\ +S{c}^{*}\frac{\partial \phi (\eta ;J)}{\partial \eta }f(\eta ;J)-S{c}^{*}{S}_3\frac{\partial f(\eta ;J)}{\partial \eta }-\mathrm{Sc}\phi (\eta ;J)\frac{\partial f(\eta ;J)}{\partial \eta }\hfill \\ +\mathrm{Pe}[\phi (\eta ;J)\frac{{\partial }^2\varphi (\eta ;J)}{\partial {\eta }^2}+\frac{\partial \varphi (\eta ;J)}{\partial \eta }\frac{\partial \phi (\eta ;J)}{\partial \eta }+\Omega \frac{{\partial }^2\varphi (\eta ;J)}{\partial {\eta }^2}]\hfill \end{array}$$

(29)

The HAM elementary concept is defined in Refs. 29–35, from equations (9)–(12) the zeroth -order problems are:

$$\begin{array}{c}{L}_f[f(\eta ;J)-f_0\left(\eta \right)](1-J)\hfill \\ =p{\hslash }_f{N}_f[\theta (\eta ;J),\phi (\eta ;J),\varphi (\eta ;J),f(\eta ;J)]\hfill \end{array}$$

(30)

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

(31)

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

(32)

$$L_{\phi }[\phi (\eta ;J)-{\phi }_0\left(\eta \right)](1-J)=p{\hslash }_{\phi }{N}_{\phi }[\varphi (\eta ;J),\phi (\eta ;J),f(\eta ;J)]$$

(33)

The corresponding BCs:

$$\begin{array}{cc}{{\frac{\partial f(\eta ;J)}{\partial \eta }|}_{\eta =0}=1,f(\eta ;J)|}_{\eta =0}=S,{\frac{\partial f(\eta ;J)}{\partial \eta }|}_{\eta \to \infty }=0,\hfill & \hfill \\ \multicolumn{1}{c}{{\theta (\eta ;J)|}_{\eta \to \infty }=0,{\theta (\eta ;J)|}_{\eta =0}=1,}& {{\phi (\eta ;J)|}_{\eta =0}=1,\phi (\eta ;J)|}_{\eta \to \infty }=0\hfill \\ \multicolumn{1}{c}{{{\varphi (\eta ;J)|}_{\eta \to 0}=1,\varphi (\eta ;J)|}_{\eta \to \infty }=0,}& \hfill \end{array}$$

(34)

Where imbedding parameter is $\overline{p}\in [0,1]$ , To control solution convergence ${\hslash }_f,{\hslash }_{\theta },{\hslash }_{\varphi },{\hslash }_{\phi }$ are used. When $J=0\mathrm{and}J=1$ we have:

$$\begin{array}{cc}\multicolumn{1}{c}{f\left(\eta \right)=f(\eta ;1),\theta (\eta ;1)}& =\theta \left(\eta \right),\phi (\eta ;1)=\phi \left(\eta \right),\varphi (\eta ;1)\hfill \\ =\varphi \left(\eta \right),\hfill \end{array}$$

(35)

Taylor’s series expansions of $f(\eta ;J),\theta (\eta ;J),\varphi (\eta ;J),\phi (\eta ;J)$ round $J=0$

$$\begin{array}{c}\left(\sum _{m=1}^{\infty }{f}_m\left(\eta \right)J^m\right)+f_0\left(\eta \right)=f(\eta ;J),\left(\sum _{m=1}^{\infty }{\theta }_m\left(\eta \right)J^m\right)\hfill \\ +{\theta }_0\left(\eta \right)=\theta (\eta ;J),\left(\sum _{m=1}^{\infty }{J}^m{\varphi }_m\left(\eta \right)\right)+{\varphi }_0\left(\eta \right)=\varphi (\eta ;J),\hfill \\ \left(\sum _{m=1}^{\infty }{J}^m{\phi }_m\left(\eta \right)\right)+{\phi }_0\left(\eta \right)=\phi (\eta ;J).\hfill \end{array}$$

(36)

