A Framework for the Magnetic Dipole Effect on the Thixotropic Nanofluid Flow Past a Continuous Curved Stretched Surface

Abstract

The magnetic dipole effect for thixotropic nanofluid with heat and mass transfer, as well as microorganism concentration past a curved stretching surface, is discussed. The flow is in a porous medium, which describes the Darcy–Forchheimer model. Through similarity transformations, the governing equations of the problem are transformed into non-linear ordinary differential equations, which are then processed using an efficient and powerful method known as the homotopy analysis method. All the embedded parameters are considered when analyzing the problem through solution. The dipole and porosity effects reduce the velocity, while the thixotropic nanofluid parameter increases the velocity. Through the dipole and radiation effects, the temperature is enhanced. The nanoparticles concentration increases as the Biot number and curvature, solutal, chemical reaction parameters increase, while it decreases with increasing Schmidt number. The microorganism motile density decreases as the Peclet and Lewis numbers increase. Streamlines demonstrate that the trapping on the curved stretched surface is uniform.

1. Introduction

Non-Newtonian fluid flows have already captivated the attention of researchers. These materials are used extensively in bioengineering, geophysics, pharmaceuticals, chemical and nuclear industries, polymer solutions, cosmetics, oil storage engineering, paper manufacturing, and other fields. Clearly, no single constitutive relationship can account for all non-Newtonian materials based on behavioral shear stresses. It is distinct from Newtonian and creeping viscous fluids [1]. As a result, several non-Newtonian fluid models have been proposed [2,3,4,5]. One such model is the thixotropic fluid model. The shear thinning fluid differs from the thixotropic fluid in that the shear thinning fluid has less viscosity and its shear rate increases over time, whereas the viscosity of a thixotropic fluid decreases with a constant shear rate. A few studies on thixotropic and non-Newtonian fluid models can be found in the references [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21].

The suitability of the flow of porous media for a wide range of practical industrial applications, including crude oil extraction, food storage, fossil fuels, geothermal systems, porous insulation, packaged beds, petroleum technology and waste disposal, etc., has attracted considerable interest among scientists. Various models, such as Darcy–Forchheimer and Darcy and Brinkman, have been introduced in the literature. Researchers are interested in studying porous media models because of its importance. These models may be established in the light of Darcy’s law, where the pressure gradient is directly related to the average velocity of the volume. Darcy’s formula may be slow and porous with no effect of inertia, porosity variable, solid boundary or thermal dispersion. To achieve the desired accurate results, the presence of non-Darcian effects is crucial for the porous media analysis as discussed by Nield and Bejan [22]. These effects are presented by Forchheimer [23], with response of the square velocity term to the Darcian velocity term. Subsequently, Morris [24] coined the term “Forchheimer”, applicable to the high Reynolds number. Kishan and Maripala [25] investigated the effects of viscous dissipation and thermophoresis analysis on the mixed convection in Darcy–Forchheimer MHD fluid via porous saturated media. Rauf et al. [26] investigated the thermal radiation viscous fluid flow in Darcy–Forchheimer porous space over a curved moving surface. Jagadha [27] studied the Darcy–Forchheimer mixed convection MHD boundary layer flow with viscous dissipation in nanofluid saturated with porous media.

The ferrofluids describe a specific category of magnetizable fluids with interesting effects that have a tremendous technological impact. The ferrofluid is often a single magnetic particle domain colloidal suspension with a size of approximately 10 nm. Aerodynamics and computer peripherals, avionics, cooling agents, crystal processing, filtration, fiber optics, loudspeakers, laser based operational devices, nuclear power plants, robotics, semiconductor processing, refrigeration, plastic drawing, etc. regularly use ferrofluids in a number of industrial applications. Like these innumerable applications, a number of studies of ferrofluids have been carried out by researchers and scientists. Andersson [28] investigated the ferrofluid with special effects of magnetic dipole. Hayat et al. [29] investigated the magnetic dipole effect on radiative ferromagnetic Williamson fluid flow. Some important studies in connection with ferrofluid are presented in the references [30,31,32].

For industrial, chemical applications and bio-engineering, such as drying, energy transport between desert coolers and cooling towers, food processing, production of polymers, evaporation, and metal work, the study of chemical reactions (productive/destructive) are essential for stretching surfaces. Extrusion on the stretched surface, heat transfer in the MHD stagnation point flow under the effect of chemical reactions and transpiration are analyzed by Mabood et al. [33]. Narayana and Babu [34] presented a study of MHD Jeffrey fluid flow with the chemical reaction effects over a stretching sheet numerically. Mixed peristaltic convective flow of Prandtl fluid to Hall current and chemical reaction effects is investigated by Hayat et al. [35]. Hayat et al. [36] studied the hydromagnetic flow of viscous fluid with chemical reaction and thermal radiation through a curved stretching sheet. The others relevent and stretching surfaces studies can be seen in the references [37,38,39,40,41,42,43].

Bioconvection is a common phenomenon that occurs in suspensions due to the up-swimming of microorganisms that have a marginally higher density than water. When the upper surface of the suspensions becomes too dense due to microorganism proliferation, it becomes porous and microorganisms collapse, resulting in bioconvection. These microorganisms may exhibit gravitaxis, gyrotaxis, or oxytaxis. Supporting gyrotactic microorganisms for fluids aids in mass conversion, mixing micro-scales, and increasing fluid stability, particularly in micro-volumes. A number of researchers have investigated its various effects on fluid flow. Chamkha et al. [44] investigated the radiating effects on gyrotactic microorganisms on a vertical plate with fluid variability of temperature in natural bioconvection flow. Raju and Sandeep [45] proposed a mathematical model to study bioconvection through the use of non-linear chemical and thermal radiation in a rotational fluid. Hady et al. [46] studied the unsteady bioconvection thermal boundary layer flow in the presence of gyrotactic microorganisms on a stretching plate and a vertical cone in a porous medium. Recent investigations on bioconvection can be found in the references [47,48,49,50,51,52,53,54,55,56].

