Entropy Analysis of Sutterby Nanofluid Flow over a Riga Sheet with Gyrotactic Microorganisms and Cattaneo–Christov Double Diffusion

Abstract

In this article, a Riga plate is exhibited with an electric magnetization actuator consisting of permanent magnets and electrodes assembled alternatively. This exhibition produces electromagnetic hydrodynamic phenomena over a fluid flow. A new study model is formed with the Sutterby nanofluid flow through the Riga plate, which is crucial to the structure of several industrial and entering advancements, including thermal nuclear reactors, flow metres and nuclear reactor design. This article addresses the entropy analysis of Sutterby nanofluid flow over the Riga plate. The Cattaneo–Christov heat and mass flux were used to examine the behaviour of heat and mass relaxation time. The bioconvective motile microorganisms and nanoparticles are taken into consideration. The system of equations for the current flow problems is converted from a highly non-linear partial system to an ordinary system through an appropriate transformation. The effect of the obtained variables on velocity, temperature, concentration and motile microorganism distributions are elaborated through the plots in detail. Further, the velocity distribution is enhanced for a greater Deborah number value and it is reduced for a higher Reynolds number for the two cases of pseudoplastic and dilatant flows. Microorganism distribution decreases with the increased magnitude of Peclet number, Bioconvection Lewis number and microorganism concentration difference number. Two types of graphical outputs are presented for the Sutterby fluid parameter (β = −2.5, β = 2.5). Finally, the validation of the present model is achieved with the previously available literature.

1. Introduction

The rate of heat transport characteristics has received increasing attention from various scientists owing to its tremendous industrial features, for example, in mechanical, optical, electrical and cooling instruments. The rate of heat transport increment is very crucial in depositing energy. Therefore, researchers have focused on the investigation of a new type of fluid that is a mixture of nanoparticles with a size of 100 nm and larger thermophysical properties than ordinary fluids, known as nanoliquids. A nanoliquid is a colloidal suspension of the nanoparticle’s thermal behaviour in the ordinary fluid. The first attempt was conducted by Choi et al. [1] in 1995. They showed the thermal conductivity of nanoliquids by adding nanosized particles. Later, Buongiorno [2] used this understanding of nanofluid to achieve a mathematical form by adding Brownian and Thermophoretic terms. A mixed convection nanofluid flow with different geometries was presented by Hussain et al. [3,4]. Haq et al. [5] studied the second law analysis on a cross nanofluid. The MHD mixed convective flow of CNTs/Al₂O₃ nanofluid in water past a heated flexible plate with injection/suction and radiation was studied by Prabakaran et al. [6]. Mankiw et al. [7] analysed the MHD time-dependent flow of nanofluid with variable properties due to an inclined stretching sheet under thermal radiation. Shahid [8] studied the effect of an upper convective Maxwell fluid over a permeable surface near the stagnation point. Rafique et al. [9] addressed the stratified micropolar nanofluid flow past the Riga surface. The unsteady viscous flow of the nanofluid flow over the Riga plate using a rotating system was investigated by Parvine et al. [10]. Abbas et al. [11] studied entropy production over the Riga plate with the suction case. Mohamed et al. [12] using a non-homogeneous dynamic model, which is physically more accurate in describing nanofluids than homogeneous ones. They numerically examined the free convective flow in a cubical cavity filled with a copper–water nanofluid. Aziz et al. [13] discussed the characteristics of nanoparticles with Lorentz and Coriolis forces. More developments of nanofluids are in [14,15,16,17].

Different fluid forms, such as polymer melts, colloidal suspensions and organic chain mixes, are used in a wide range of industrial and production processes. The rheological behaviour of these fluids cannot be described well by the Naiver–Stokes equation. Therefore, several nonlinear fluid models have been proposed to represent the rheological characteristics of complicated fluids. One of the non-Newtonian fluid models is used to examine the key characteristics of pseudoplastic and dilatant fluids, which is known as Sutterby fluid model. Numerous experts have studied the flow of the Sutterby liquid extensively. Waqas et al. [18] inspected the Sutterby nanofluid flow using two rotating disks. Yahya et al. [19] investigated Williamson Sutterby nanoparticles under the Cattaneo–Christov heat flux. The effect of MHD on Sutterby nanoparticles due to porous movable sheets was discovered by Fayydh et al. [20]. Gowda et al. [21] examined the Cattaneo–Christof theory of heat diffusion in a Sutterby nanofluid. The thermal aspect of Sutterby nanofluid containing the microorganisms due the stretched cylinder was examined by Aldabesh et al. [22]. Hayat et al. [23] investigated the Sutterby fluid with thermal radiation due to a rotating disk. Fujii et al. [24] addressed a Sutterby fluid with natural convection flow due to a vertical plate. Darcy surface with MHD flow of Sutterby fluid was reported by Bilal et al. [25]. The bioconvection flow of a Sutterby nanofluid due to a rotating disk is described by Khan et al. [26]. Sohail et al. [27] designed the free convection flow of a Sutterby fluid with Cattaneo–Christov theory. The heat generation/absorption in the thermally stratified flow of a Sutterby fluid through a linearly stretched plate is analysed by Saif et al. [28]. Usman et al. [29] investigated the two-dimensional stagnant flow of a Sutterby nanofluid across a stretching wedge with porous media. Ali et al. [30] discussed the heat and mass transportation of a Sutterby nanofluid due to a horizontally stretching surface with bioconvection of microorganisms. The influence of homogeneous–heterogeneous reaction on Sutterby fluid flow through a disk with Cattaneo–Christov heat flux was studied by Khan et al. [31].

In the modern period, research on bioconvection exists due to the upwards motion of microorganisms, whose microorganisms are denser than water. The upward surface of the fluid develops thickness due to the collection of the microorganisms. Because of this, the upper surface becomes disturbed and microorganisms are fall down, which develops the bioconvection. Bioconvection phenomena have been continuously researched due to their applications in the clinical area, manufacturing process and biofuel production. Bioconvection can be organised in the motion of direction with enormous microorganism species. In this way, gyrotactic microorganisms are among those whose swimming directional is based on viscous and gravitational force. Kuznetsov et al. [32,33] reported the investigation of bioconvection in a mixed suspension of nanoparticles with gyrotactic microorganisms. Kotha et al. [34] examined the MHD flow of nanofluids with motile gyrotactic microorganisms over a vertical plate. Siddiq et al. [35] analysed numerically, through the bvp4c method, the bioconvection of micropolar nanofluid flow through a stretchable disk. Ali et al. [36] studied the effect of bioconvection and Cattaneo–Christov heat flux effects of a micropolar-type nanofluid past a vertical stretching sheet. Azam et al. [37] investigated the effect of bioconvection flow for a Sutterby nanoliquid with nonlinear radiation. Khashi’ie et al. [38] studied a hybrid nanofluid having bioconvection with gyrotactic microorganisms. Azam [39] explored the time-dependent flow of the chemically reactive Sutterby nanofluid and the influence of gyrotactic microorganisms. Hayat et al. [40] operated the bioconvection flow of nanomaterial subject to the melting effect. They addressed thermal nonlinear radiation and Joule heating for heat distribution characteristics. Reddy et al. [41] analysed the time-dependent flow of a cross nanofluid comprising the gyrostatic microorganisms due to slip velocity. Sarkar et al. [42] defined a Sutterby nanofluid flow having motile gyrotactic microorganisms over the Riga plate. Syed et al. [43] described the biocovective phenomena of a Prandtl hybrid nanofluid over a stretched surface.