Where

$$\begin{array}{c}{f}_m\left(\eta \right)={\frac{\partial f(\eta ;J)}{\partial \eta }|}_{J=0}\frac{1}{m!},{\theta }_m\left(\eta \right)={\frac{\partial \theta (\eta ;J)}{\partial \eta }|}_{J=0}\frac{1}{m!},\hfill \\ {\varphi }_m\left(\eta \right)={\frac{\partial \varphi (\eta ;J)}{\partial \eta }|}_{J=0}\frac{1}{m!},{\phi }_m\left(\eta \right)={\frac{\partial \phi (\eta ;J)}{\partial \eta }|}_{J=0}\frac{1}{m!}.\hfill \end{array}$$

(37)

${\hslash }_f,{\hslash }_{\theta },{\hslash }_{\varphi },{\hslash }_{\phi }$ are secondary constraints selected in a way such as it converges the series (36) at $J=1$ , swapping $J=1$ in (36), we get:

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

$$\begin{array}{c}\multicolumn{1}{c}{\varphi \left(\eta \right)=\left(\sum _{m=1}^{\infty }{\varphi }_m\left(\eta \right)\right)+{\varphi }_0\left(\eta \right),\phi \left(\eta \right)=\left(\sum _{m=1}^{\infty }{\phi }_m\left(\eta \right)\right)}\\ +{\phi }_0\left(\eta \right),\hfill \end{array}$$

(38)

The problem of $m$ th-order satisfies the following:

$$\begin{array}{c}{L}_f[-f_{m-1}\left(\eta \right){\chi }_m+f_m\left(\eta \right)]\hfill \\ =R_m^f\left(\eta \right){\hslash }_f,L_{\theta }[-{\theta }_{m-1}\left(\eta \right){\chi }_m+{\theta }_m\left(\eta \right)]=R_m^{\theta }\left(\eta \right){\hslash }_{\theta },\hfill \end{array}$$

$$\begin{array}{c}{L}_{\varphi }[-{\varphi }_{m-1}\left(\eta \right){\chi }_m+{\varphi }_m\left(\eta \right)]\hfill \\ =R_m^{\varphi }\left(\eta \right){\hslash }_{\varphi },L_{\phi }[-{\phi }_{m-1}\left(\eta \right){\chi }_m+{\phi }_m\left(\eta \right)]=R_m^{\phi }\left(\eta \right){\hslash }_{\phi }\hfill \end{array}$$

(39)

The corresponding BCs are:

$$\begin{array}{c}\multicolumn{1}{c}{f_m\prime \left(0\right)=f_m\left(0\right)={\theta }_m\left(0\right)={\varphi }_m\left(0\right)={\phi }_m\left(0\right)=0,}\\ \multicolumn{1}{c}{f_m\prime \left(\infty \right)={\theta }_m\left(\infty \right)={\varphi }_m\left(\infty \right)={\phi }_m\left(\infty \right)=0}\end{array}$$

(40)

Here

$$\begin{array}{c}\multicolumn{1}{c}{R_m^f\left(\eta \right)=(1+\frac{1}{\beta })f_{m-1}″\prime -\sum _{k=0}^{m-1}{f}_{m-1-k}\prime f_k\prime +\sum _{k=0}^{m-1}{f}_{m-1-k}{f}_k″}\\ -{\mathrm{Mf}}_{m-1}\prime +\lambda ({\theta }_{m-1}-N{\varphi }_{m-1}-R_b{\phi }_{m-1})\hfill \end{array}$$

(41)

$$\begin{array}{c}\multicolumn{1}{c}{R_m^{\theta }\left(\eta \right)=(1+\frac{4}{3}R){\theta }_{m-1}″+\mathrm{Pr}(\sum _{k=0}^{m-1}{f}_{m-1-k}{\theta }_k\prime -\sum _{k=0}^{m-1}{f}_{m-1-k}\prime {\theta }_k)}\\ +\mathrm{Pr}\mathrm{Ec}(1+\frac{1}{\beta })\sum _{k=0}^{m-1}{f}_{m-1-k}″f_k″+\mathrm{Pr}\mathrm{Ec}M\sum _{k=0}^{m-1}{f}_{m-1-k}\prime f_k\prime \hfill \\ +\mathrm{Pr}{N}_b\sum _{k=0}^{m-1}{\theta }_{m-1-k}\prime {\varphi }_k\prime +\mathrm{Pr}{N}_t\sum _{k=0}^{m-1}{\theta }_{m-1-k}\prime {\theta }_k\prime -\mathrm{Pr}{S}_1{f}_{m-1}\prime \hfill \\ -\mathrm{Pr}{\beta }_1^{*}\left(\begin{array}{c}\sum _{k=0}^{m-1}{f}_{m-1-k}\sum _{l=0}^k{f}_{k-l}{\theta }_l″+S_1\sum _{k=0}^{m-1}{f}_{m-1-k}\prime f_k\prime \hfill \\ +\sum _{k=0}^{m-1}{f}_{m-1-k}\prime \sum _{l=0}^k{f}_l\prime {\theta }_l-S_1\sum _{k=0}^{m-1}{f}_{m-1-k}{f}_k″\hfill \\ -\sum _{k=0}^{m-1}{f}_{m-1-k}\sum _{l=0}^k{f}_{k-l}″{\theta }_l-\sum _{k=0}^{m-1}{f}_{m-1-k}\sum _{l=0}^k{f}_{k-l}\prime {\theta }_l\prime \hfill \end{array}\right)\hfill \end{array}$$

