Numerical Scrutinization of Ternary Nanofluid Flow over an Exponentially Stretching Sheet with Gyrotactic Microorganisms

Abstract

In the modern age, the study of nanofluids over the stretching sheet has received much attention from researchers due to its significant role in the polymer industry, for instance in the production of fibre sheets and the extrusion of molten polymers through a slit die. Due to these affordable applications, the current study focusses on the motion of metallic ternary nanofluids (Ag-Au-Cu/H₂O) past an exponential stretching sheet, taking diverse effects such as gyrotactic microorganisms, activation energy, buoyancy forces and thermal radiation into consideration. The model was created with the complex system of partial differential equations. Suitable similarity transformations and non-dimensional quantities were utilized to transform the complex system of partial differential equations to a set of ordinary differential equations. The resultant system is solved with the help of Matlab software. The computational outcomes are presented through the tables and pictorial notations. It is observed from the current analysis that the nanoparticle temperature of the ternary nanofluid enhances with the enhancement of activation energy and Brownian motion parameters. For the rising values of Lewis and thermophoresis numbers there is a declination in the nanoparticle concentration distribution. The Brownian motion and radiation effects increase the microorganism profile.

1. Introduction

According to Choi’s [1] theory, the addition of nanoparticles (metallic or non-metallic) in the carrier or base fluid (blood, engine oil, ethylene glycol and water) is termed as nanofluid. The nanoparticles are customarily utilized in agriculture, nuclear reactors, medicine and food items. The various kinds of nanofluids improve the thermal conductivities of base liquids, which results in a remarkable influence on the heat transfer performance. In view of a variety of applications, many investigators have proposed the nanofluid flows in various practical geometries. Mahdavi et al. [2] presented numerical results (with the help of ANSYS Fluent 19.3) to discuss the Poiseuille flow of a nanofluid with uniform heat flux. Alghamdi et al. [3] used the homotopy analysis method (HAM) to investigate the motion of nanofluid in a rectangular domain with heat source/sink. Asif and Dhiman [4] discussed the flow of nano-liquid through multiple square cylinders using finite volume methodology. Nayak et al. [5] analysed the water–ethylene glycol nanofluid flow over a vertical thin needle with the differential quadrature algorithm. Nadeem and Lee [6] presented the HAM solution for the nanofluid flow over an exponential stretching surface. Suresh and Panda [7] studied the Casson liquid aluminium oxide nanofluid flow in a wavy trapezoidal enclosure using the Galerkin finite element method. Ramezan et al. [8] investigated the motion of an MWCNT/water nano-liquid along microchannels. Rafique et al. [9] discussed the Casson nanofluid flow past an inclined stretching surface using the Keller-box method. Some more important studies in the direction of nanofluid flow through various geometries can be seen in references [10,11,12] and the references therein.

The next aspect in the nanofluid domain is the creation of a hybrid nanofluid. The amalgamation of two kinds of nanoparticles in the same carrier fluid is called a hybrid nanofluid. This type of hybrid nanofluid flow has attracted many investigators due to its significant features. Hybrid nanofluid flows improve the physical and chemical properties of materials at the same time. Hybrid nanofluids have efficiently improved the heat transfer rate as compared with binary nanofluids, because the hybrid nanofluids have more thermal conductivities and can be moulded as per the requirement. These fluids have notable applications in many fields, such as engineering and medical sciences. Waini et al. [13] discussed the hybrid nano-liquid motion over a stretching surface via the bvp4c solver in Matlab software. Kot and Elmaboud [14] presented the analytical solutions for the flow of hybrid nanofluids in the diseased coronary artery. Sulochana and Prasanna Kumar [15] used the shooting algorithm to obtain the results of boundary layer motion of a hybrid nanofluid past a stretching surface. Bouslimi et al. [16] discussed the Sutterby hybrid nanofluid motion using the finite element method. Bilal et al. [17] gave shooting technique results for the hybrid nanofluid flow through the circular cylindrical microchannels. Alzahrani et al. [18] discussed the motion of a hybrid nanofluid over a flat plate using the homotopy technique. Ramzan et al. [19] presented bvp4c solutions for the propulsion of hybrid nanofluids amidst two rotating disks. Zhang et al. [20] utilized Matlab to investigate the flow of hybrid nanofluid towards an elastic surface. Salman et al. [21] discussed the review of the propulsion of a hybrid nanofluid. Waini et al. [22] presented the bvp4c solutions for the motion of hybrid nanofluids over a stretching sheet.