The current investigation aims to express the Sutterby nanofluid flow over a Riga plate with Cattaneo–Christov double diffusion and gyrotactic microorganisms. Chemical reactions and heat source-sink are considered. The main intention of this work is the inclusive analysis of this flow problem. The governing systems are designed as a coupled partial system. The flow problems are altered into the nonlinear ordinary system by applying suitable transformations. Further, the solution of ordinary differential equations is computed via the homotopy analysis method (HAM). The novel outcomes of the current work are obtained through different parameters and explained in detail with graphs and tables.

2. Description of the Physical Model

Consider the incompressible and steady flow of a Sutterby nanofluid over the Riga plate containing gyrotactic microorganisms. Cattaneo–Christov with heat and mass flux were also incorporated into the temperature and concentration equation. The $x$-axis is considered along with the sheet and the $y$-axis is taken perpendicular to the sheet. Moreover, the velocity of the sheet is taken as $U_w=ax$. The temperature of the surface, the concentration of the surface and the microorganism of the surface are represented by $T_w$, $C_w$ and${\chi }_w$, respectively. Furthermore, it was assumed that the fluid contains gyrotactic bacteria. The fluids’ microorganisms gravitate towards the light. Gyrotactic phenomena, or movement against gravity, are made possible by the “bottom heavy” bulk microorganism, which orients their bodies. The existence of microorganisms is advantageous for the suspension of the nanoparticles. The motion of microorganisms is taken, irrespective of that of the nanoparticles, to ensure the stability of convection. The flow of a double-diffusive fluid across a Riga plate containing gyrotactic microorganisms has not been investigated, and this study aims to fill that gap with the simplification of unsteady boundary layer approximation expressions provided in [22]. Figure 1 describes the physical model of the present problem.

Figure 1. Physical configuration of the flow problem.

Fluid Model

The Cauchy stress tensor $\tau$ for the Sutterby fluid [44] is defined as

$$\tau =\mu \left(\dot{\gamma }\right)A_1-\mathit{pI},$$

(1)

where the Sutterby viscosity model is represented as

$$\mu ={\mu }_{\circ }{\left[\left(\frac{{\mathit{sinh}}^{-1}\left(\dot{\beta }\dot{\gamma }\right)}{\left(\dot{\beta }\dot{\gamma }\right)}\right)\right]}^n,$$

(2)

where $n$, ${\mu }_{\circ } \mathrm{and}$ $\dot{\beta }$ are the power law index, zero share rate viscosity and time material constant.

Introducing Equation (2) into Equation (1), then we have

$$\tau ={\mu }_{\circ }{\left[\left(\frac{{\mathit{sinh}}^{-1}\left(\dot{\beta }\dot{\gamma }\right)}{\left(\dot{\beta }\dot{\gamma }\right)}\right)\right]}^n{A}_1-p.$$

(3)

The governing equations are illustrated in the following form [21]:

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

(4)

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{{\mu }_0}{\rho }\left[\frac{{\partial }^{2}u}{\partial y^2}+\frac{\beta B^2}{2}{\left(\frac{\partial u}{\partial y}\right)}^2\frac{{\partial }^{2}u}{\partial y^2}\right]+\frac{\pi j_0{M}_0}{8\rho }\mathit{exp}\left(-\frac{\pi }{d}y\right)$$

(5)

$$\begin{gathered} u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+{\Phi }_E{\Omega }_E=\frac{k}{\rho C_p}\left(\frac{{\partial }^{2}T}{\partial y^2}\right)+\tau \left(D_B\frac{\partial C}{\partial y}\frac{\partial T}{\partial y}+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial y}\right)}^2\right) \\ -\frac{1}{\rho C_p}\frac{\partial q_r}{\partial y}+\frac{Q_0}{\rho C_p}\left(T-T_{\infty }\right), \end{gathered}$$

(6)

$$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}+{\Phi }_C{\Omega }_C=D_B\frac{{\partial }^{2}c}{\partial y^2}+\frac{D_T}{T_{\infty }}\left(\frac{{\partial }^{2}T}{\partial y^2}\right)-K_0\left(C-C_{\infty }\right),$$

(7)

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

(8)

where u and v are velocity components in the x and y directions. $\upsilon$ represents the kinematic viscosity of the fluid, $\rho$represents the density of the fluid, $\alpha$ represents the thermal diffusivity, $C$ is the concentration, $D_B \mathrm{and} D_t$ represent Brownian diffusion and thermophoretic diffusion (respectively), $C_p$ denotes volumetric expansion, $D_m$ represents microorganism coefficient, and ${\Omega }_E$ and ${\Omega }_C$ are the fluid relaxation time.

In the above equations, the terms ${\Omega }_E$ and ${\Omega }_C$ are stated as

$${\Omega }_E=u\frac{\partial u}{\partial x}\frac{\partial T}{\partial x}+v\frac{\partial u}{\partial y}\frac{\partial T}{\partial x}+u\frac{\partial v}{\partial x}\frac{\partial T}{\partial y}+v\frac{\partial v}{\partial y}\frac{\partial T}{\partial y}+u^2\frac{{\partial }^{2}T}{\partial x^2}+2uv\frac{{\partial }^{2}T}{\partial x\partial y}+v^2\frac{{\partial }^{2}T}{\partial y^2}$$

(9)

$${\Omega }_C=u\frac{\partial u}{\partial x}\frac{\partial C}{\partial x}+v\frac{\partial u}{\partial y}\frac{\partial C}{\partial x}+u\frac{\partial v}{\partial x}\frac{\partial C}{\partial y}+v\frac{\partial v}{\partial y}\frac{\partial C}{\partial y}+u^2\frac{{\partial }^{2}C}{\partial x^2}+2uv\frac{{\partial }^{2}C}{\partial x\partial y}+v^2\frac{{\partial }^{2}C}{\partial y^2}$$

(10)