The current study discusses the magnetic dipole effect on thixotropic fluid with heat and mass transfer, as well as microorganism concentration passing through a curved stretching surface. The Darcy–Forchheimer model is used to describe the flow in a porous medium. Thermal radiation and viscous dissipation effects are also taken into consideration. Through appropriate similarity transformations, partial differential equations are transformed into ordinary differential equations and solved using a well-known technique, namely homotopy analysis method HAM [57,58,59]. Many researchers [40,47,60,61,62,63] have used HAM to solve their research problems. The results obtained are used to discuss graphically the effects of all the relevant parameters on all dimensionless profiles.

2. Methods

Two-dimensional hydrodynamic incompressible ferromagnetic thixotropic nanofluid past a stretched curved sheet under the influence of magnetic dipole is considered. x and y are used for curvilinear coordinates. The stretching surface is curled in a radius circle $R^{\prime }$. Based on the linear velocity $u=Ax$ (A is constant), the sheet is stretched in the x-direction and y-direction, which is transverse to x-direction. The magnetic field of strength $B_0$ is perpendicular to the flow direction. The surface is submerged in a non-Darcy porous medium. As the Reynolds number (due to a magnet) is smaller in the present problem, the electrical and induced magnetic fields are ignored. Convective heat and mass transfer conditions are observed. In addition, a chemical reaction of the first order is also considered.

In conjunction with the above assumptions, the boundary layer of the equations involved are governed by the following terms [7,26,27,29,30]

$$\frac{\partial \left\{(y+R^{\prime })v\right\}}{\partial y}+R^{\prime }\frac{\partial u}{\partial x}=0,$$

(1)

$$\frac{u^2}{y+R^{\prime }}=\frac{1}{\rho }\frac{\partial p}{\partial y},$$

(2)

$$\begin{array}{cc}\hfill & \rho (v\frac{\partial u}{\partial y}+\frac{R^{\prime }u}{y+R^{\prime }}\frac{\partial u}{\partial x}+\frac{uv}{y+R^{\prime }})=\frac{R^{\prime }}{y+R^{\prime }}\frac{\partial p}{\partial x}+\mu \left(\frac{{\partial }^{2}u}{\partial y^2}-\frac{u}{{(y+R^{\prime })}^2}+\frac{1}{y+R^{\prime }}\frac{\partial u}{\partial y}\right)\hfill \\ & -6R_1{\left(\frac{\partial u}{\partial y}\right)}^2\left(\frac{{\partial }^{2}u}{\partial y^2}\right)+4R_2[\frac{\partial u}{\partial y}\frac{{\partial }^{2}u}{\partial y^2}\left(u\frac{{\partial }^{2}u}{\partial x\partial y}+v\frac{{\partial }^{2}u}{\partial y^2}\right)\hfill \\ & +{\left(\frac{\partial u}{\partial y}\right)}^2\left(u\frac{{\partial }^{3}u}{\partial x\partial y^2}+v\frac{{\partial }^{3}u}{\partial y^3}+\frac{\partial u{\partial }^{2}u}{\partial y\partial x\partial y}+\frac{\partial v{\partial }^{2}u}{\partial y\partial y^2}\right)]-\frac{\mu S_1}{k_o}u-\frac{\rho C_b{S}_1}{\sqrt{k_o}}{u}^2+{\mu }_{o}M\frac{\partial H}{\partial x},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & \left(\rho c_p\right)\left(\frac{R^{\prime }u}{y+R^{\prime }}\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)=\frac{k_T}{y+R^{\prime }}\left[\frac{\partial T}{\partial y}+(y+R^{\prime })\frac{{\partial }^{2}T}{\partial y^2}\right]-2\mu R_1{\left(\frac{\partial u}{\partial y}\right)}^4+4\mu R_{2}u\frac{{\partial }^{2}u}{\partial x\partial y}{\left(\frac{\partial u}{\partial y}\right)}^3\hfill \\ & +4\mu R_{2}v\frac{{\partial }^{2}u}{\partial y^2}{\left(\frac{\partial u}{\partial y}\right)}^3+\left(u\frac{\partial H}{\partial x}+v\frac{\partial H}{\partial y}\right){\mu }_{o}T\frac{\partial M}{\partial T}-\frac{k_T}{y+R^{\prime }}\left(\frac{\partial q_r}{\partial y}(y+R^{\prime })\right),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & \frac{R^{\prime }}{y+R^{\prime }}u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=\frac{D}{y+R^{\prime }}\left(\frac{\partial C}{\partial y}+(y+R^{\prime })\frac{{\partial }^{2}C}{\partial y^2}\right)-K_c(C-C_{\infty }),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & \left(\frac{R^{\prime }}{y+R^{\prime }}\right)u\frac{\partial N}{\partial x}+v\frac{\partial N}{\partial y}+\frac{b{W}_c}{C_w-C_{\infty }}\frac{\partial \left(N\frac{\partial C}{\partial y}\right)}{\partial y}=D_m\frac{{\partial }^{2}N}{\partial y^2},\hfill \end{array}$$

(6)

with boundary conditions

$$\begin{array}{cc}\hfill & u=Ax=U_w\left(x\right),v=0,-k_T\frac{\partial T}{\partial y}=h_1\left(T_w-T\right),-D\frac{\partial C}{\partial y}=k_m\left(C_w-C\right),N=N_{w}aty=0,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & u\to 0,\frac{\partial u}{\partial y}\to 0,v\to 0,T\to T_{\infty },C\to C_{\infty },N\to N_{\infty },asy\to \infty ,\hfill \end{array}$$

(8)

where velocity components are (u,v) in the radial (x-direction) and transverse (y-direction), $k_m$ is the mass transfer coefficient, $h_1$ is the convective heat transfer coefficient, $R_1$ and $R_2$ are the material constants, diffusion coefficient is D, constant fluid density is $\rho$, $k_T$ is the thermal conductivity, $\sigma$ is the electrical conductivity, $k_o$ is permeability of porous medium, the effective dynamic viscosity is $\mu$, magnetic permeability is ${\mu }_o$, heat capacitance is $\left(\rho c_p\right)$, first order chemical reaction parameter is $K_c$, microorganisms diffusion is $D_m$, speed of gyrotactic cell is $W_c$, b is the chemotaxis, $C_b$ is the drag coefficient, $S_1$ is the porosity of porous medium, T is the temperature, C is the concentration, N is the gyrotactic microorganisms concentration, and $C_{\infty }$, $T_{\infty }$, and $N_{\infty }$, respectively, stand for the nanoparticles concentration, temperature, and density of microorganisms far away from the surface.

