Numerical Calculation of Thermal Radiative Boundary Layer Nanofluid Flow across an Extending Inclined Cylinder

Abstract

This research presents the numerical analysis of the fluid flow containing the micro gyrotactic organism with heat and mass transfer. The flow is allowed to pass through an inclined stretching cylinder with the effects of heat generation/a heat source and activation energy subject to the symmetric boundary conditions at the cylinder walls. Similarity transformation is employed in the system of PDEs (partial differential equations) to transform them into non-dimensional ODEs (ordinary differential equations). The solution to the proposed problem is obtained by using the bvp4c (numerical scheme). The graphical results are plotted for various flow parameters in order to show their impact on the flow, mass, energy, and motile microorganism profiles. Moreover, the angle of inclination disturbs the flow within an inclined cylinder and slows down the fluid motion, while it elevates the energy of the fluid inside an inclined cylinder. Similarly, the curvature effect is also highlighted in the dynamics of fluid velocity, temperature, and the motile microorganism profile. From the obtained results, it is elucidated that growing values of the curvature factor accelerate the temperature, velocity, and motile microbes’ profiles. Finally, some engineering quantities are calculated in terms of skin friction, the Nusselt and Sherwood number, and the density of motile microbes. The acquired results are also displayed in tabular form.

1. Introduction

The problems of energy and mass transference are used in a variety of physical situations due to their stretching surfaces and them having useful industrial and engineering applications. For instance, they have applications in heat systems in industry, condensation processes, industrial cooling applications, and many other daily life applications. Rohsenow et al. [1] explained the applications of heat transfer in their book. A few years later Bejan [2] discussed the novel applications of energy transport by putting forward an idea of designing a thermodynamic system for various fluid flow problems. Motivated by the emerging applications and the needs of modern research, many scientists have reported on it. For example, Saltiel and Datta [3] inspected heat and mass transport properties and discussed their useful applications in microwave processing. Similarly, Bigdeli et al. [4] carried out research on energy transference by collecting some novel applications in various cooling systems, and their research focused on automotive applications. Vadász [5] investigated some emerging topics on energy transference. In addition to this, some fundamental applications of mass and energy transmission were presented in the book of Bergman et al. [6]. The research of heat and mass transport is famous and has widely used applications in different operating systems, some of which are given in References [7,8,9].

The research of the fluid flow across a stretching cylinder and motile microorganism is of great interest in various fields of engineering sciences; stretching cylinders have useful applications in extrusion processes, fiber technology, polymer production processes, and plastic films, and have many other practical engineering uses. Keeping these applications in mind, many researchers have presented their work. Recently, Hayat et al. [10] studied the effect of fluid that passes through the stretching cylinder and addressed some advanced physical applications of using second grade fluid. Kumar et al. [11] established a fluid model describing the hybrid fluid that is allowed to pass through the stretch cylinder for thermal applications in different industries and modern sciences. In another paper, Hayat et al. [12] carried out research on fluid flow through a stretching inclined cylinder with the effect of mass and energy transfer. In addition to this, the authors addressed the impact of viscous dissipation, the Soret and Dufour effect, and thermal radiation. Naqvi et al. [13] inspected the applications of the stretching cylinder by considering blood as the base fluid and arteries as an example of the stretching cylinder for various biomedical applications. Moreover, more recently, the authors have elucidated the impact of MHD (magnetohydrodynamics) by choosing the boundary conditions as a moving cylinder. The MHD Casson fluid flowing through a stretched cylinder was investigated by Ahmad et al. [14]. Similarly, Alharbi et al. [15], Raizah et al. [16], and Usman et al. [17] studied different features of the fluid passing through the stretching cylinder and discussed its various physical applications. In the above literature, the authors studied the results of the stretching cylinder and their importance.

The microscopic view of the fluid motion can be highlighted by considering the mechanism of the motile gyrotactic microorganism, which has useful applications in industrial science and production technology. Sampath et al. [18] inspected the presence of microbes and calculated the thermal performance of the fluid containing nanoparticles in the presence of second-order slip and convective conditions. Waqas et al. [19] discussed the numerical analysis of fluid moving on a rotating disk and highlighted the impact of microorganisms on the fluid containing nanoparticles for advanced cooling applications. Sreenivasulu et al. [20] studied the fluid flow where the fluid contained gyrotactic microorganisms under the influence of nanoparticles’ flux for some new advanced applications in modern science. Furthermore, research containing gyrotactic microorganisms is famous in various industrial and practical applications. Motivated by these applications, Kotnurkar and Giddaiah [21] highlighted the flow dynamics of peristaltic nano Eyring–Powell fluid by considering the gyrotactic microorganism. Some recent important research studies on the topic of motile gyrotactic microorganisms and nanofluid can be seen in the research of Elsebaee et al. [22], Algehyne et al. [23], Ferdows et al. [24], and Alrabaiah et al. [25].