The relevant boundary conditions are assumed to be:

$$\begin{array}{l}u=u_w\left(x\right), v=0,-k\frac{\partial T}{\partial y}=h\left(T_w-T_{\infty }\right), C=C_w, \chi ={\chi }_w \mathit{as} y=0,\\ u\to 0, T\to T_{\infty }, C\to C_{\infty }, \chi \to {\chi }_{\infty } \mathit{at} y\to \infty .\end{array}\}$$

(11)

$$q_r=\frac{4{\sigma }_1}{3k^{\ast }}\frac{\partial T^4}{\partial y}=-\frac{16{\sigma }^{\ast }}{3k^{\ast }}{T}^3\frac{\partial T^4}{\partial y},$$

(12)

where $T^4$ can be expanded as follows:

$$T^4 \cong 4T^3{}_{\infty }T-3T^4{}_{\infty }.$$

(13)

Replacing Equation (12) into Equation (13),

$$q_r=\frac{16\sigma \ast T^3{}_{\infty }}{3k^{\ast }}\frac{\partial T}{\partial y}.$$

(14)

Introducing the variables [14]

$$\left.\begin{array}{l}u=ax{f}^{\prime }(\eta ),v=-\sqrt{av}f(\eta ),\eta =y\sqrt{\frac{a}{v}},\\ \theta =\frac{T-T_{\infty }}{T_w-T_{\infty }},\varphi =\frac{C-C_{\infty }}{C_w-C_{\infty }},W=\frac{\chi -{\chi }_{\infty }}{{\chi }_w-{\chi }_{\infty }},\end{array}\right\}$$

(15)

Using Equation (15), Equations (4)–(6) become

$$f^{‴}+f f^{″}-f^{\prime }{}^2+\frac{1}{2}\beta \delta {\mathit{Re}}_{\gamma }{f}^{″}{}^2{f}^{‴}+Z{e}^{-A\eta }=0,$$

(16)

$${\theta }^{″}\left(1+\frac{4}{3}Rd\right)+\mathit{Pr}Nt {\theta }^{\prime }{}^2+\mathit{Pr}Nb {\theta }^{\prime }{\varphi }^{\prime }-\mathit{Pr}{\lambda }_1\left(f{f}^{\prime }{\theta }^{\prime }+f^2{\theta }^{″}\right)+\mathit{Pr}\mathsf{\Upsilon }\theta =0$$

(17)

$${\varphi }^{″}+Scf{\varphi }^{\prime }+\left(\frac{Nt}{Nb}\right){\theta }^{″}-\mathit{Pr}Sc{\lambda }_2\left(f{f}^{\prime }{\varphi }^{\prime }+f^2{\varphi }^{″}\right)-ScCr\varphi =0,$$

(18)

$$W^{″}-Pe\left[{\varphi }^{″}\left(W+\varpi \right)+{\varphi }^{\prime }{W}^{\prime }\right]-Lbf{W}^{\prime }=0,$$

(19)

The corresponding boundary conditions are

$$\begin{array}{l}f\left(0\right)=0,f^{\prime }\left(0\right)=1,{\theta }^{\prime }\left(0\right)=Bi\left(\theta \left(0\right)\right)-Bi,\varphi \left(0\right)=1\\ f^{\prime }(\infty )=0,\theta (\infty )=0,\varphi (\infty )=0,W(\infty )=0.\end{array}\}$$

(20)

where $Z=\frac{\pi J_{\circ }{M}_{\circ }h}{8\rho U_{\circ }^2}$ is the modified Hartmann number; ${\mathit{Re}}_{\gamma }=\frac{a{x}^2}{v}$ is the Reynolds number; $\delta =\frac{B^2{a}^2}{v}$ is the Deborah number; $Rd=\frac{4{\sigma }^{\ast }{T}_{\infty }^3}{k{k}^{\ast }}$ is the radiation parameter; $\mathit{Pr}=\frac{v}{a}$ is the Prandtl number; $Nb=\frac{\tau D_B\left(C_w-C_{\infty }\right)}{v}$ is the Brownian motion parameter; $Nt=\frac{\tau D_t\left(T_w-T_{\infty }\right)}{T_{\infty }v}$ is the thermophoresis parameter; $Sc=\frac{v}{D_B}$ is the Schmidt number; $\mathit{Bi}=\frac{h_f}{k}\sqrt{\frac{v}{a}}$ is the Biot number; $Pe=\frac{b{W}_c}{D_m}$ is the Peclet number; $Lb=\frac{v}{Dm}$ is the bioconvection Lewis number; and $Cr=\frac{K_{\circ }}{c}$ is the chemical reaction.

The thermofluidic quantities of engineering interest in this study are skin friction, $C{f}_x$, heat transfer rate, $N{u}_x$, mass transfer rate, $S{h}_x$, and motile density, $W{h}_x$.

$$C_{fx}=\frac{{\tau }_w}{p{u}_w^2},N{u}_x=\frac{x{q}_w}{k\left(T_w-T_{\infty }\right)},S{h}_x=\frac{x{q}_m}{D_b\left(C_w-C_{\infty }\right)} \mathrm{and} W{h}_x=\frac{x{q}_n}{D_m\left(X_w-X_{\infty }\right)}\}$$

(21)

where ${\tau }_w$ is the surface shear stress, $q_w$ is the surface heat flux, $q_m$ is the surface mass flux and $q_n$ is the motile density, which are presented by the following expressions:

$$\begin{array}{l}{\tau }_w=-{\mu }_0{\left[\frac{\partial u}{\partial y}+\frac{1}{6}\beta \delta {\mathit{Re}}_{\gamma }{\left(\frac{\partial u}{\partial y}\right)}^3\right]}_{y=0},q_w={-k\left(\frac{\partial T}{\partial y}\right)|}_{y=0},\\ q_m={-k\left(\frac{\partial C}{\partial y}\right)|}_{y=0}\mathrm{and} q_n={-D_m\left(\frac{\partial \chi }{\partial y}\right)|}_{y=0}.\end{array}\}$$

(22)

The dimensionless form of the above parameters is expressed as

$$\begin{array}{l}{C}_{fx}{\mathit{Re}}_x^{0.5}=\left[f^{″}\left(0\right)+\frac{1}{6}\beta \delta {\mathit{Re}}_{\gamma }{\left(f^{″}\left(0\right)\right)}^3\right],\frac{N{u}_x}{R{e}_x^{1/2}}=-\left[1+\frac{4}{3}Rd\right]{\theta }^{\prime }\left(0\right),\\ \frac{S{h}_x}{R{e}_x^{1/2}}=-{\varphi }^{\prime }\left(0\right) \mathrm{and} \frac{W{h}_x}{R{e}_x^{1/2}}=-W^{\prime }\left(0\right).\end{array}\}$$

(23)

where ${\mathit{Re}}_x=\frac{x{U}_w}{v}$ is the local Reynolds number.