The further improvement in the hybrid nanofluids has received much attention recently. The investigators found a new kind of nanofluid by adding three diverse nanoparticles in the base fluid and named it as ternary hybrid nanofluid. The ternary hybrid nanofluids have received significant attention from recent investigators in view of their enhanced thermo-physical properties. As compared with binary and hybrid nanofluids, the ternary hybrid nanofluids have excellent heat transfer properties. The flow of ternary hybrid nanofluids has received much attention from researchers, and they started working on variety of ternary nanofluid flows in various situations. Ramesh et al. [23] investigated the ternary nanofluid motion through the slipped surface. Goud et al. [24] used Runge–Kutta–Fehlberg’s methodology to discuss the motion of ternary hybrid nanofluid in a dovetail fin. Sohail et al. [25] presented a computational technique to investigate the propulsion of a ternary hybrid nanofluid due to a stretching sheet using finite element analysis. Oke [26] analysed the flow of aa ternary hybrid nanofluid over a rotating surface using the bvp4c solver. Saleem et al. [27] presented Mathematica’s NDSolve solutions for the ternary nanofluid flow between two cilia-carpeted walls. Adun et al. [28] studied the motion of ternary nanofluids in a photovoltaic thermal collector. Elnaqeeb et al. [29] provided bvp4c simulations to investigate the ternary hybrid nano-liquid motion with various shapes and densities. Boudraa and Bessaïh [30] investigated the ternary hybrid nanofluid motion over a heated block using the SIMPLE algorithm. Shah et al. [31] studied second-grade fluid flow with ternary nanoparticle suspension in a vertical plate using the Laplace transform technique.

The transport phenomena due to the moving surface are encountered in various important engineering applications such as drawing of plastic films and annealing of copper wires. Very few works have been seen in the direction of ternary hybrid nanofluids over a stretching sheet [25,32,33,34]. To the best of the authors’ knowledge, no work has been conducted to study the ternary nanofluid flow over an exponentially stretching plate with gyrotactic microorganisms. The current article deals with the metallic ternary nanofluid flow over an exponentially stretching sheet with diverse effects such as radiation and gyrotactic microorganisms. The computational problem is simplified with the help of suitable similarity transformations and dimensionless quantities. The bvp4c technique in Matlab software is utilized to study the numerical simulations.

2. Mathematical Modelling

Consider an incompressible ternary nanofluid fluid (base fluid as water and nanoparticles as metallic particles) flow over a sheet surface under the action of gyrotactic microorganisms and radiation. The Riga plate is assumed as an infinitely long surface at $y=0$ and the x-axis is normal to it (see Figure 1). It is also assumed that the Riga plate surface is maintained at velocity, temperature, concentration and microorganisms $u_w, T_w, C_w$$\mathrm{and} N_w$, respectively, and the same quantities at the boundary layer region are $u_{\infty }, T_{\infty }, C_{\infty }$$\mathrm{and} N_{\infty }$, respectively. The flow is characterized by velocity, which is governed by a randomly expanding surface, and the velocity of the stretching sheet is considered as $u_w\left(x\right)=U_0{e}^{x/l}.$

The basic constitutive equations of the current model can be expressed as [35,36,37]

$$\nabla \cdot \overline{q}=0,$$

(1)

$${\rho }_{tn}\frac{D\overline{q}}{Dt}=-\nabla p+{\mu }_{tn}{\nabla }^2\overline{q}+\overline{b},$$