(42)

$$\begin{array}{c}\multicolumn{1}{c}{R_m^{\varphi }\left(\eta \right)={\varphi }_{m-1}″+\mathrm{Sc}\sum _{k=0}^{m-1}{f}_{m-1-k}{\varphi }_k\prime -\mathrm{Sc}\sum _{k=0}^{m-1}{f}_{m-1-k}\prime {\varphi }_k}\\ -\mathrm{Sc}{S}_2{f}_{m-1}\prime +\frac{N_t}{N_b}{\theta }_{m-1}″-{\beta }_2^{*}\hfill \\ \left(\begin{array}{c}\sum _{k=0}^{m-1}{f}_{m-1-k}\sum _{l=0}^k{f}_{k-l}{\varphi }_l″+S_2\sum _{k=0}^{m-1}{f}_{m-1-k}\prime f_k\prime \hfill \\ +\sum _{k=0}^{m-1}{f}_{m-1-k}\prime \sum _{l=0}^k{f}_{k-l}\prime {\varphi }_l-S_2\sum _{k=0}^{m-1}{f}_{m-1-k}{f}_k″\hfill \\ -\sum _{k=0}^{m-1}{f}_{m-1-k}\sum _{l=0}^k{f}_{k-l}″{\varphi }_l-\sum _{k=0}^{m-1}{f}_{m-1-k}\sum _{l=0}^k{f}_{k-l}\prime {\varphi }_l\prime \hfill \end{array}\right)\hfill \\ -\sigma ({\varphi }_{m-1}+\delta \sum _{k=0}^{m-1}{\theta }_{m-1-k}\varphi -E{\varphi }_{m-1})\hfill \end{array}$$

(43)

$$\begin{array}{c}\multicolumn{1}{c}{R_m^{\phi }\left(\eta \right)={\phi }_{m-1}″+Sc\sum _{k=0}^{m-1}{f}_{m-1-k}{\phi }_k\prime -Sc{S}_3{f}_{m-1}\prime }\\ -\mathrm{Sc}\sum _{k=0}^{m-1}{f}_{m-1-k}\prime {\phi }_k\mathrm{Pe}\hfill \\ (\sum _{k=0}^{m-1}{\varphi }_{m-1-k}\prime {\phi }_k\prime +\sum _{k=0}^{m-1}{\varphi }_{m-1-k}″{\phi }_k+\Omega {\varphi }_{m-1}″)\hfill \end{array}$$

(44)

Where,

$${\chi }_m=\{\begin{array}{c}1,\mathrm{if}p>1\hfill \\ 0,\mathrm{if}p\le 1\hfill \end{array}$$

(45)

Convergence analysis

The series convergence given in equation $\left(38\right),f\left(\eta \right),\theta \left(\eta \right),\varphi \left(\eta \right),\mathrm{and}\phi \left(\eta \right)$ totally depend on the supplementary parameters ${\hslash }_f,{\hslash }_{\theta },{\hslash }_{\varphi }\mathrm{and}{\hslash }_{\phi }$ known as $\hslash$ -curve. This is chosen so that the series solution is controlled and converges. The section possible for $\hslash$ can be determined by plotting $\hslash$ -curves of $f″\left(\eta \right),\theta \prime \left(\eta \right),\varphi \prime \left(\eta \right),\mathrm{and}\phi \prime \left(\eta \right)$ for 24th order HAM approximated solution. The effective regions for $\hslash$ are $-0.19<{\hslash }_f<0.09,-0.2<{\hslash }_{\theta }<0.09,-0.21<{\hslash }_{\varphi }<0.09,\mathrm{and}-0.22<{\hslash }_{\varphi }<0.09,$ and is plotted in Figure 2. Table 1 are prepared for convergence of series solutions at different values of physical parameters. It is noted that the 40th order approximation.