3. Entropy Generation Analysis

The entropy generation with a Sutterby nanofluid is communicated as [37]:

$$\begin{array}{l}{S}_{gen}^{‴}=\frac{k}{T_{\circ }}\left({\left[\frac{\partial T}{\partial x}\right]}^2+{\left[\frac{\partial T}{\partial y}\right]}^2+\frac{16{\sigma }^{\ast }{T}_{\infty }^3}{3k{k}^{\ast }}{\left(\frac{\partial T}{\partial y}\right)}^2\right)+\frac{\mu }{T_{\infty }}{\left(\frac{\partial u}{\partial y}\right)}^2\left[1+\frac{\beta \delta R{e}_{\gamma }}{6}{\left(\frac{\partial u}{\partial y}\right)}^2\right]\\ +\frac{R{D}_m}{C_{\infty }}{\left(\frac{\partial C}{\partial y}\right)}^2+\frac{R{D}_m}{T_{\infty }}\left(\frac{\partial T}{\partial y}\right)\left(\frac{\partial C}{\partial y}\right)+\frac{R{D}_m}{{\chi }_{\infty }}{\left(\frac{\partial \chi }{\partial y}\right)}^2+\frac{R{D}_m}{T_{\infty }}\left(\frac{\partial \chi }{\partial y}\right)\left(\frac{\partial T}{\partial y}\right).\end{array}$$

(24)

The significance of the entropy production can be written as

$$S_{\circ }^{‴}=\frac{\kappa {\left(\mathsf{\Delta }T\right)}^2}{L^2{T}_{\infty }^2}$$

(25)

Using Equation (15), the rate of entropy Equation (24) can be converted as:

$$\begin{array}{l}{N}_G=\frac{S_{gen}^{‴}}{S_{\circ }^{‴}}={\mathit{Re}}_{\gamma }\left(1+Rd\right){\theta }^{\prime }{}^2+{\mathit{Re}}_{\gamma }\frac{Br}{\Pi }\left[1+\frac{\beta \delta R{e}_{\gamma }}{3}{\left(f^{″}\right)}^2\right]+{\mathit{Re}}_{\gamma }{\left(\frac{\Gamma }{\Pi }\right)}^2{\varphi }^{\prime }{}^2\\ +{\mathit{Re}}_{\gamma }\left(\frac{\Gamma }{\Pi }\right){\varphi }^{\prime }{\theta }^{\prime }+{\mathit{Re}}_{\gamma }{\left(\frac{\xi }{\Pi }\right)}^2{W}^{\prime }{}^2+{\mathit{Re}}_{\gamma }\left(\frac{\xi }{\Pi }\right)W^{\prime }{\theta }^{\prime }.\end{array}$$

(26)

The Bejan number $Be$ is defined as the ratio over the entropy generation with heat transport, $S_T$, and the total entropy production, $N_G$, and it can be written as:

$$\begin{array}{l}Be=\frac{S_T}{N_G}\\ Be=\frac{{\mathit{Re}}_{\gamma }(1+Rd){\theta }^{\prime }{}^2+{\mathit{Re}}_{\gamma }{\left(\frac{\Gamma }{\Pi }\right)}^2{\varphi }^{\prime }{}^2+{\mathit{Re}}_{\gamma }\left(\frac{\Gamma }{\Pi }\right){\varphi }^{\prime }{\theta }^{\prime }+{\mathit{Re}}_{\gamma }{\left(\frac{\xi }{\Pi }\right)}^2{W}^{\prime }{}^2+{\mathit{Re}}_{\gamma }\left(\frac{\xi }{\Pi }\right)W^{\prime }{\theta }^{\prime }}{\begin{array}{l}{\mathit{Re}}_{\gamma }(1+Rd){\theta }^{\prime }{}^2+R{e}_{\gamma }\frac{Br}{\Pi }\left[1+\frac{\beta \delta R{e}_{\gamma }}{3}{\left(f^{″}\right)}^2\right]+{\mathit{Re}}_{\gamma }{\left(\frac{\Gamma }{\Pi }\right)}^2{\varphi }^{\prime }{}^2\\ +R{e}_{\gamma }\left(\frac{\Gamma }{\Pi }\right){\varphi }^{\prime }{\theta }^{\prime }+R{e}_{\gamma }{\left(\frac{\xi }{\Pi }\right)}^2{W}^{\prime }{}^2+R{e}_{\gamma }\left(\frac{\xi }{\Pi }\right)W^{\prime }{\theta }^{\prime }\end{array}}\end{array}$$

(27)

4. Homotopy Expression

Nonlinearity issues are solved using a variety of numerical approaches. The HAM [45,46,47,48,49,50,51,52] technique, which is the most successful semi-analytically approach and applied to utilize these greatly nonlinear equations. These variables are used to calculate the approximation rate of this solution. The flow map of HAM process is given in Chart 1. Furthermore, the user can select the starting assumptions for the solutions. The higher order non-linear ODE Equations (16)–(20) are solved through this HAM technique.

$$f_{\circ }\left(\eta \right)=\left(1-e^{-\eta }\right),{\theta }_{\circ }\left(\eta \right)=\left[\frac{Bi}{1+Bi}\right]e^{-\eta },{\varphi }_{\circ }\left(\eta \right)=e^{-\eta },W_{\circ }\left(\eta \right)=e^{-\eta },$$

(28)

$${\widehat{L}}_f=f^{‴}-f^{\prime }, {\widehat{L}}_{\theta }={\theta }^{″}-\theta , {\widehat{L}}_{\varphi }={\varphi }^{″}-\varphi , {\widehat{L}}_W=W^{″}-W,$$

(29)

with the property

$$\begin{array}{l}{\widehat{L}}_f\left(R_1+R_2{e}^{-\eta }+R_3{e}^{-\eta }\right)=0,\\ {\widehat{L}}_{\theta }\left(R_4+R_5{e}^{-\eta }\right)=0,\\ {\widehat{L}}_{\varphi }\left(R_6+R_7{e}^{-\eta }\right)=0,\\ {\widehat{L}}_W\left(R_8+R_9{e}^{-\eta }\right)=0,\end{array}\}$$

(30)

in which $Bi\left(i=1–9\right)$ are the constants.

Chart 1. Flow chart of HAM expression.