Recently, several researchers have employed the idea of activation energy (AE) in fluid flow. Yan et al. [26] inspected the activation energy for advanced applications in modern sciences. Ma et al. [27] investigated 2D fluid flow with the AE model for the applications of pyrolysis of solid fuels. Danish et al. [28] discussed the upshot of AE for various physical and real-life applications. In their study, they considered Williamson nanofluid model with the electrical MHD for advanced cooling applications. After this, Arenas et al. [29] studied the activation energy models and their daily life applications. Nazrin et al. [30] calculated the AE for various applications. Similarly, Huang et al. [31] and Satyayani et al. [32] inspected the upshot of AE and its useful applications in the solid fuel processes and the coefficient of diffusion. Some other applications of AE can be found in References [33,34,35].

In modern science, the research of thermal radiation has a broad range of uses in industry and cooling systems. Motivated by the unique applications of thermal radiation, many scholars have considered the impact of radiation on various physical situations. In the early days, the thermal impacts of radiation on the atmosphere were investigated by Idso and Jackson [36]. In another paper, Dicke et al. [37] calculated the measurement of the efficiency of thermal radiation in microwave applications using a simple thermodynamic model. Rehman et al. [38] highlighted the thermal radiation effect of the Darcy nanoliquid flow subject to heat flux. Recently, Howell et al. [39] investigated some advanced applications of heat radiation in energy transfer phenomena. Kumar et al. [40] investigated the technological processes with the improvement of heat and its prospects. Shah et al. [41] debated the upshot of heat radiation and highlighted the dynamics of the fluid containing carbon nanotubes (CNTs) over an elongating sheet. Similarly, Bafakeeh et al. [42], Assiri et al. [43], and Haq et al. [44] discussed the impact of thermal radiation in various physical situations.

There are various physical situations where the chemical reaction (CR) shows a vital role. The chemical reaction is used in chemical engineering, many drug delivery systems, and pharmaceutical applications. Initially, Ewell [45] discussed the chemical reaction and its reaction rate, and some of its useful applications in various industries. Vogt et al. [46] described and discussed the applications of chemical reactions in space and on Earth. Suleimanov et al. [47] scrutinized the phenomena of chemical reaction rate coefficients and explained their useful applications in ring polymer and molecular dynamics. Moreover, in this research, the authors focused on the CR in the prototypical gas phase and explained chemical reactions in multifarious cases. Shi et al. [48] investigated chemical reactions and their applications in modern science. Similarly, Raza et al. [49] studied the nano and hybrid nanoliquid flow with the influence of a chemical reaction, magnetic field, and activation energy.

Motivated by the above-mentioned literature, in the current research article, we focused on investigating the numerical analysis of the fluid flow containing the micro gyrotactic organism. The flow is allowed to pass through an inclined stretching cylinder with the effects of heat generation/a heat source and AE. Such types of effects have not been considered before for a thermally radiative nanofluid flow subject to the symmetric boundary conditions. The symmetric boundary conditions presume that identical physical processes exist on both sides of the cylinder. At the same distance from the boundary, all variables have the same values and gradients. As a result, it works as a mirror, reflecting all flow patterns to the other side. The similarity transformation is employed on the system of PDEs to transform them into non-dimensional ODEs; then, solutions are obtained to the proposed problem by using the bvp4c numerical scheme. The graphical results are plotted for various flow parameters in order to show their impact on the flow, energy, and mass profiles. Finally, some engineering quantities are calculated in tabular form.

2. Mathematical Formulation

We considered the mixed convection laminar thermally radiative fluid flow with convective conditions across an inclined stretching cylinder. The fluid flow is comprised of motile microbes over an elongating surface. The cylinder is stretched with velocity $U_w$ horizontally having radius $a$ as displayed in Figure 1. Here, $T$ and $C$ are the fluid temperature and concentration, respectively. The surface of the cylinder is exposed to activation energy.