(2)

$${\left(\rho c_p\right)}_{tn}\frac{DT}{Dt}=K_{tn}{\nabla }^{2}T-\nabla .q_r+{\left(\rho c_p\right)}_p\left(D_B\nabla C\cdot \nabla T+\frac{D_T}{T_{\infty }}\nabla T\cdot \nabla T\right),$$

(3)

$$\frac{DC}{Dt}=D_B{\nabla }^{2}C+\frac{D_T}{T_{\infty }}{\nabla }^{2}T-k_r^2\left(C-C_{\infty }\right){\left(\frac{T}{T_{\infty }}\right)}^n{e}^{\frac{-Ea}{\omega T}},$$

(4)

$$\frac{DN}{Dt}=D_m{\nabla }^{2}N+\frac{b^{*}We}{\left(C_w-C_{\infty }\right)}\nabla \left(N\cdot \nabla C\right),$$

(5)

where $\overline{q}$ is the velocity vector, ${\rho }_{tn}$ is the density of the ternary nanofluid, ${\mu }_{tn}$ is the viscosity of the ternary nanofluid, $\overline{b}\left(=F+g\left(\left(1-C_{\infty }\right){\rho }_{tn}{\beta }_1\left(T-T_{\infty }\right)-\left({\rho }_p-{\rho }_f\right)\left(C-C_{\infty }\right)-\left({\rho }_m-{\rho }_p\right){\beta }_3\left(N-N_{\infty }\right)\right)\right)$ is the body force term, $F$ is the force density, $g$ is the gravitational force term, ${\beta }_1$ is thermal expansion, $T$ is temperature, ${\rho }_p$ is the density of nanoparticles, ${\rho }_f$ is the density of fluid, $C$ is concentration, ${\rho }_m$ is the density of motile organisms, ${\beta }_3$ is the average volume of the microorganism, $N$ is the concentration of microorganisms, $c_p$ is specific heat, $K_{tn}$ is thermal conductivity, $q_r$ is radiative heat flux, $D_B$ is the Brownian motion parameter, $D_T$ is the thermophoresis parameter, $k_r$ is the reaction rate, $Ea$ is the activation energy, $\omega$ is the Boltzmann constant, $D_m$ is the diffusivity of microorganisms, $b^{*}$ is the chemotaxis constant and $We$ is the maximum cell swimming speed.

The Lorentz force acts in the direction of the array of electrodes, which is only in the x-direction. There is a strong variation in the forces, which decreases rapidly by increasing the distance y. Then, the force density becomes an exponentially decreasing function of y [38], which is given by $F=\frac{\pi J_0{M}_0}{8}{e}^{\frac{-\pi y}{a}},$ in which $J_0$ is the current density, $M_0$ is the magnetization of the permanent magnets and $a$ is the width of magnets and electrodes.

Under the steady state and the choice of velocity field $\overline{q}=\left(u, v\right)$, the Equations (1)–(5) can be converted as

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

(6)

$${\rho }_{tn}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)={\mu }_{tn}\frac{{\partial }^{2}u}{\partial y^2}+\frac{\pi J_0{M}_0}{8}{e}^{\frac{-\pi y}{a}}+g\left(\begin{array}{l}\left(1-C_{\infty }\right){\rho }_{tn}{\beta }_1\left(T-T_{\infty }\right)\\ -\left({\rho }_p-{\rho }_f\right)\left(C-C_{\infty }\right)\\ -\left({\rho }_m-{\rho }_p\right){\beta }_3\left(N-N_{\infty }\right)\end{array}\right),$$

(7)

$${\left(\rho c_p\right)}_{tn}\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)=\left(K_{tn}+\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}}\right)\frac{{\partial }^{2}T}{\partial y^2}+{\left(\rho c_p\right)}_p\left(D_B\frac{\partial T}{\partial y}\frac{\partial C}{\partial y}+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial y}\right)}^2\right),$$