• Zero^th order formulation

$$\left.\begin{array}{l}(1-s){\widehat{L}}_f\left[f(\eta ;s)-f_{\circ }(\eta )\right]=s{ħ}_f{\underset{¯}{N}}_f[f(\eta ;s),\theta (\eta ,s),\varphi (\eta ,s),W(\eta ,s)],\\ (1-s){\widehat{L}}_{\theta }\left[\theta (\eta ;s)-{\theta }_{\circ }(\eta )\right]=s{ħ}_{\theta }{\underset{¯}{N}}_{\theta }[f(\eta ;s),\theta (\eta ,s),\varphi (\eta ,s)],\\ (1-s){\widehat{L}}_{\varphi }\left[\varphi (\eta ;s)-{\varphi }_{\circ }(\eta )\right]=s{ħ}_{\varphi }{\underset{¯}{N}}_{\varphi }[f(\eta ;s),\theta (\eta ,s),\varphi (\eta ,s)],\\ (1-s){\widehat{L}}_W\left[W(\eta ;s)-W_{\circ }(\eta )\right]=s{ħ}_W{\underset{¯}{N}}_W[f(\eta ;s),\theta (\eta ,s),\varphi (\eta ,s)],W(\eta ,s)].\end{array}\right\}$$

(31)

$$\begin{array}{l}f\left(0,s\right)=0, f^{\prime }\left(0,s\right)=1, f^{\prime }\left(\infty ,s\right)=0,\\ {\theta }^{\prime }\left(0,s\right)=Bi\left[\theta \left(0,s\right)-1\right], \theta \left(\infty ,s\right)=0,\\ \varphi \left(0,s\right)=1, \varphi \left(\infty ,s\right)=0,\\ W\left(0,s\right)=0, W\left(\infty ,s\right)=0,\end{array}\}$$

(32)

where ${\underset{¯}{N}}_f$, ${\underset{¯}{N}}_{\theta }$, ${\underset{¯}{N}}_{\varphi }$ and ${\underset{¯}{N}}_W$ are defined below:

$$\begin{array}{l}{\underset{¯}{N}}_f\left[f\left(\eta ,s\right)\right]=\frac{{\partial }^{3}f\left(\eta ,s\right)}{\partial {\eta }^3}+f\left(\eta ,s\right)\frac{{\partial }^{2}f\left(\eta ,s\right)}{\partial {\eta }^2}-{\left[\frac{\partial f\left(\eta ,s\right)}{\partial \eta }\right]}^2\\ +\frac{1}{2}\beta \delta {\mathit{Re}}_{\gamma }{\left[\frac{\partial f\left(\eta ,s\right)}{\partial \eta }\right]}^2\left[\frac{{\partial }^{3}f\left(\eta ,s\right)}{\partial {\eta }^3}\right]+Z{e}^{-A\eta },\end{array}$$

(33)

$$\begin{array}{l}{\underset{¯}{N}}_{\theta }\left[f\left(\eta ,s\right),\theta \left(\eta ,s\right)\right]=\frac{{\partial }^2\theta \left(\eta ,s\right)}{\partial {\eta }^2}\left(1+\frac{4}{3}Rd\right)+\mathit{Pr}Nt{\left[\frac{\partial \theta \left(\eta ,s\right)}{\partial \eta }\right]}^2\\ +\mathit{Pr}Nb\left[\frac{\partial \theta \left(\eta ,s\right)}{\partial \eta }\right]\left[\frac{\partial \varphi \left(\eta ,s\right)}{\partial \eta }\right]-\mathit{Pr}{\lambda }_1\left(\begin{array}{l}f\left(\eta ,s\right)\left[\frac{\partial f\left(\eta ,s\right)}{\partial \eta }\right]\left[\frac{\partial \theta \left(\eta ,s\right)}{\partial \eta }\right]\\ +{\left[f\left(\eta ,s\right)\right]}^2\frac{{\partial }^2\theta \left(\eta ,s\right)}{\partial {\eta }^2}\end{array}\right)\\ +\mathit{Pr}\mathsf{\Upsilon }\theta \left(\eta ,s\right),\end{array}$$

(34)

$$\begin{array}{l}{\underset{¯}{N}}_{\varphi }\left[f\left(\eta ,s\right),\theta \left(\eta ,s\right),\varphi \left(\eta ,s\right)\right]=\frac{{\partial }^2\varphi \left(\eta ,s\right)}{\partial {\eta }^2}+Scf\left(\eta ,s\right)\left[\frac{\partial \varphi \left(\eta ,s\right)}{\partial \eta }\right]\\ +\left(\frac{Nt}{Nb}\right)\left[\frac{{\partial }^2\theta \left(\eta ,s\right)}{\partial {\eta }^2}\right]-\mathit{Pr}Sc{\lambda }_2\left(\begin{array}{l}f\left(\eta ,s\right)\frac{\partial f\left(\eta ,s\right)}{\partial \eta }\frac{\partial \varphi \left(\eta ,s\right)}{\partial \eta }\\ +f^2\left(\eta ,s\right)\frac{{\partial }^2\varphi \left(\eta ,s\right)}{\partial {\eta }^2}\end{array}\right)-ScCr\varphi \left(\eta ,s\right),\end{array}$$

(35)

$${\underset{¯}{N}}_W\left[f\left(\eta ,s\right),\varphi \left(\eta ,s\right),W\left(\eta ,s\right),\right]=\frac{{\partial }^{2}W\left(\eta ,s\right)}{\partial {\eta }^2}-Pe\left[\begin{array}{l}\frac{{\partial }^2\varphi \left(\eta ,s\right)}{\partial {\eta }^2}\left(W+\varpi \right)\\ +\frac{\partial \varphi \left(\eta ,s\right)}{\partial \eta }\frac{\partial W\left(\eta ,s\right)}{\partial \eta }\end{array}\right]-Lbf\left(\eta ,s\right)\frac{\partial W\left(\eta ,s\right)}{\partial \eta }.$$

(36)

For $s=0$ and $s=1$, the results are achieved:

$$\begin{array}{l}f\left(\eta ;0\right)=f_0\left(\eta \right), \theta \left(\eta ;0\right)={\theta }_0\left(\eta \right), \varphi \left(\eta ;0\right)={\varphi }_0\left(\eta \right), W\left(\eta ;0\right)=W_0\left(\eta \right)\\ f\left(\eta ;1\right)=f\left(\eta \right), \theta \left(\eta ;1\right)=\theta \left(\eta \right), \varphi \left(\eta ;1\right)=\varphi \left(\eta \right), W\left(\eta ;1\right)=W\left(\eta \right)\end{array}\}$$

(37)