The temperature difference and gravity force between the cylinder surface and those neighboring cause the buoyancy effect. The modeled equations are formulated as [12,50,51]:

$$\frac{\partial }{\partial x}\left(ru\right)+\frac{\partial }{\partial r}\left(rv\right)=0,$$

(1)

$$\begin{array}{c}{\rho }_{f\mathrm{\infty }}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial r}\right)-\mu \left(\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial u}{\partial r}\right)\right)-\left(1-C_{\mathrm{\infty }}\right){\rho }_{f\mathrm{\infty }}\left(T-T_{\mathrm{\infty }}\right)\beta g\mathit{cos}{\alpha }_1+\left({\rho }_p-{\rho }_{f\mathrm{\infty }}\right)g\left(C-C_{\mathrm{\infty }}\right)\mathit{cos}{\alpha }_1\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left({\rho }_{n\mathrm{\infty }}-{\rho }_{f\mathrm{\infty }}\right)g{\gamma }_1\left(n-n_{\mathrm{\infty }}\right)\mathit{cos}{\alpha }_1=0,\hfill \end{array}$$

(2)

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}={\alpha }^{\mathrm{*}}\left(1+\frac{4}{3}R\right)\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial T}{\partial r}\right)+\tau \left[D_B\left(\frac{\partial T}{\partial r}\frac{\partial C}{\partial r}+\frac{\partial T}{\partial x}\frac{\partial C}{\partial x}\right)+\frac{D_T}{T_{\mathrm{\infty }}}\left({\left(\frac{\partial T}{\partial r}\right)}^2+{\left(\frac{\partial T}{\partial x}\right)}^2\right)\right]+Q_0\left(T-T_{\mathrm{\infty }}\right),$$

(3)

$$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial r}=\frac{D_B}{r}\frac{\partial }{\partial r}\left(r\frac{\partial C}{\partial r}\right)+\frac{D_T}{T_{\mathrm{\infty }}}\frac{1}{r}\frac{\partial }{\partial x}\left(r\frac{\partial C}{\partial r}\right)-k_r^2\left(C-C_0\right){\left(\frac{T}{T_{\mathrm{\infty }}}\right)}^n\mathit{exp}\left(-\frac{E_a}{\kappa T}\right),$$

(4)

$$u\frac{\partial n}{\partial x}+v\frac{\partial n}{\partial r}=\frac{D_n}{r}\frac{\partial }{\partial r}\left(r\frac{\partial n}{\partial r}\right)-\frac{b_c{W}_c}{\left(C_w-C_{\mathrm{\infty }}\right)}\frac{1}{r}\frac{\partial }{\partial r}n\left(r\frac{\partial C}{\partial r}\right).$$

(5)

The boundary conditions are:

$$\left.\begin{array}{c}u=U_0\left(\frac{x}{l}\right)=U_w,-k\frac{\partial T}{\partial r}=h\left(T_f-T\right),n=n_w,v=0,-D_m\frac{\partial C}{\partial r}=k_m\left(C_w-C\right) at r=a,\\ u\to 0,n\to n_{\mathrm{\infty }},v\to 0,C\to C_{\mathrm{\infty }},T\to T_{\mathrm{\infty }} as r\to \infty .\hfill \end{array}\right\}$$

(6)

Here, $u$, $v$ are the components of velocity. ${\rho }_n,{ \rho }_p,{ \rho }_f$ are the microorganism’s density, fluid density, and nanoparticles’ density, respectively. $E_a$ is the activation energy, $k_r^2$ is the 2nd-order chemical reaction. Furthermore, $n$, ${\gamma }_1$, $\beta$, $g$, and ${\alpha }_1$ are the motile microbe density, average volume of microorganisms, volume expansion, gravity acceleration, and inclination angle of stretching cylinder, respectively; $\tau$ is heat capacitance, $D_n$, is the microorganism diffusivity. The similarity variables are [51]:

$$\eta =\frac{r^2-a^2}{2a}\sqrt{\frac{U_W}{vx}},\psi =a\sqrt{v{U}_{w}x}f\left(\eta \right),\mathrm{\hslash }=\frac{n-n_{\mathrm{\infty }}}{n_w-n_{\mathrm{\infty }}},\varphi \left(\eta \right)=\frac{C-C_{\mathrm{\infty }}}{C_w-C_{\mathrm{\infty }}},\theta \left(\eta \right)=\frac{T-T_{\mathrm{\infty }}}{T_w-T_{\mathrm{\infty }}}.$$