(8)

$$\left(u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}\right)=D_B\frac{{\partial }^{2}C}{\partial y^2}+\frac{D_T}{T_{\infty }}\frac{{\partial }^{2}T}{\partial y^2}-k_r^2\left(C-C_{\infty }\right){\left(\frac{T}{T_{\infty }}\right)}^n{e}^{\frac{-Ea}{\omega T}},$$

(9)

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

(10)

with the suitable boundary conditions

$$u=u_w\left(x\right), v=-v_w\left(x\right), T=T_w\left(x\right), C=C_w\left(x\right), N=N_w\left(x\right) \mathrm{at} y=0,$$

(11)

$$u\to 0, v\to 0, T\to T_{\infty }\left(x\right), C\to C_{\infty }\left(x\right), N\to N_{\infty }\left(x\right) \mathrm{at} y\to \infty ,$$

(12)

where $u_w\left(x\right)=U_0{e}^{\frac{x}{l}}, T_w\left(x\right)=T_{\infty }+T_0{e}^{\frac{x}{2l}}, C_w\left(x\right)=C_{\infty }+C_0{e}^{\frac{x}{2l}}, N_w\left(x\right)=N_{\infty }+N_0{e}^{\frac{x}{2l}}.$

Applying the following similarity transformations and non-dimensional quantities [39]

$$\begin{array}{l}u=U_0{e}^{\frac{x}{l}}{f}^{\prime }\left(\eta \right), v=-\sqrt{\frac{{\mu }_f{U}_0}{2l{\rho }_f}}{e}^{\frac{x}{2l}}\left(f\left(\eta \right)+\eta f^{\prime }\left(\eta \right)\right), \eta =\sqrt{\frac{U_0{\rho }_f}{2{\mu }_{f}l}}{e}^{\frac{x}{2l}}y,\\ \theta =\frac{T-T_{\infty }}{T_w-T_{\infty }}, \sigma =\frac{C-C_{\infty }}{C_w-C_{\infty }}, \chi =\frac{N-N_{\infty }}{N_w-N_{\infty }}, M=\frac{\pi J_0{M}_{0}l}{4{\rho }_f{U}_0^2{e}^{\frac{2x}{l}}}, m=\frac{\pi }{a{e}^{\frac{x}{2l}}}\sqrt{\frac{2{\mu }_{f}l}{U_0{\rho }_f}},\\ \lambda =\frac{2\lg {\beta }_1\left(1-C_{\infty }\right)\left(T_w-T_{\infty }\right)}{U_0^2{e}^{\frac{2x}{l}}}, Nr=\frac{\left({\rho }_p-{\rho }_f\right)\left(C_w-C_{\infty }\right)}{{\rho }_f{\beta }_1\left(1-C_{\infty }\right)\left(T_w-T_{\infty }\right)}, Rb=\frac{{\beta }_3\left({\rho }_m-{\rho }_f\right)\left(N_w-N_{\infty }\right)}{{\rho }_f{\beta }_1\left(1-C_{\infty }\right)\left(T_w-T_{\infty }\right)},\\ \mathrm{Pr}=\frac{{\left(\mu c_p\right)}_f}{k_f}, Rn=\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}{k}_f}, Nb=\frac{D_B{\left(\mu c_p\right)}_p\left(C_w-C_{\infty }\right)}{{\nu }_f{\left(\mu c_p\right)}_f}, Nt=\frac{D_T{\left(\mu c_p\right)}_p\left(T_w-T_{\infty }\right)}{{\nu }_f{\left(\mu c_p\right)}_f{T}_{\infty }},\\ {\sigma }_m=\frac{2l{k}_r^2}{U_0{e}^{\frac{x}{l}}}, \Gamma =\frac{T_w-T_{\infty }}{T_{\infty }}, E=\frac{Ea}{\omega T_{\infty }}, Le=\frac{{\nu }_f}{D_B}, Lb=\frac{{\nu }_f}{D_m}, Pe=\frac{b^{*}We}{D_m}, \delta =\frac{N_{\infty }}{\left(N_w-N_{\infty }\right)},\end{array}$$

in the Equations (7)–(12), the resulting dimensionless system can be put in the form

$$\left({\rho }_{tn}/{\rho }_f\right)\left(2{f^{\prime }}^2-f{f}^{″}\right)-\left({\mu }_{tn}/{\mu }_f\right)f^{‴}-M{e}^{-m\eta }-\lambda \left(\left({\beta }_{tn}/{\beta }_f\right)\theta -Nr\sigma -Rb\chi \right)=0,$$