• m^th order formulation

The $m$^th order deformation can be presented in the following forms:

$$\begin{array}{l}{\widehat{L}}_f\left[f_m\left(\eta ,s\right)-{\chi }_m{f}_{m-1}\left(\eta \right)\right]={\hslash }_f{R}_{f,m}\left(\eta \right),\\ {\widehat{L}}_{\theta }\left[{\theta }_m\left(\eta ,s\right)-{\chi }_m{\theta }_{m-1}\left(\eta \right)\right]={\hslash }_{\theta }{R}_{\theta ,m}\left(\eta \right),\\ {\widehat{L}}_{\varphi }\left[{\varphi }_m\left(\eta ,s\right)-{\chi }_m{\varphi }_{m-1}\left(\eta \right)\right]={\hslash }_{\varphi }{R}_{\varphi ,m}\left(\eta \right),\\ {\widehat{L}}_W\left[{\theta }_m\left(\eta ,s\right)-{\chi }_m{W}_{m-1}\left(\eta \right)\right]={\hslash }_W{R}_{W,m}\left(\eta \right).\end{array}\}$$

(38)

Boundary conditions are:

$$\begin{array}{l}{f}_m^{\prime }\left(0\right)=f_m\left(0\right)=f_m^{\prime }(\infty )={\theta }_m^{\prime }\left(0\right)-Bi{\theta }_m\left(0\right)={\theta }_m(\infty )=0,\\ {\varphi }_m\left(0\right)={\varphi }_m(\infty )=W\left(0\right)=W(\infty )=0.\end{array}\}$$

(39)

where

$$R_{f,m}\left(\eta \right)=f_{m-1}^{‴}\left(\eta \right)+{\displaystyle {\sum }_{k=0}^{m-1}}{f}_{m-1-k}{f}_k^{″}-{\displaystyle {\sum }_{k=0}^{m-1}}{f}_{m-1-k}^{\prime }{f}_k^{\prime }+\frac{1}{2}\beta \delta R{e}_{\gamma }{\displaystyle {{\displaystyle \sum }}_{k=0}^{m-1}}{f}_{m-1-k}^{″}{f}_k^{″}{f}_k^{‴}+Z{e}^{-A\eta }$$

(40)

$$\begin{gathered} R_{\theta ,m}\left(\eta \right)={\theta }_{m-1}^{″}\left(\eta \right)\left(1+\frac{4}{3}Rd\right)+\mathit{Pr}Nt{\displaystyle {\sum }_{k=0}^{m-1}}{\theta }_{m-1-k}^{\prime }{\theta }_k^{\prime }+\mathit{Pr}Nb{\displaystyle {\sum }_{k=0}^{m-1}}{\varphi }_{m-1-k}^{\prime }{\theta }_k^{\prime } \\ -\mathit{Pr}{\lambda }_1\left({\displaystyle {\sum }_{k=0}^{m-1}}\left({\displaystyle {\sum }_{r=0}^k}{f}_{m-1-k}{f}_{k-r}^{\prime }\right){\theta }_k^{\prime }+{\displaystyle {\sum }_{k=0}^{m-1}}{f}_{m-1-k}^{\prime }{f}_k^{\prime }{\theta }_k^{″}\right)+\mathit{Pr}\mathsf{\Upsilon }{\theta }_{m-1} \end{gathered}$$

(41)

$$\begin{gathered} R_{\varphi ,m}\left(\eta \right)={\varphi }_{m-1}^{″}\left(\eta \right)+Sc{\displaystyle {\sum }_{k=0}^{m-1}}{f}_{m-1-k}^{\prime }{\varphi }_k+\left(\frac{Nt}{Nb}\right){\theta }_{m-1}^{″} \\ -\mathit{Pr}Sc{\lambda }_2\left({\displaystyle {\sum }_{k=0}^{m-1}}\left({\displaystyle {\sum }_{r=0}^k}{f}_{m-1-k}{f}_{k-r}^{\prime }\right){\varphi }_k^{\prime }+{\displaystyle {\sum }_{k=0}^{m-1}}{f}_{m-1-k}^{\prime }{f}_k^{\prime }{\varphi }_k^{″}\right)-ScCr{\varphi }_{m-1} \end{gathered}$$

(42)

$$\begin{gathered} R_{W,m}\left(\eta \right)=W_{m-1}^{″}\left(\eta \right)-Pe\left({\displaystyle {\sum }_{k=0}^{m-1}}{\varphi }_{m-1-k}^{″}{W}_k+{\varphi }_{m-1}^{″}\varpi +{\displaystyle {\sum }_{k=0}^{m-1}}{\varphi }_{m-1-k}^{\prime }{W}_k^{\prime }\right)- \\ Lb{\displaystyle {\sum }_{k=0}^{m-1}}{W}_{m-1-k}^{\prime }{f}_k \end{gathered}$$

(43)

$${\eta }_m=\left\{\begin{array}{l}0,m\le 1\\ 1,m>1\end{array}\right\}.$$

(44)

The general solutions are

$$\begin{array}{l}{f}_m=f_m^{\ast }+R_1+R_2{e}^{\eta }+R_3{e}^{-\eta }\\ {\theta }_m={\theta }_m^{\ast }+R_4{e}^{\eta }+R_5{e}^{-\eta }\\ {\varphi }_m={\varphi }_m^{\ast }+R_6{e}^{\eta }+R_7{e}^{-\eta }\\ W_m=W_m^{\ast }+R_8{e}^{\eta }+R_9{e}^{-\eta }\end{array}\}$$

(45)

where $f_m^{\ast }, {\theta }_m^{\ast }, {\varphi }_m^{\ast }, W_m^{\ast }$ are the special solutions.

Convergence of Homotopy Solutions

The parameters ${\hslash }_f,{\hslash }_{\theta },{\hslash }_{\varphi } \mathrm{and} {\hslash }_W$ are the converging control of the desired series solution. For the function f″(0), θ′(0), ϕ′(0), W′(0) seek the permissible values to obtain the 25th and 30th order. Figure 1, Figure 2, Figure 3 and Figure 4 specify that the range of ${\hslash }_f,{\hslash }_{\theta },{\hslash }_{\varphi } \mathrm{and} {\hslash }_W \mathrm{as}-2.0<{\hslash }_f<-0.1,-2.0<{\hslash }_{\theta }<-1.0,-1.7<{\hslash }_{\varphi }<-1.0 \mathrm{and}-2.0<{\hslash }_W<-1.0.$ The series converges in the entire region of $\eta$ when ${\hslash }_f=-0.65, {\hslash }_{\theta }={\hslash }_{\varphi }=-0.55 \mathrm{and} {\hslash }_W=-0.7$. The order of approximation for HAM is denoted in Table 1.

Figure 2. Plots of (a) ${\hslash }_f$—curve of $f^{″}\left(0\right)$, (b) ${\hslash }_{\theta }$ —curve of ${\theta }^{\prime }\left(0\right)$, (c) ${\hslash }_{\varphi }$ —curve of ${\varphi }^{\prime }\left(0\right)$ and (d) ${\hslash }_w$ —curve of $W^{\prime }(0)$.