Figure 2.

The combined graph of $\hslash$ -curves $f″\left(\eta \right),\theta \prime \left(\eta \right),\varphi \prime \left(\eta \right),\mathrm{and}\phi \prime \left(\eta \right)$ .

Table 1.

Convergent table for $\mathrm{Pr}=6.2,M=0.1,\beta =1,R=0.5,\mathrm{Nb}=0.2$ , $\mathrm{Nt}=0.1,E=0.2,\mathrm{Pe}=0.3,\mathrm{\Omega }=0.1$ .

m	$f″(0)$	$\theta \prime (0)$	$\varphi \prime (0)$	$\phi \prime (0)$
1	$-0.956000$	$-0.948139$	$-0.991667$	$-0.988667$
4	$-0.853493$	$-0.852200$	$-0.972840$	$-0.961992$
8	$-0.763207$	$-0.800957$	$-0.956933$	$-0.939452$
12	$-0.704054$	$-0.787428$	$-0.946996$	$-0.927193$
16	$-0.662869$	$-0.788833$	$-0.940578$	$-0.921839$
20	$-0.632528$	$-0.776179$	$-0.936424$	$-0.921148$
24	$-0.609045$	$-0.771836$	$-0.933819$	$-0.923611$
28	$-0.607104$	$-0.763309$	$-0.932319$	$-0.928204$
32	$-0.606983$	$-0.759421$	$-0.928427$	$-0.927364$
36	$-0.603481$	$-0.758940$	$-0.927320$	$-0.927173$
38	$-0.603274$	$-0.757695$	$-0.927122$	$-0.927126$
40	$-0.603261$	$-0.756342$	$-0.927078$	$-0.927122$