Rosseland and Ozisik approximation allows to write the radiation heat flux $q_r$ with ${\sigma }^{*}$ Stenfan-Boltzman, and ${\beta }_R$ mean absorption coefficient [64] as:

$$q_r=-\frac{4{\sigma }^{*}}{3{\beta }_R}\frac{\partial T^4}{\partial y}=-\frac{4T_{\infty }^3{\sigma }^{*}}{3{\beta }_R}\frac{\partial T}{\partial y}.$$

(9)

Magnetic Dipole

The characteristics of the magnetic field have an effect on the flow of ferrofluid due to the magnetic dipole. Magnetic dipole effects are recognized by the magnetic scalar potential $\mathsf{\Phi }$ [29] shown in Equation (10)

$$\begin{array}{cc}\hfill & \mathsf{\Phi }=\frac{\gamma }{2\pi }\frac{x}{x^2+{(y+c)}^2},\hfill \end{array}$$

(10)

where $\gamma$ stands for magnetic field strength at the source, c is the distance of the line currents from the leading edge. $H_x$ and $H_y$ are taken as the components of magnetic field as shown in Equations (11) and (12)

$$\begin{array}{cc}\hfill & H_x=-\frac{\partial \mathsf{\Phi }}{\partial x}=\frac{\gamma }{2\pi }\frac{x^2-{(y+c)}^2}{{[x^2+{(y+c)}^2]}^2},\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & H_y=-\frac{\partial \mathsf{\Phi }}{\partial y}=\frac{\gamma }{2\pi }\frac{2x(y+c)}{{[x^2+{(y+c)}^2]}^2}.\hfill \end{array}$$

(12)

The magnetic field H is usually proportional to the components of magnetic field $H_x$ and $H_y$, gradient along x and y directions respectively. It is therefore defined in Equation (13) as

$$H=\sqrt{H_x^2+H_y^2}.$$

(13)

It is considered that the temperature-dependent variation of magnetization M is linear as shown in Equation (14)

$$\begin{array}{cc}\hfill & M=K_1(T-T_{\infty }),\hfill \end{array}$$

(14)

where $K_1$ identifies the coefficient of the ferromagnetic. The physical schematic of the heated ferrofluid can be seen in Figure 1.

Considering the following transformations [26], with $\nu$ as kinematic viscosity, A is constant:

$$\begin{array}{cc}\hfill & u=Ax{f}^{\prime }\left(\zeta \right),v=-\left(\frac{R^{\prime }}{y+R^{\prime }}\right)\sqrt{A\nu }f\left(\zeta \right),p=\rho A^2{x}^{2}p\left(\zeta \right),\zeta =y\sqrt{\frac{A}{\nu }},\hfill \\ & \theta \left(\zeta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\varphi \left(\zeta \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }},\chi \left(\zeta \right)=\frac{N-N_{\infty }}{N_w-N_{\infty }},\hfill \end{array}$$

(15)

By the application of Equation (15), Equations (2)–(7) provide the following Equations (16), (18)–(22)

$$\begin{array}{cc}\hfill & p^{\prime }=\frac{f^{\prime 2}}{\zeta +{\alpha }_1},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & f^{‴}-\left(\frac{{\alpha }_1}{\zeta +{\alpha }_1}\right)\left[f{f}^{″}-f^{\prime 2}+A_1{f}^{\prime }f\right]+N{n}_1{f}^{″2}f{}^{‴}+N{n}_2\left[f^{″4}-\frac{{\alpha }_1}{\zeta +{\alpha }_1}\left(f^{‴2}f{f}^{″}+f^{″2}f{f}^{\prime \prime \prime \prime }\right)\right]\hfill \\ & -P_1{f}^{\prime }-L_i{f}^{\prime 2}+\frac{2\beta }{{(\zeta +d)}^4}\theta +2\left(\frac{{\alpha }_1}{\zeta +{\alpha }_1}\right)p=0,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & \left(1+Rd\right)\left({\theta }^{″}+\frac{{\theta }^{\prime }}{(\zeta +{\alpha }_1)}\right)+Pr\left(\frac{{\alpha }_1}{\zeta +{\alpha }_1}\right)f{\theta }^{\prime }+\frac{1}{3}PrN{n}_{1}Ec{f}^{″4}+PrN{n}_{1}Ec\left[f^{\prime }{f}^{″}-f{f}^{‴}\right]\hfill \\ \hfill & +\frac{2\beta \lambda (\theta -ϵ)f}{{(\zeta +d)}^3}+\beta \lambda (\theta -ϵ)\left[\frac{2f^{\prime }}{{(\zeta +d)}^4}+\frac{4f}{{(\zeta +d)}^5}\right]=0,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & {\varphi }^{″}+\frac{{\varphi }^{\prime }}{(\zeta +{\alpha }_1)}+\left(\frac{{\alpha }_1}{\zeta +{\alpha }_1}\right)Scf{\varphi }^{\prime }-\delta Sc\varphi =0,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & {\chi }^{″}+Pe\left[{\varphi }^{\prime }{\chi }^{\prime }+{\varphi }^{″}\chi +N_{\delta }{\varphi }^{″}\right]+Le\left(\frac{{\alpha }_1}{\zeta +{\alpha }_1}\right)f{\chi }^{\prime }=0.\hfill \end{array}$$

(20)

To eliminate the pressure term, integrating (16) to get p and replacing it, then (17) becomes

$$\begin{array}{cc}\hfill & f^{‴}-\left(\frac{{\alpha }_1}{\zeta +{\alpha }_1}\right)\left[f{f}^{″}-f^{\prime 2}+A_1{f}^{\prime }f\right]+N{n}_1{f}^{″2}f{}^{‴}+N{n}_2\left[f^{″4}-\frac{{\alpha }_1}{\zeta +{\alpha }_1}\left(f^{‴2}f{f}^{″}+f^{″2}f{f}^{\prime \prime \prime \prime }\right)\right]\hfill \\ & -P_1{f}^{\prime }-L_i{f}^{\prime 2}+\frac{2\beta }{{(\zeta +d)}^4}\theta +\left(\frac{{\alpha }_1}{{(\zeta +{\alpha }_1)}^2}\right)(2f{f}^{″}-f^{\prime 2})=0,\hfill \end{array}$$