Figure 3. (a–d). The impact of the $\mathrm{numerous} \mathrm{variables} \mathrm{of} f^{\prime }\left(\eta \right)$: (a)$\delta$, (b) $R{e}_{\gamma }$, (c) $\beta$ and (d) Z.

Figure 4. (a–f). The impact of the numerous variables of $\theta \left(\eta \right):$ (a) ${\lambda }_1$, (b) $Rd$, (c) $Nt,$ (d) $Nb,$ (e) $Bi$ and (f) $Pr$.

Table 1. Convergence solution of HAM.

Order of HAM
Approximation	−f″(0)	−θ′(0)	−ϕ′(0)	−W′(0)
1	0.874702	0.166821	1.21667	0.955833
5	0.768677	0.167973	1.41759	1.038657
10	0.765186	0.168223	1.42596	1.07814
15	0.765357	0.168144	1.42606	1.08236
20	0.765347	0.168158	1.42609	1.08264
25	0.765341	0.168158	1.42608	1.08261
30	0.765341	0.168157	1.42608	1.08258
35	0.765341	0.168157	1.42608	1.08258
40	0.765341	0.168157	1.42608	1.08258

5. Results and Discussion

The system of Equations (16)–(19) subject to the boundary condition (20) was addressed through the homotopy analysis method (HAM). To discuss the performance of the physical significance against the velocity field $f^{\prime }\left(\eta \right)$, temperature distribution $\theta \left(\eta \right)$, concentration field $\varphi \left(\eta \right)$, motile microorganism profile $W\left(\eta \right)$, entropy production $N_G$, Bejan number $Be$, as well as skin friction, Nusselt number, Sherwood number and motile density microorganism were delineated, as seen in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9. Table 2 verifies $-\theta \left(0\right)$ in accordance with Wang [52], Gorla and Sidawi [53], and Khan and Pop [54], with the limiting case $Nt=Nb=Rd={\lambda }_1=\mathsf{\Upsilon }=0$ and found a good agreement.

Figure 5. (a–d) The impact of the numerous variables of $\varphi \left(\eta \right):$ (a), (b) $Nt$, (c) $Cr$ and (d) $Sc$.

Figure 6. (a–c) The impact of the numerous variables of $\varphi \left(\eta \right)$: (a) $Nb$, (b) $Nb$ and (c) $Cr$.

Figure 7. (a–c) The impact of the numerous variables of $N_G$: (a) $\delta$, (b) $R{e}_{\gamma }$ and (c) $Br$.

Figure 8. (a–c) The impact of the numerous variables of $Be:$ (a)$\delta$, (b) $R{e}_{\gamma }$ and (c) $Br$.

Figure 9. (a–d) Variation in the numerous variables of ${\mathit{Re}}_x^{1/2}Cf,{\mathit{Re}}_x^{-1/2}Nu,{\mathit{Re}}_x^{-1/2}Sh,{\mathit{Re}}_x^{-1/2}W{h}_x:$ (a) $Z,\delta$, (b) $Nb,\mathsf{\Upsilon }$, (c) $Nb,Nt$ and (d) $Lb,Pe$.

Table 2. Comparison of the obtained values of the Nusselt number $-\theta \left(0\right)$ with those of Wang [52], Gorla and Sidawi [53], and Khan and Pop 45 [54], when $Nt=Nb=Rd={\lambda }_1=\mathsf{\Upsilon }=0$.

Pr	Wang [52]	Gorla and Sidawi [53]	Khan and Pop [54]	Present
0.07	0.0663	0.0663	0.0663	0.0663
0.20	0.1691	0.1691	0.1691	0.1691
0.70	0.4539	0.4539	0.4539	0.4539
2.00	0.9113	0.9113	0.9113	0.9113

5.1. Velocity Profile

The effects of different numerous parameters over the velocity distribution $f^{\prime }\left(\eta \right)$ are discussed in Figure 3a–d. Figure 3a shows that the velocity of the fluid diminishes with the superior values of Deborah number $\delta$, for the case of $\beta <0$, and the velocity field is enhanced for the rising value of Deborah number $\delta$, for the case of $\beta >0$. Figure 3b shows that the velocity field $f^{\prime }\left(\eta \right)$ is reduced for higher Reynolds number $R{e}_{\gamma }$values, in the case of $\beta <0$; a higher $R{e}_{\gamma }$tends to diminish the viscous force and the fluidity decreases for the pseudoplastic fluid. For shear thicking fluid, the velocity field enhances as $R{e}_{\gamma }$ increases, for the $\beta <0$ case. Figure 3c shows that the effect of the augmentation in the power law index parameter $\beta$ causes that the velocity profile rises for shear thickening fluid. Figure 3d demonstrates the significance of the Hartmann number $Z$ on the velocity field, for the two cases of $\beta <0$ and $\beta >0$. It was revealed that the strength of $Z$ changes and the velocity of the fluid escalates in both cases. Physically, an increment in $Z$ corresponds to enhancing the external electric field that constructs the wall-parallel Lorentz force. Therefore, $f^{\prime }\left(\eta \right)$ increases.

5.2. Temperature Profile

Figure 4a–f plot the consequences of temperature $\theta \left(\eta \right)$ against different values of the involved parameters over the temperature field. The thermal relaxation time parameter impacts on $\theta \left(\eta \right)$ are demonstrated in Figure 4a. It can be noted that the thermal relaxation parameter tends to decrease the temperature profile for both the dilatants and pseudoplastic cases. Figure 4b reveals the inclination of $\theta \left(\eta \right)$ for specific values of the thermal radiation parameter, for $\beta >0$ and $\beta <0$. The temperature of the fluid increases due to the enlargement in the radiation parameter $Rd$ in both cases. Figure 4c reports the variation of thermophoresis $Nt$ over temperature. This is due to the nanoparticles move from the hotter surface to the colder surface. Figure 4d demonstrates the influence of the Brownian motion parameter $Nb$ on the temperature profile. This is due to Brownian motion, which is the erratic movement of the particles suspended in the fluid. The random collision of particles suspended in the fluid increases the temperature of the fluid, which further contributes to the anticipated improvement in the temperature profile θ(η). Figure 4e depicts the impact of the Biot number $\theta \left(\eta \right)$. From the figure, it can be seen that the temperature field is boosted by enhancing the value of $Bi$. Actually, the Biot number $Bi$ means the ratio of convection proportion of conducting the inner side of the boundary at the surface. Figure 4f displays variations of the Prandtl number against $\theta \left(\eta \right)$. The temperature is maintained in light of a higher $Pr$.