Results and discussion

The consequence of the $M$ on the velocity profile is displayed in Figure 3. It can be realized that as the $M$ value increases, the velocity of fluid decreases. This is owing to the creation of Lorentz force by the magnetic field, which outcomes a decrease in the velocity of fluid. As resistive force is the Lorentz force, it decreases the velocity profile, and subsequently, the motion of particles decreases. The profile of velocity decreases when the Casson parameter β increases, as seen in Figure 4. since the fluid’s viscosity rises along with the Casson parameter, causing the fluid’s velocity to decrease. Figure 5 shows the radiation parameter $R$ effect of on $\theta \left(\eta \right)$ . An rise in $R$ yields rises in $\theta \left(\eta \right)$ . As a results energy is released and these results in temperature increase. This result is consistent with the fact that adding more heat to a flow system the temperature and the thickness of boundry layer are intensified. Figure 6 displays the effect of $\mathrm{Pr}$ , on temperature profiles. Temperature usually decays as values of $\mathrm{Pr}$ increase. $\mathrm{Pr}$ and thermal diffusivity are inversely proportional to each other. As a consequence, the thermal diffusivity is very small for larger values of $\mathrm{Pr}$ . For higher $\mathrm{Pr}$ values the fluid becomes more viscous, whereas the viscosity of fluid is lesser for the low $\mathrm{Pr}$ values. As a consequence of increasing the values of $\mathrm{Pr}$ , the fluid temperature decreases. Figure 7 demonstrates the thermophoresis parameter $\mathrm{Nt}$ significances on the temperature $\theta \left(\eta \right)$ . It is detected that the profile of temperature $\theta \left(\eta \right)$ increases slowly with the increasing the values of $\mathrm{Nt}$ . The heated particles are move from the hotter region into the colder region due to Nt, raising the fluid temperature. Figure 8 shows the influence of Brownian parameter $\mathrm{Nb}$ on temperature profile $\theta \left(\eta \right)$ . From the figure it can be seen that the temperature profile increases for increasing the value of $\mathrm{Nb}$ . This is because the temperature of the fluid rises as fluid particles collide with one another. The influence of Deborah number on the temperature profile is seen in Figure 9. It has been seen that the fluid’s temperature rises in parallel with the Deborah number. The effect of thermophoresis parameter $\mathrm{Nt}$ on $\varphi \left(\eta \right)$ are revealed in Figure 10. From this illustration, it can be seen that the concentration profile $\varphi \left(\eta \right)$ increasing along with the increasing values of $\mathrm{Nt}$ . The significant effects of the Brownian motion parameter $\mathrm{Nb}$ on the concentration profile $\varphi \left(\eta \right)$ of nanoparticles are shown in Figure 11. It is inferred that when $\mathrm{Nb}$ values rises, the concentration profile ϕ(η) of nanoparticles declines. Since the Brownian motion warms the boundary layer making the particles moves away from the fluid regime. Figure 12 shows the consequence of activation energy parameter $E$ on the concentration profiles. This result demonstrates that the nanofluids concentration increases somewhat as the activation energy parameter E increases. Figure 13 shows the effect of $S_3$ on $\phi \left(\eta \right)$ . Here $\phi \left(\eta \right)$ declines when the value of $S_3$ is increases. Physically, the change between concentration of microorganisms’ and away from moving surface decrease which leads to smaller $\phi \left(\eta \right)$ . The influence of $\mathrm{Pe}$ on $\phi \left(\eta \right)$ is shown in Figure 14. A decrease in $\phi \left(\eta \right)$ correlates with an increase in $\mathrm{Pe}$ . Pe is physically different with motile density. Greater Pe values result in a decrease in motile density, which in turn lowers the concentration of motile microorganisms. The effects of $\Omega$ on $\phi \left(\eta \right)$ are seen in Figure 15. An increase in $\Omega$ yields a decay in $\phi \left(\eta \right)$ . Figure 16 displays the consequence of $M$ and $\beta$ on skin friction. For growing values of $M$ and $\beta$ correspond to higher skin friction. The consequence of $\mathrm{EC}$ and $R$ on Nusselt number is revealed in Figure 17. The Nusselt number declines for growing value of $\mathrm{EC}$ while opposite trend is shown for $R$ . Figure 18 displays the effect of Schmidt number on mass transfer rate. The rate of mass transfer is enhanced with an increase in $\mathrm{Sc}$ values. The consequence of peclet number on density of microorganism is revealed in Figure 19. The microorganism density is growing for growing the value of $\mathrm{Pe}$ . The variations of $f″$ (0) for increasing constraints M, β are designed in Table 2. It is established that variability of $f″$ (0) improves by raising the value of the factors M, β. Table 3 presents the impact of many significant factors on the Local Nusselt number. This table demonstrates that when the values of $\mathrm{Pr},R,N,{\beta }_1$ increases, the local Nusselt number also increase. However, for increasing the value of Brownian motion parameter and thermophoresis parameter cause decay in the local Nusselt number value. The several important factors that have an impact on the Local Sherwood number are listed in Table 4. The table shows that for increasing the values of $\mathrm{Nb},\mathrm{Sc},{\beta }_2$ , the local Sherwood number also increases. However, for increasing the activation energy and thermophoresis parameter cause decrease in the local Sherwood number. Table 5 display the numerical value of density of microorganisms. It shows that for the increasing the bioconvection Peclet number $\left(\mathrm{Pe}\right)$ value and microorganisms Concentration difference factor ( $\Omega$ ) will upsurge the value of microorganism’s density.

Figure 3.

Effect of M on f′(η).

Figure 4.

Effect of β on f′(η).

Figure 5.

Effect of R on θ(η).

Figure 6.

Effect of Pr on θ(η).

Figure 7.

Effect of Nt on θ(η).

Figure 8.

Effect of Nb on θ(η).

Figure 9.

Effect of β^*₁ on θ(η).

Figure 10.

Effect of Nt on φ(η).

Figure 11.

Effect of Nb on φ(η).

Figure 12.

Effect of E on φ(η).

Figure 13.

Effect of S₃ on ϕ(η).

Figure 14.

Effect of Pe on ϕ(η).

Figure 15.

Effect of Ω on ϕ(η).

Figure 16.