(21)

and the boundary conditions become

$$\begin{array}{cc}\hfill & f^{\prime }\left(0\right)=1,f\left(0\right)=0,f(\infty )=0,f^{″}(\infty )=0,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & {\theta }^{\prime }\left(0\right)=-B{i}_1[1-\theta \left(0\right)],\theta (\infty )=0,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & {\varphi }^{\prime }\left(0\right)=-B{i}_2[1-\varphi \left(0\right)],\varphi (\infty )=0,\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & {\chi }^{\prime }\left(0\right)=1,\chi (\infty )=0,\hfill \end{array}$$

(25)

where $A_1$ is the ratio of rate constants, ${\alpha }_1$ is the curvature parameter, d is the dimensionless distance, $N{n}_1$ and $N{n}_2$ are the non-Newtonian parameters, $\beta$ is the ferrohydrodynamic interaction parameter, heat dissipation parameter is $\lambda$, $\epsilon$ is the curie temperature, Prandtl number is $Pr$, radiation parameter is $Rd$, Eckert number is $Ec$, chemical reaction parameter is $\delta$, the Schmidt number is $Sc$, local inertia parameter is $Li$, porosity parameter is $P_1$, Lewis number is $Pe$, Lewis number is $Le$, thermal Biot number is $B{i}_1$ and concentration Biot number is $B{i}_2$, which are defined by

$$\begin{array}{cc}\hfill & A_1=\sqrt{\frac{\mu }{\rho A}},{\alpha }_1=R^{\prime }\sqrt{\frac{A}{\nu }},N{n}_1=\frac{4R_2{A}^4{x}^2}{\rho {\nu }^2},N{n}_2=\frac{-6R_1{A}^4{x}^2}{\rho \nu },P_1=\frac{\mu S_1}{\rho A{k}_o^{*}},L_i=\frac{C_b{S}_1}{\sqrt{k_o^{*}}},\hfill \\ & \beta =\frac{\gamma {\mu }_o{K}_1\rho (T_w-T_{\infty })}{2\pi {\mu }^2},Pr=\frac{\mu C_P}{k_T},Ec=\frac{{\left(Ax\right)}^2}{C_P(T_w-T_{\infty })},\lambda =\frac{A{\mu }^2}{\rho (T_w-T_{\infty })k_T},d=\sqrt{\frac{A{c}^2}{\nu }},\hfill \\ & \delta =\frac{A{K}_c}{\nu },Rd=\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}},Sc=\frac{\nu }{D},Pe=\frac{b{W}_c}{D_m},Le=\frac{\nu }{D_n},B{i}_1=\frac{h_1}{k_T}\sqrt{\frac{\nu }{A}},B{i}_2=\frac{k_m}{k_T}\sqrt{\frac{\nu }{A}},\hfill \\ & ϵ=\frac{T_{\infty }}{T_{\infty }-T_w},N_{\delta }=\frac{N_{\infty }}{N_w-N_{\infty }}.\hfill \end{array}$$

(26)

The quantities of interest, such as coefficient of skin friction, local Nusselt, Sherwood and local density numbers, are determined by

$$C_f=\frac{{\tau }_{yx}}{\rho {\left(Ax\right)}^2},N{u}_x=\frac{-x{q}_w}{k_T(T_w-T_{\infty })},S{h}_x=\frac{-x{q}_m}{D(C_w-C_{\infty })},S{n}_x=\frac{-x{q}_n}{D(N_w-N_{\infty })},$$

(27)

where

$${\tau }_{yx}=\mu u_y{|}_{y=0},q_w=-k_T{T}_y{|}_{y=0}-\frac{4T_{\infty }^3{\sigma }^{*}}{3{\beta }_R}\frac{\partial T}{\partial y}{|}_{y=0},q_m=-D{C}_y{|}_{y=0},q_n=D_m{N}_y{|}_{y=0}.$$

(28)

By putting values from Equation (28) in Equation (27), it is obtained that

$$\begin{array}{cc}\hfill & C_f=\frac{1}{R{e}_x}\left(f^{″}\left(0\right)-\frac{f^{\prime }\left(0\right)}{{\alpha }_1}\right),Nu=-R{e}_x^{0.5}(1+Rd){\theta }^{\prime }\left(0\right),\hfill \\ & Sh=-R{e}_x^{0.5}{\varphi }^{\prime }\left(0\right),Sn=-R{e}_x^{0.5}{\chi }^{\prime }\left(0\right).\hfill \end{array}$$

(29)

3. HAM Solution

The initial guesses and the linear operators are taken as

$$f_0\left(\zeta \right)=A\zeta +(1-A)(1-e^{-\zeta }),{\theta }_0\left(\zeta \right)=\frac{B{i}_1}{1+B{i}_1}{e}^{-\zeta },{\varphi }_0\left(\zeta \right)=\frac{B{i}_2}{1+B{i}_2}{e}^{-\zeta },{\chi }_0=e^{-\zeta }.$$

(30)

Equation (30) satisfies the properties as given below

$$\begin{array}{cc}\hfill & L_f(E_1+E_2{e}^{\zeta }+E_3{e}^{-\zeta })=0,L_{\theta }(E_4{e}^{\zeta }+E_5{e}^{-\zeta })=0,\hfill \\ & L_{\varphi }(E_6{e}^{\zeta }+E_7{e}^{-\zeta })=0,L_{\chi }(E_8{e}^{\zeta }+E_9{e}^{-\zeta })=0,\hfill \end{array}$$

(31)

where $E_i(i=1,$…$,9)$ indicates the arbitrary constants.