5.3. Nanoparticle Concentration Profile

The outcomes of the different leading parameters $\varphi \left(\eta \right)$ are presented in Figure 5a–d. Figure 5a shows the characteristics of $Nb$ on $\varphi \left(\eta \right)$. The concentration distribution depletes with a rising $Nb$. Brownian motion’s relationship with the Brownian diffusion coefficient, which causes the concentration field to decrease, is the cause of this phenomenon. The influence of the thermophoresis variable on $\varphi \left(\eta \right)$ is rendered in Figure 5b. An augmentation in $Nt$ leads to a reduction in concentration. One can notice, from that graph, the upsurge in Nt improves the mass transfer. The chemical reaction influences the profile of concentration, as seen in Figure 5c. The enhanced values $Cr$ result in a fluid particle break near the surface, which reduces the concentration and the corresponding boundary layer thickness. The Schmidt number effect against the concentration profile is displayed in Figure 5d. Clearly, a depreciation in concentration is noted for the greater $Sc$, due to the reduction in mass diffusion.

5.4. Microorganism Profile

The effect of different influential variables on the microorganism’s field is shown in Figure 6a–c. Variations in motile microorganisms against the biocovection Lewis number $Lb$ for various values are seen in Figure 6a. Therefore, the greater values of $Lb$ reduces the microorganism field. Actually, $Lb$ has an opposite trend with thermal diffusivity as an escalation in $Lb$decreases the thermal diffusivity in regards to a decline in motile density. From Figure 6b, it can be seen that the higher Peclet number in the microorganism field produces a reduction in $W\left(\eta \right)$. It has a direct relation with cell swimming speed; therefore, the climbing $Pe$ improves the cell speed of micro-best stumbling microorganism diffusivity. As a result, $W\left(\eta \right)$ declines with the rising values of $Pe$. Figure 6c examines the characteristics of $W\left(\eta \right),$ the opposite $\varpi$. It can be seen that the motile density shrinks for the larger $\varpi$. In fact, improving the values of $\varpi$ escalates the concentration of microorganisms in ambient concentration. Finally, $W\left(\eta \right)$ declines.

5.5. Entropy Generation Profile

Figure 7a–c examine the performance of numerous variable parameters $\delta , R{e}_{\gamma }$ and $Br$ entropy production $N_G$. Figure 7a sketched the effect of Deborah number $\delta$ over $N_G$. This shows the enhancement in entropy production close to the wall for dilatant $\beta >0$ and deduction close to the wall for pseudoplastic fluid $\beta <0$. Increasing the influence in the Reynolds number $R{e}_{\gamma }$ entropy generation is studied for dilatant and pseudoplastic fluid fluids plots in Figure 7b. The disparity in $N_G$ against $Br$ is plotted in Figure 7c. Entropy generation increases with increasing value $Br$ in both cases. Subsequently, $Br$ attributes the proportion of free heat through viscous heating with the molecular condition. Therefore, heat is created in the system for increased values of $Br$ as well as disorder increasing in the system, which explains the upsurges in the entropy of the system.

5.6. Bejan Number Profile

The performance $Be$ is opposite to the variations in the variables $\delta ,R{e}_{\gamma }$ and $Br$; the plots $Be$ are shown in Figure 8a–c. Figure 8a,b show the behaviour of the physical parameters of the Deborah and Reynolds numbers on the Bejan number. It examined that the Bejan number declines with the larger values of Deborah and Reynolds numbers for shear thickening and increases for both numbers for the shear-thinning fluid. Furthermore, Figure 8c shows that the influence of the Bejan number Be reduces $\left(\beta >0, \beta <0\right)$ with the growing values of $Br$.

5.7. Physical Entitles

Figure 9a reports that the skin friction coefficient ${\mathit{Re}}_x^{1/2}C{f}_x$ is deformed in both cases for the larger values of the parameters $Z$ and $\delta$ for $\beta <0 \mathrm{and} \beta >0$. The influence of the non-Newtonian nanofluid parameter on the Nusselt number ${\mathit{Re}}_x^{-1/2}Nu$ against the thermal radiation $Rd$ is highlighted in Figure 9b. The heat transport gradient increases when rising the $Rd$ and $\mathsf{\Upsilon }$. The significance of $Nt$ and $Nb$ on the Sherwood number is shown in Figure 9c. It is determined that there is amplification in the Sherwood number ${\mathit{Re}}_x^{-1/2}Sh$ for raised values of fluid parameters. Figure 9d elucidates the substantial rescaled density number of motile microorganisms. The rescaled density number of motile microorganisms is voluminous for higher variations of $Pe$ and $Lb$. Figure 10a–d shows the 3D representation of skin friction, Nusselt number, Sherwood number and motile density, respectively.

Figure 10. (a–d) Three-dimensional graphs; (a) Skin friction for $\beta$ and $Z$; (b) Nusselt number for $Rd$ and $\mathsf{\Upsilon };$ (c) Sherwood number for $Nt$ and $Nb$; and (d) motile density for $Pe$ and $Lb$.

5.8. Stream Line and Isotherm Line

Figure 11b shows the behaviour of the stream function for the current flow. The patterns depict that the streamlines are more obscured and split into two sections, pseudo-plastic $\beta <0$ and dilatant $\beta >0$; the shape is modest and fills the flow field. Figure 12a,b show the behaviour of the isotherm line for the present flow for both the cases.

Figure 11. (a,b) Streamline for (a) $\beta =-2.5$ and (b) $\beta =2.5$.

Figure 12. (a,b) Isotherm lines (a) $\beta =-2.5$ and (b) $\beta =-2.5$.

6. Major Outcomes

This investigation examined a Sutterby nanofluid with Cattaneo–Christov double diffusion theory over a Riga plate. Additionally, the bioconvection of the motile microorganisms and the chemical reaction was included. To obtain a non-linear system of ordinary differential problems, appropriate transformations were used. The non-linear systems were computed through the HAM technique. The main findings of the present study were as follows:

• The Deborah and Reynolds numbers produce the opposite behaviour in the flow field for the different cases of $\beta =-2.5$ and $\beta =2.5$.
• The velocity shows the continuous improvement with increasing the Hartman number in both dilatant and pseudoplastic fluid cases.
• A larger chemical reaction reveals a decrement in the concentration, while the thermophoresis parameter $Nt$ lead to the expansion in concentration.
• The microorganism field deteriorated for the higher values of $Pe$ and microorganism difference parameter.
• The entropy generation number presented an increasing magnitude for large valuesof the Reynolds and Brinkman numbers, for the cases of pseudoplastic and dilatants fluid. Large values of entropy generation number appear in the area of the sheet due to the high viscous effects.
• Enhancing the value of the Deborah and Reynolds numbers results in the decrease in the Bejan profile in the case of the dilatant fluid, while the opposite effect is observed in the case of shear thinning.