M and β effect of on Cf_x Re_x^-1/2.

Figure 17.

Ec and R effect of on Nu_x Re_x^-1/2.

Figure 18.

Sc and h effect of on Sh_x Re_x^-1/2.

Figure 19.

Pe and h effect of on Nh_x Re_x^-1/2.

Table 2.

$\mathrm{Local}\mathrm{Skin}\mathrm{friction}\mathrm{coefficients}\mathrm{for}\mathrm{distinct}\mathrm{parameter}$ $M\mathrm{and}\mathrm{\beta }$ .

$M$	$\beta$	$C{f}_x{\mathrm{Re}}_x^{\frac{1}{2}}$
0.1	0.09	$1.57579$
0.15		$1.51523$
0.2		$1.45468$
0.1	0.1	$0.09900$
	0.11	$0.03915$
	0.12	$0.00369$

Table 3.

Local Nusselt number for distinct parameters $\mathrm{Pr},\mathrm{Nb},\mathrm{Nt},R,\mathrm{and}{\beta }_1^{*}$ .

$\mathrm{pr}$	$\mathrm{Nb}$	$\mathrm{Nt}$	$R$	${\beta }_1^{*}$	$N{u}_x{\mathrm{Re}}_x^{-\frac{1}{2}}$
$2.0$	$0.2$	$0.3$	$0.5$	$0.2$	$1.59101$
$3.0$					$1.80979$
$3.5$					$1.92547$
$2.0$	$0.1$				$1.62407$
	$0.4$				$1.52599$
	$0.8$				$1.40046$
	$0.2$	$0.2$			$1.61774$
		$0.4$			$1.56502$
		$0.6$			$1.51529$
		$0.3$	$0.1$		$1.12072$
			$0.3$		$1.35844$
			$0.6$		$1.70606$
			$0.5$	0.1	$1.54275$
				0.3	$1.63935$
				0.5	$1.73631$

Table 4.

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

$\mathrm{Nt}$	$\mathrm{Nb}$	$E$	$S{h}_x{\mathrm{Re}}_x^{-\frac{1}{2}}$
$0.3$	$0.2$	$0.6$	$0.859641$
$0.4$			$0.828415$
$0.8$			$0.736512$
$0.3$	$0.1$		$0.736233$
	$0.4$		$0.921356$
	$0.8$		$0.952236$
	$0.2$	$0.4$	$0.867019$
		$0.5$	$0.863331$
		$0.7$	$0.855949$

Table 5.

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

$\mathrm{Pe}$	$\mathrm{\Omega }$	$N{n}_x{\mathrm{Re}}_x^{-\frac{1}{2}}$
$0.2$		$0.997537$
$0.6$		$1.02436$
$1.2$		$1.06316$
$0.1$	$0.5$	$0.992413$
	$0.6$	$0.994117$
	$0.7$	$0.995821$

Conclusion

The current study examines steady-state visco-plastic Casson nanofluid flow over an electrically conducting vertically stretched porous surface. The effect of bioconvection, joule heating, thermal radiation, viscous dissipation, thermo-solutal stratifications, and thermal robin condition are into account. By using appropriate similarity transformations, the governing equations are transformed into ordinary differential equations (ODEs). The Homotopy Analysis Method (HAM) is useful to attain the results. The possessions of different parameters that control the flow, such as velocity, nanoparticle volume fraction, temperature, gyrotactic microorganisms, local Sherwood number, shear stress rate, Nusselt numbers, and density of microorganisms, are numerically and graphically presented. Some of key finding are as follows:

• The fluid velocity decreases as both the Casson parameters and magnetic parameters increase.

• Increasing the values of $\mathrm{Pr}$ decreases the temperature of the fluid. Conversely, increasing radiation parameter rises the fluid temperature.

• Raising the thermophoresis parameter significantly quickens the fluid’s concentration and the temperature of the nanoparticles. On the other hand, raising the Brownian motion parameter raises the fluid’s temperature and reduces the nanoparticle concentration.

• The fluid’s temperature is marginally raised by raising the values of Deborah’s number and activation energy.

• Motile microorganisms’ field $\phi \left(\eta \right)$ reduces when motile density increases.

• The material features are improved by minimizing the profiles of motile microorganisms. The peclet number Pe and the microbe concentration differential parameter Ω.