The corresponding zeroth order form of the problems are

$$\begin{array}{cc}\hfill & (1-q)L_f[f(\zeta ,q)-f_0\left(\zeta \right)]=q{h}_f{N}_f[f(\zeta ,q),\theta (\zeta ,q)],\hfill \\ & (1-q)L_{\theta }[\theta (\zeta ,q)-{\theta }_0\left(\zeta \right)]=q{h}_{\theta }{N}_{\theta }[\theta (\zeta ,q),f(\zeta ,q)],\hfill \\ & (1-q)L_{\varphi }[\varphi (\zeta ,q)-{\varphi }_0\left(\zeta \right)]=q{h}_{\varphi }{N}_{\varphi }[\varphi (\zeta ,q),f(\zeta ,q)],\hfill \\ & (1-q)L_{\chi }[\chi (\zeta ,q)-{\chi }_0\left(\zeta \right)]=q{h}_{\chi }{N}_{\chi }[\chi (\zeta ,q),\varphi (\zeta ,q),f(\zeta ,q)],\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & f(0,q)=0,f^{\prime }(0,q)=1,f^{\prime }(\infty ,q)=A,{\theta }^{\prime }(0,q)=-B{i}_1(1-\theta (0,q)),\theta (\infty ,q)=0,\hfill \\ & {\varphi }^{\prime }(0,q)=-B{i}_2(1-\varphi (0,q)),\varphi (\infty ,q)=0{\chi }^{\prime }(0,q)=1,\chi (\infty ,q)=0,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & {\mathbf{N}}_f\left[f(\zeta ,q)\right]=\frac{{\partial }^{3}f(\zeta ,q)}{\partial {\zeta }^3}-\left(\frac{{\alpha }_1}{\zeta +{\alpha }_1}\right)\left[f(\zeta ,q)\frac{{\partial }^{2}f(\zeta ,q)}{\partial {\zeta }^2}-{\left(\frac{\partial f(\zeta ,q)}{\partial \zeta }\right)}^2+A_1\frac{\partial f(\zeta ,q)}{\partial \zeta }f(\zeta ,q)\right]\hfill \\ & +N{n}_1{\left(\frac{{\partial }^{2}f(\zeta ,q)}{\partial {\zeta }^2}\right)}^2\frac{{\partial }^{3}f(\zeta ,q)}{\partial {\zeta }^3}+N{n}_2[{\left(\frac{{\partial }^{2}f(\zeta ,q)}{\partial {\zeta }^2}\right)}^4-\frac{{\alpha }_1}{\zeta +{\alpha }_1}({\left(\frac{{\partial }^{3}f(\zeta ,q)}{\partial {\zeta }^3}\right)}^{2}f(\zeta ,q)\frac{{\partial }^{2}f(\zeta ,q)}{\partial {\zeta }^2}\hfill \\ & +{\left(\frac{{\partial }^{2}f(\zeta ,q)}{\partial {\zeta }^2}\right)}^{2}f(\zeta ,q)\frac{{\partial }^{4}f(\zeta ,q)}{\partial {\zeta }^4}\left)\right]-P_1\frac{\partial f(\zeta ,q)}{\partial \zeta }-L_i{\left(\frac{\partial f(\zeta ,q)}{\partial \zeta }\right)}^2\hfill \\ & +\frac{2\beta }{{(\zeta +d)}^4}\theta (\zeta ,q)+\left(\frac{{\alpha }_1}{{(\zeta +{\alpha }_1)}^2}\right)\left(2f(\zeta ,q)\frac{{\partial }^{2}f(\zeta ,q)}{\partial {\zeta }^2}-{\left(\frac{\partial f(\zeta ,q)}{\partial \zeta }\right)}^2\right),\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & {\mathbf{N}}_{\theta }\left[\theta (\zeta ,q)\right]=\left(1+Rd\right)\left(\frac{{\partial }^2\theta (\zeta ,q)}{\partial {\zeta }^2}+\frac{1}{(\zeta +{\alpha }_1)}\frac{\partial \theta (\zeta ,q)}{\partial \zeta }\right)+Pr\left(\frac{{\alpha }_1}{\zeta +{\alpha }_1}\right)f(\zeta ,q)\frac{\partial \theta (\zeta ,q)}{\partial \zeta }\hfill \\ & +\frac{1}{3}PrN{n}_{1}Ec{\left(\frac{{\partial }^{2}f(\zeta ,q)}{\partial {\zeta }^2}\right)}^4+PrN{n}_{2}Ec\left[\frac{\partial f(\zeta ,q)}{\partial \zeta }\frac{{\partial }^{2}f(\zeta ,q)}{\partial {\zeta }^2}-f(\zeta ,q)\frac{{\partial }^{3}f(\zeta ,q)}{\partial {\zeta }^3}\right]\hfill \\ & +\frac{2\beta \lambda \left(\theta \right(\zeta ,q)-ϵ)f(\zeta ,q)}{{(\zeta +d)}^3}+\beta \lambda (\theta -ϵ)\left[\frac{2}{{(\zeta +d)}^4}\frac{\partial f(\zeta ,q)}{\partial \zeta }+\frac{4f(\zeta ,q)}{{(\zeta +d)}^5}\right],\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & {\mathbf{N}}_{\varphi }\left[\varphi (\zeta ,q)\right]=\frac{{\partial }^2\varphi (\zeta ,q)}{\partial {\zeta }^2}+\frac{1}{(\zeta +{\alpha }_1)}\frac{\partial \varphi (\zeta ,q)}{\partial \zeta }+\left(\frac{{\alpha }_1}{\zeta +{\alpha }_1}\right)Scf(\zeta ,q)\frac{\partial \varphi (\zeta ,q)}{\partial \zeta }-\delta Sc\varphi (\zeta ,q),\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill & {\mathbf{N}}_{\chi }\left[\chi (\zeta ,q)\right]=\frac{{\partial }^2\chi (\zeta ,q)}{\partial {\zeta }^2}+Pe\left[\frac{\partial \varphi (\zeta ,q)}{\partial \zeta }\frac{\partial \chi (\zeta ,q)}{\partial \zeta }+\frac{{\partial }^2\varphi (\zeta ,q)}{\partial {\zeta }^2}\chi (\zeta ,q)+N_{\delta }\frac{{\partial }^2\varphi (\zeta ,q)}{\partial {\zeta }^2}\right]\hfill \\ & +Le\left(\frac{{\alpha }_1}{\zeta +{\alpha }_1}\right)f(\zeta ,q)\frac{\partial \varphi (\zeta ,q)}{\partial \zeta },\hfill \end{array}$$

(37)

where $q\in [0,1]$ is the embedding parameter while ${\mathbf{N}}_f$, ${\mathbf{N}}_{\theta }$, ${\mathbf{N}}_{\varphi }$, and ${\mathbf{N}}_{\chi }$ are the nonlinear operators.