(13)

$$\left({\left(\rho c_p\right)}_{tn}/{\left(\rho c_p\right)}_f\right)\mathrm{Pr}\left(f^{\prime }\theta -f{\theta }^{\prime }\right)-\left(\left(k_{tn}/k_f\right)+Rn\right){\theta }^{″}-Nb\mathrm{Pr}{\theta }^{\prime }{\sigma }^{\prime }-Nt\mathrm{Pr}{{\theta }^{\prime }}^2=0,$$

(14)

$$NbLe\left(f^{\prime }\sigma -f{\sigma }^{\prime }\right)-Nb{\sigma }^{″}-Nt{\theta }^{″}+Nb{\sigma }_m\sigma {\left(1+\Gamma \theta \right)}^n{e}^{\frac{-E}{1+\Gamma \theta }}=0,$$

(15)

$$Lb\left(f^{\prime }\chi -f{\chi }^{\prime }\right)-{\chi }^{″}+Pe\left(\left(\chi +\delta \right){\sigma }^{″}+{\chi }^{\prime }{\sigma }^{\prime }\right)=0,$$

(16)

with the corresponding boundary conditions

$$f(\eta )=0, f^{\prime }\left(\eta \right)=1, \theta \left(\eta \right)=1, \sigma \left(\eta \right)=1, \chi \left(\eta \right)=1 \mathrm{at} \eta =0,$$

(17)

$$f^{\prime }\left(\eta \right)\to 0, \theta \left(\eta \right)\to 0, \sigma \left(\eta \right)\to 0, \chi \left(\eta \right)\to 0 \mathrm{at} \eta \to \infty .$$

(18)

The thermophysical properties of ternary nanofluids can be written as

$${\rho }_{tn}={\rho }_1{\varphi }_1+{\rho }_2{\varphi }_2+{\rho }_3{\varphi }_3+\left(1-\varphi \right){\rho }_f,$$

(19)

$${\mu }_{tn}={\mu }_f/{\left(1-{\varphi }_1\right)}^{2.5}{\left(1-{\varphi }_2\right)}^{2.5}{\left(1-{\varphi }_3\right)}^{2.5},$$

(20)

$${\left(\rho c_p\right)}_{tf}={\rho }_1{c}_{p,1}{\varphi }_1+{\rho }_2{c}_{p,2}{\varphi }_2+{\rho }_3{c}_{p,3}{\varphi }_3+\left(1-\varphi \right){\left(\rho c_p\right)}_f,$$

(21)

$${\beta }_{tn}={\beta }_{p1}{\varphi }_1+{\beta }_{p2}{\varphi }_2+{\beta }_{p3}{\varphi }_3+\left(1-\varphi \right){\beta }_f,$$

(22)

$$\begin{array}{l}{k}_{tn}=\left(\frac{k_1+2k_{hn}-2{\varphi }_1\left(k_{hn}-k_1\right)}{k_1+2k_{hn}+{\varphi }_1\left(k_{hn}-k_1\right)}\right)k_{hn}, k_{hn}=\left(\frac{k_2+2k_n-2{\varphi }_2\left(k_n-k_2\right)}{k_2+2k_n+{\varphi }_2\left(k_n-k_2\right)}\right)k_n, \\ k_n=\left(\frac{k_3+2k_f-2{\varphi }_3\left(k_f-k_3\right)}{k_3+2k_f+{\varphi }_3\left(k_f-k_3\right)}\right)k_f,\end{array}$$