The m-$th$ order deformation problems are as follows

$$L_f[f_m(\zeta ,q)-{\eta }_m{f}_{m-1}\left(\zeta \right)]=h_f{\mathcal{R}}_{f,m}\left(\zeta \right),$$

(38)

$$L_{\theta }[{\theta }_m(\zeta ,q)-{\eta }_m{\theta }_{m-1}\left(\zeta \right)]=h_{\theta }{\mathcal{R}}_{\theta ,m}\left(\zeta \right),$$

(39)

$${\mathcal{L}}_{\varphi }[{\varphi }_m(\zeta ,q)-{\eta }_m{\varphi }_{m-1}\left(\zeta \right)]=h_{\varphi }{\mathcal{R}}_{\varphi ,m}\left(\zeta \right),$$

(40)

$$L_{\chi }[{\chi }_m(\zeta ,q)-{\eta }_m{\chi }_{m-1}\left(\zeta \right)]=h_{\chi }{\mathcal{R}}_{\chi ,m}\left(\zeta \right),$$

(41)

$$\begin{array}{cc}\hfill & f_m\left(0\right)=f_m^{\prime }\left(0\right)=f_m^{\prime }(\infty )=0,\hfill \\ \hfill & {\theta }_m^{\prime }\left(0\right)-B{i}_1{\theta }_m\left(0\right)={\theta }_m(\infty )=0,\hfill \\ \hfill & {\varphi }_m^{\prime }\left(0\right)-B{i}_2{\varphi }_m\left(0\right)={\varphi }_m(\infty )=0,\hfill \\ \hfill & {\chi }_m^{\prime }\left(0\right)={\chi }_m\left(0\right)={\chi }_m(\infty )=0,\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill {\mathcal{R}}_f^m\left(\zeta \right)& =f_{m-1}^{‴}-\left(\frac{{\alpha }_1}{\zeta +{\alpha }_1}\right)\left[\sum _{r=0}^{m-1}{f}_{m-1-r}{f}_r^{″}-\sum _{r=0}^{m-1}{f}_{m-1-r}^{\prime }{f}_r^{\prime }+A_1\sum _{r=0}^{m-1}{f}_{m-1-r}{f}_r^{\prime }\right]\hfill \\ & +N{n}_1\sum _{r=0}^{m-1}\left(\sum _{k=0}^r{f}_{m-1-r}^{″}{f}_{r-k}^{″}\right)f_k^{‴}+N{n}_2[\sum _{r=0}^{m-1}\left(\sum _{k=0}^r\left(\sum _{p=0}^k{f}_{m-1-r}^{″}{f}_{r-k}^{″}\right)f_{k-p}^{″}\right)f_p^{″}\hfill \\ & -\frac{{\alpha }_1}{\zeta +{\alpha }_1}(\sum _{r=0}^{m-1}\left(\sum _{k=0}^r\left(\sum _{p=0}^k{f}_{m-1-r}{f}_{r-k}^{‴}\right)f_{k-p}^{‴}\right)f_p^{\prime \prime \prime \prime }\hfill \\ & +\sum _{r=0}^{m-1}\left(\sum _{k=0}^r\left(\sum _{p=0}^k{f}_{m-1-r}{f}_{r-k}^{″}\right)f_{k-p}^{″}\right)f_p^{\prime \prime \prime \prime }\left)\right]-P_1{f}_{m-1}^{\prime }-L_i\sum _{r=0}^{m-1}{f}_{m-1-r}^{\prime }{f}_r^{\prime }\hfill \\ & +\frac{2\beta }{{(\zeta +d)}^4}{\theta }_{m-1}+\frac{{\alpha }_1}{{(\zeta +{\alpha }_1)}^2}\left(2\sum _{r=0}^{m-1}{f}_{m-1-r}{f}_r^{″}-\sum _{r=0}^{m-1}{f}_{m-1-r}^{\prime }{f}_r^{\prime }\right),\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill {\mathcal{R}}_{\theta }^m\left(\zeta \right)& =\left(1+Rd\right)\left({\theta }_{m-1}^{″}+\frac{{\theta }_{m-1}^{\prime }}{(\zeta +{\alpha }_1)}\right)+Pr\left(\frac{{\alpha }_1}{\zeta +{\alpha }_1}\right)\sum _{r=0}^{m-1}{f}_{m-1-r}{\theta }_r^{\prime }\hfill \\ & +\frac{1}{3}PrN{n}_{1}Ec\sum _{r=0}^{m-1}\left(\sum _{k=0}^r\left(\sum _{p=0}^k{f}_{m-1-r}^{″}{f}_{r-k}^{″}\right)f_{k-p}^{″}\right)f_p^{″}\hfill \\ & +PrN{n}_{2}Ec\left[\sum _{r=0}^{m-1}{f}_{m-1-r}^{\prime }{f}_r^{″}-\sum _{r=0}^{m-1}{f}_{m-1-r}{f}_r^{‴}\right]+\frac{2\beta \lambda ({\theta }_{m-1}-ϵ)f_{m-1}}{{(\zeta +d)}^3}\hfill \\ & +\beta \lambda ({\theta }_{m-1}-ϵ)\left[\frac{2f_{m-1}^{\prime }}{{(\zeta +d)}^4}+\frac{4f_{m-1}}{{(\zeta +d)}^5}\right],\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill {\mathcal{R}}_{\varphi }^m\left(\zeta \right)& ={\varphi }_{m-1}^{″}+\frac{{\varphi }_{m-1}^{\prime }}{(\zeta +{\alpha }_1)}+\left(\frac{{\alpha }_1}{\zeta +{\alpha }_1}\right)Sc\sum _{r=0}^{m-1}{f}_{m-1-r}{\varphi }_r^{\prime }-\delta Sc{\varphi }_{m-1},\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill {\mathcal{R}}_{\chi }^m\left(\zeta \right)& ={\chi }_{m-1}^{″}+Pe\left[\sum _{r=0}^{m-1}{\varphi }_{m-1-r}^{\prime }{\chi }_r^{\prime }+\sum _{r=0}^{m-1}{\varphi }_{m-1-r}^{″}{\chi }_r+N_{\delta }{\varphi }_{m-1}^{″}\right]\hfill \\ & +Le\left(\frac{{\alpha }_1}{\zeta +{\alpha }_1}\right)\sum _{r=0}^{m-1}{f}_{m-1-r}{\chi }_r^{\prime },\hfill \end{array}$$