(23)

where $\varphi ={\varphi }_1+{\varphi }_2+{\varphi }_3.$

The non-dimensional skin friction and local Nusselt number expressions can be written as

$$c_{fx}{\mathrm{Re}}_x^{1/2}=-\frac{{\mu }_{tn}}{{\mu }_f}f″(0) \mathrm{and} N{u}_{fx}{\mathrm{Re}}_x^{-1/2}=-\left(\frac{k_{tn}}{k_f}+Rn\right)\theta \prime (0).$$

3. Numerical Simulations

The present considered problem deals with the system of Equations (13)–(16) along with the boundary conditions (17) and (18). The exact solutions are not possible for the system of equations due to the complex nature of the problem. To evaluate the system of equations, the well-known computational Matlab (bvp4c package) software was used. The bvp4c is a finite difference code that implements the three-stage Lobatto IIIa formula. This is a collocation formula, and the collocation polynomial provides a C¹-continuous solution that is fourth-order accurate uniformly in the interval of integration. For better understanding, the flow chart of the scheme is presented in Figure 2.

4. Discussion of Results

The current section deals with the numerical simulations of velocity, temperature, nanoparticle concentration and microorganism profiles with respect to various fluid flow parameters through pictorial representations. The following values are kept constant for the analysis of the fluid flow: $M=m_1=\lambda =Nr=Rb=Rn=Nb=Nt=Le={\sigma }_m=\Gamma =n=E=Pe=Lb=\delta =0.5.$$\mathrm{Pr}=6.8,$${\varphi }_1={\varphi }_2={\varphi }_3=0.05.$ 

Table 1. Thermophysical aspects of base fluid and nanoparticles [40,41].

Properties	H2O	Ag	Au	Cu
$\rho \left(kg/m^3\right)$	997.1	10,500	19,300	8933
$c_p \left(J/kgK\right)$	4179	235	129	385
$k \left(W/mK\right)$	0.613	429	318	401
$\beta \left(1/K\right)\times {10}^5$	21	1.89	1.4	1.67

displays the thermophysical properties of the base fluid and nanoparticles.

Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7 are plotted to see the variations in the velocity profile for the diverse fluid parameters of interest, such as Lewis number, microorganisms’ parameter, radiation parameter, Brownian motion parameter and thermophoresis parameter. It is noted from Figure 3 that the velocity enhances with rising values of the Lewis number. This is in good agreement with the existing literature [42]. It is also observed that in the absence of the Lewis number, the lower velocities can be reached, which means that the momentum boundary layers are enhanced as the Lewis number increases. It is observed that for increasing values of the microorganisms’ parameter the velocity rises (see Figure 4). It is clear from Figure 5 that with increasing radiation parameter the velocity rises in the boundary layer flow. A similar trend is noticed in [43]. The velocity is an increasing function of the Brownian motion parameter in the boundary layer flow (see Figure 6). It is noticed from Figure 7 that the boundary later thickness decreases with enhancing values of the thermophoresis parameter due to that the velocity profile declines in the boundary layer flow. The nanoparticle temperature distributions were plotted for different parameters such as the microorganisms’ parameter, activation energy parameter, radiation parameter, Brownian motion parameter, thermophoresis parameter and Lewis number in Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13. It is depicted in Figure 8 that the thermal boundary layer thickness declines with the increment of the microorganisms’ parameter and decreases the nanoparticle temperature profile. Activation energy decreases the temperature of the fluid in the boundary layer flow to a certain position, and later it behaves in an opposite trend (see Figure 9). It is clear from Figure 10 that the nanoparticle temperature is an increasing function of the radiation parameter. Similar behaviour is observed in [44]. The Brownian motion and thermophoresis parameters enhance the thickness of the thermal boundary layer, which thereby enhances the nanoparticle temperature (see Figure 11 and Figure 12). The opposite trend is noticed with rising Lewis number regarding the activation energy (see Figure 13).