(46)

where

$${\eta }_m=\left\{\begin{array}{c}0,\mathrm{if}m\le 1\hfill \\ 1,\mathrm{if}m>1.\hfill \end{array}\right.$$

(47)

The general solutions are given by

$$\begin{array}{cc}\hfill f_m\left(\zeta \right)& =f_m^{*}\left(\zeta \right)+E_1+E_2{e}^{\zeta }+E_3{e}^{-\zeta },\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill {\theta }_m\left(\zeta \right)& ={\theta }_m^{*}\left(\zeta \right)+E_4{e}^{\zeta }+E_5{e}^{-\zeta },\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill {\varphi }_m\left(\zeta \right)& ={\varphi }_m^{*}\left(\zeta \right)+E_6{e}^{\zeta }+E_7{e}^{-\zeta },\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill {\chi }_m\left(\zeta \right)& ={\chi }_m^{*}\left(\zeta \right)+E_8{e}^{\zeta }+E_9{e}^{-\zeta },\hfill \end{array}$$

(51)

where $(f_m^{*}\left(\zeta \right),{\theta }_m^{*}\left(\zeta \right),{\varphi }_m^{*}\left(\zeta \right),{\chi }_m^{*}\left(\zeta \right))$ are special solutions.

4. Convergence Analysis of the Homotopy Solution

The nonzero auxiliary parameters are involved in the homotopy solution. These parameters are extremely important in controlling and adjusting the convergence acquired by the homotopic series solutions. The h-curves at the 15th order of approximations are sketched to show the acceptable approximate region of convergence. Figure 2 depicts the region as falling within the ranges $-1.8\le h_f\le 0.2$,$-2.5\le h_{\theta }\le 0.8$,$-4.2\le h_{\varphi }\le 1.1$,$-0.4\le h_{\chi }\le 0.4$. The values of parameters used are $A_1=d=1,\lambda =\beta =P_1=Li=Sc=0.3,\epsilon =\delta =Ec=0.1,Pe=Le=N_{\delta }=0.2,N{n}_1=N{n}_2=0.5,Pr=6.8,Re=0.7,{\alpha }_1=Rd=B{i}_1=B{i}_2=0.4$.

5. Discussion

The velocity behavior with the ferromagnetic hydrodynamic interaction parameter $\beta$ can be seen in Figure 3. It demonstrates that the velocity decreases as $\beta$ increases. Ideally, the resistance force known as Lorentz force [65] increases with the $\beta$ increase, and the velocity field decreases. Figure 4 is used to investigate the effect of curvature parameter ${\alpha }_1$ on the velocity profile. It is clearly shown in the figure that the velocity component decreases for larger ${\alpha }_1$. Figure 5 and Figure 6 describe the effects of the thixotropic parameters $N{n}_1$ and $N{n}_2$ on the velocity profile. From these figures, it is observed that $N{n}_1$ and $N{n}_2$ result in an increase in fluid velocity. Ideally, $N{n}_1$ and $N{n}_2$ are associated with the properties of shear thinning, which show a time-dependent changes in viscosity. The higher the fluid under shear stress, the lower the viscosity of nanofluid, which will ultimately lead to an increase in fluid velocity. Figure 7 is used to present the velocity behavior with the porosity parameter $P_1$. The presence of porous medium slows down the field of the flow, resulting in an increase in shear stress on the curved surface, and therefore the velocity profile shows a declining trend by increasing the values of $P_1$. In contrast to the effect seen with $P_1$, change in local inertia parameter $Li$ results in an increase in velocity as shown in Figure 8.

Figure 9 is used used to examine the effect of $\beta$ on temperature. Here, temperature increases with higher values of $\beta$. The temperature profile behavior relating to the higher values of thermal Biot number $B{i}_1$ is shown in Figure 10. The parameter $B{i}_1$ significantly promotes the temperature field in a positive manner attributable to the effective convective heat effects. It is also observed that there is no heat transfer at $B{i}_1=0$. The effect of the heat dissipation parameter $\lambda$ on temperature is shown in Figure 11. The temperature is a decreasing function of $\lambda$. Physically thermal conductivity of liquid decreases with larger $\lambda$, and therefore the temperature decreases. The Eckert number $Ec$ attributes to the temperature profile is shown in Figure 12. For larger $Ec$, temperature and thermal boundary layer thickness were observed to be effected with the increase in $Ec$. In this phenomenon, the heat energy stored in the fluid is caused by friction forces that increase the temperature. The Curie temperature parameter $\epsilon$ effect on temperatureprofile is shown in Figure 13. The temperature decreases through larger values of $\epsilon$. Thermal conductivity of the liquid increases with the larger $\epsilon$. The effect of Prandtl number $Pr$ on temperature profile is shown in Figure 14. The temperature distribution and thermal boundary layer are reduced by higher values of $Pr$, due to which thermal diffusion is reduced. In addition, fluids with a smaller values of $Pr$ slowly decay compared to liquids with larger values of $Pr$. The effect of radiation parameter $Rd$ on temperature profile is discussed in Figure 15. The increase in temperature curves with a larger boundary layer thickness is determined by an increase in $Rd$. Usually, mean absorption coefficient decays for higher estimation of $Rd$ and diffusion flux occurs as a consequence of the temperature gradient, which therefore increases the temperature.

The effect of the concentration Biot number $B{i}_2$ on the nanoparticles concentration profile is shown in Figure 16. In this case, the concentration is increased in response to increase in the $B{i}_2$ values. Figure 17 shows the effect of the $Sc$ on concentration profile. Since $Sc$ is the ratio of momentum to mass diffusivity, the increase in $Sc$ causes a decay in mass diffusivity, thus leading to a decrease in nanoparticles concentration. Figure 18 shows the effect of the curvature parameter ${\alpha }_1$ on the nanoparticles concentration profile. The increase in the curvature parameter results in an increase in the concentration. Figure 19 shows the effect of the chemical reaction parameter $\delta$ on the concentration profile. The nanoparticles concentration is observed to increase for the higher estimates of $\delta$. In fact, the consumption of reactive species rapidly declines as $\delta$ becomes larger.