The nanoparticle concentration profiles are presented in Figure 14, Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19 for various involved parameters such as the microorganisms’ parameter, activation energy parameter, radiation parameter, Brownian motion parameter, thermophoresis parameter and Lewis number. For the given set of parameters, the nanoparticle concentration affectively varies. It is noted from Figure 14 that the nanoparticle concentration decreases with the enhancement in the microorganisms’ parameter. The nanoparticle concentration boundary layer thickness enhances for the activation energy parameter (see Figure 15). It is noticed from Figure 16 and Figure 17 that the nanoparticle concentration of the ternary hybrid nanofluid is a decreasing function of radiation and Brownian motion parameters. The concentration boundary layer thickness increases with rising values of the thermophoresis parameter (see Figure 18). The Lewis number reduces the nanoparticle concentration (see Figure 19). Figure 20, Figure 21, Figure 22, Figure 23, Figure 24 and Figure 25 are drawn to see the microorganism profiles for various involved parameters. Figure 20 displays the effect of the microorganisms’ parameter on the microorganism profile. It is depicted that the parameter declines the profile of microorganisms, and the microorganism boundary layer thickness reduces. The activation energy parameter shows mixed behaviour in the microorganism distribution based on the distance of the Riga plate (see Figure 21). The microorganism profile declines with rising values of radiation and Brownian motion parameters (see Figure 22 and Figure 23). It is observed from Figure 24 that the thermophoresis parameter enhances the microorganism profile, and Figure 25 displays the influence of Lewis number on the microorganism profile. Therefore, it is concluded that the Lewis number decreases the microorganism boundary layer thickness.

Figure 26, Figure 27, Figure 28 and Figure 29 were drawn to see the effect of diverse parameters such as the radiation parameter, thermophoresis parameter, Brownian motion parameter and microorganisms’ parameter on the skin friction coefficient against the activation energy. It is depicted from these figures that the skin friction coefficient rises slowly as the activation energy enhances. Increasing values of the radiation parameter decline the skin friction coefficient (see Figure 26). It is noted from Figure 27 that the skin friction coefficient increases with enhancing values of the thermophoresis parameter. The skin friction coefficient is a decreasing function of the Brownian motion parameter and microorganisms’ parameter (see Figure 28 and Figure 29). Figure 30 and Figure 31 were prepared to see the effect of the Brownian motion parameter and radiation parameter on the local Nusselt number. It is noted from Figure 28 that the local Nusselt number is a decreasing function of the Brownian motion parameter, and it increases slowly with activation energy parameter. The stronger radiation effects result in the enhancement of the local Nusselt number (see Figure 31). Figure 32 displays the comparison of the temperature profiles for base fluid (water), mono nanofluid (Ag/water), hybrid nanofluid (Ag-Au/water) and ternary nanofluid (Ag-Au-Cu/water). It is noted from this figure that the temperature enhancement is more for the ternary nanofluid, and it is less for pure water. It concludes that when we suspend a greater number of nanoparticles in the base fluid it improves the performance of the system.

5. Conclusions

In the current study, the ternary nanofluid (by considering the base fluid as water and nanoparticles as gold, silver and copper) motion past a stretching Riga plate was considered. The effects of gyrotactic microorganisms and radiation were taken into account. The suitable similarity transformations and non-dimensional quantities were utilized to simplify the system of governing equations. The simplified system is not able to obtain the exact solutions. In view of this, the computational software Mathematica was used to analyse the current analysis. The main findings of the current study are as follows:

• The nanoparticle temperature of ternary hybrid nanofluid enhances with activation energy and Brownian motion parameters.
• The radiation effects reduce the thermal boundary layer thickness.
• The Lewis number and thermophoresis number decrease the nanoparticle concentration.
• The radiation parameter enhances the concentration of the boundary layer thickness.
• The Brownian motion and radiation effects increase the microorganism profile.
• The microorganism profile is a decreasing function of the thermophoresis parameter and Lewis number.
• Higher temperatures can be found in ternary nanofluids as compared to classical fluid and mono and hybrid nanofluids.