Figure 20 shows the effect of Peclet number $Pe$ on the microorganisms profile. There is a clear relationship between the reduced density of the microorganisms and the increase in $Pe$. The higher values of $Pe$ indicate the minimum motile diffusivity. Figure 21 shows the impact of Lewis number $Le$ on microorganisms concentration profile. The decrease in the concentration distribution is shown as the Lewis number increases, since it is inversely proportional to the mass diffusion.

The effect of the dimensionless variable $\zeta$ on the streamlines is shown in Figure 22 and Figure 23. It is shown that the number of the trapped boluses increases as the values of $\zeta$ increase, and the streamlines have also been identified to be perpendicular to the surface. The increase in the $\zeta$ increases the shearing motion, which, in fact, results in a higher precession of the flow to the stretching surface.

Table 1. Numerical values of skin friction coefficient with varying values of the parameters $\beta ,{\alpha }_1,P_1,Li,N{n}_1,N{n}_2$.

$\mathit{\beta }$	${\mathit{\alpha }}_1$	${\mathit{P}}_1$	$\mathit{Li}$	${\mathit{Nn}}_1$	${\mathit{Nn}}_2$	$-{\mathit{C}}_{\mathit{f}}$
0.3	0.4	0.3	0.3	0.5	0.5	1.24238
0.7						1.24698
1.1						1.25158
	0.7					1.24467
	1.0					1.23511
	1.3					1.22220
		0.6				1.33715
		0.9				1.43382
		1.2				1.53241
			0.6			1.28757
			0.9			1.33233
			1.2			1.37976
				1.0		1.19970
				1.5		1.16111
				2.0		1.12627
					1.0	1.20631
					1.5	1.17235
					2.0	1.14041

shows a numerical analysis of the skin friction coefficient for $\beta ,{\alpha }_1,P_1,Li,N{n}_1,N{n}_2$. It is discovered that the skin friction coefficient increases with the increasing values of $\beta ,P_1,Li,N{n}_2$, while a reverse trend is observed for ${\alpha }_1$ and $N{n}_1$.

Table 2. Comparative numerical values of the skin friction coefficient with published result with changing values of ${\alpha }_1$ with $P_1=Li=\beta =A_1=N{n}_1=N{n}_2=0$.

${\mathit{\alpha }}_1$	Published Work [66]	Present Work
5	0.7577	0.7569
10	0.8735	0.8736
15	0.9357	0.9357

cross-checks the accurateness of the homotopic solution used in the present investigation. A comparison of skin friction coefficient for the different values of ${\alpha }_1$ with the study [66] is shown for $P_1=Li=\beta =A_1=N{n}_1=N{n}_2=0$ at $\zeta =0$.

Table 3. Numerical values of Nusselt number with varying values of the parameters $\beta ,{\alpha }_1,\lambda ,Pr,Rd,\epsilon ,Ec,N{n}_1,N{n}_2$.

${\mathit{\alpha }}_1$	$\mathit{\lambda }$	$\mathit{\beta }$	$\mathit{Pr}$	$\mathit{Rd}$	$\mathit{\epsilon }$	$\mathit{Ec}$	${\mathit{Nn}}_1$	${\mathit{Nn}}_2$	$\mathit{Nu}$
0.4	0.3	0.3	6.8	0.4	0.1	0.1	0.5	0.5	0.331075
0.7									0.330973
1.0									0.330021
	0.8								0.332224
	1.3								0.332115
	1.8								0.332006
		0.7							0.332245
		1.1							0.332158
		1.5							0.332071
			6.9						0.333827
			10.0						0.335319
			10.11						0.336809
				0.6					0.379260
				0.8					0.426063
				1.0					0.472745
					0.4				0.332614
					0.7				0.332896
					1.0				0.333177
						0.4			0.334739
						0.7			0.337145
						1.0			0.339552
							1.0		0.332088
							1.5		0.331853
							2.0		0.331625
								1.0	0.333376
								1.5	0.334405
								2.0	0.335419

shows the numerical assessment of the local Nusselt number for various values of $\beta ,{\alpha }_1,\lambda ,Pr,Rd,\epsilon ,Ec,N{n}_1,N{n}_2$. It is observed that the local Nusselt number decreases with increasing values of $\beta ,{\alpha }_1,\lambda ,N{n}_1$.

Table 4. Numerical values of Sherwood number with varying values of the parameters ${\alpha }_1,Sc,\delta$.

${\mathit{\alpha }}_1$	$\mathit{Sc}$	$\mathit{\delta }$	$\mathit{Sh}$
0.4	0.3	0.1	0.23643
0.7			0.23658
1.0			0.23661
	0.8		0.23775
	1.3		0.238104
	1.8		0.238455
		0.5	0.238244
		0.9	0.239083
		1.3	0.239915

shows the numerical values of the local Sherwood number for various values of ${\alpha }_1,Sc,\delta$. It is observed that the local Sherwood number decreases with the increasing values of parameters. The tables clearly show that the current findings are completely consistent.

6. Conclusions

The Darcy–Forchheimer hydromagnetic flow of thixotropic nanofluid through a curved stretching sheet with thermal radiation and chemical reaction in the presence of heat and mass transfer, gyrotactic microorganisms, and magnetic dipole is explored. The present study contributes to the findings set out below.

• The velocity decreases with increasing values of ferromagnetic parameter $\beta$ and a curvature parameter ${\alpha }_1$, while it increases with increasing values of $N{n}_1$, $N{n}_2$ and $P_1$.
• The temperature increases with increasing values of $Ec$, $\beta$, and $Rd$ and decays with increasing values of $Pr$.
• The nanoparticles concentration decreases with increasing values of $Sc$ and $\delta$, while it increases with increasing values of $B{i}_2$ and ${\alpha }_1$.
• The distribution of the microorganism is decreased with increasing values of $Pe$ and $Le$.
• The non-Newtonian parameters $N{n}_1$ and $N{n}_2$ have the same decreasing effects on the skin friction coefficient, while $N{n}_1$ decreases and $N{n}_2$ increases the heat transfer rate.