(7)

Incorporating Equation (7) in Equations (1)–(5):

$$\left(2\gamma \eta +1\right)f^{‴}+\left(2\gamma +f\right)f^{″}-f^{\prime 2}+Ri\left(\theta -N_r\varphi -R_b\mathrm{\hslash }\right)\mathit{cos}{\alpha }_1=0,$$

(8)

$$\left(1+\frac{4}{3}R\right)\left(1+2\gamma \eta \right){\theta }^{″}+\left(2\gamma +Prf\right){\theta }^{\prime }+\left(2\gamma \eta +1\right)Pr\left(N_B{\varphi }^{\prime }+N_T{\varphi }^{\prime }\right){\varphi }^{\prime }+Pr\zeta \theta =0,$$

(9)

$$\left(2\gamma \eta +1\right)N_B{\varphi }^{″}+N_B\left(2\gamma +Scf\right){\varphi }^{\prime }+N_T\left(\left(2\gamma \eta +1\right){\theta }^{″}+2\gamma {\varphi }^{\prime }\right)-Kr{\left(1+\epsilon \delta \right)}^n\varphi \mathit{exp}\left(-\frac{E}{1+\epsilon \delta }\right)=0,$$

(10)

$$\left(2\gamma \eta +1\right){\mathrm{\hslash }}^{‴}+2\gamma {\mathrm{\hslash }}^{\prime }+LbPr{\mathrm{\hslash }}^{\prime }-Pe\left[\begin{array}{c}\sigma \gamma {\varphi }^{\prime }+\sigma \left(2\gamma \eta +1\right){\varphi }^{″}+\gamma {\varphi }^{\prime }\hslash +\\ \left(2\gamma \eta +1\right){\varphi }^{″}\hslash +\left(2\gamma \eta +1\right){\varphi }^{\prime }{\mathrm{\hslash }}^{\prime }\end{array}\right]=0,$$

(11)

with boundary conditions

$$\begin{array}{c}f\left(0\right)=0,f^{\prime }\left(0\right)=1,\mathrm{\hslash }\left(0\right)=1,{\theta }^{\prime }\left(0\right)=-B{i}_c\left(1-\theta \left(0\right)\right),{\varphi }^{\prime }\left(0\right)=-B{i}_t\left(1-\varphi \left(0\right)\right)\eta \to \mathrm{\infty }:f^{\prime }\to 0,\theta \to 0,\varphi \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\to 0,\mathrm{\hslash }\to 0\hfill \end{array}$$

(12)

where $\gamma =\frac{1}{a}\sqrt{\frac{vl}{U_0}}$ is the curvature parameter, $N_r=\frac{\left({\rho }_p-{\rho }_{\mathrm{\infty }}\right)\mathsf{\Delta }C}{\left(1-C_{\mathrm{\infty }}\right){\rho }_{\mathrm{\infty }}\beta \mathsf{\Delta }T}$ is buoyancy ratio factor, $R_b=\frac{\left({\rho }_{m\mathrm{\infty }}-{\rho }_{\mathrm{\infty }}\right){\gamma }_1\mathsf{\Delta }n}{\left(1-C_{\mathrm{\infty }}\right){\rho }_{\mathrm{\infty }}\beta \mathsf{\Delta }T}$ is the Rayleigh number, $Ri=G{r}_x/{\mathit{Re}}_x^2=g{\beta }_T{l}^2\mathsf{\Delta }T/\left(U_0^{2}x\right)$ is Richardson number, $Pr=v/\alpha$ is Prandtl number, $N_B=\tau D_B\mathsf{\Delta }C/\left(v{T}_{\mathrm{\infty }}\right)$ is Brownian motion factor, $\zeta =Q_{0}l/\left(\rho C_p{U}_0\right)$ is heat generation/absorption term, $N_T=\tau D_T\mathsf{\Delta }T/\left(v{T}_{\mathrm{\infty }}\right)$ is thermophoresis constraint, $Lb=D_B/D_n$ is Lewis number, $Pe$ is the Peclet number, $Sc=v/D_B$ is Schmidt number, $E=\frac{E_a}{\kappa T_{\mathrm{\infty }}}$ is the activation energy, $B{i}_c=\frac{k_m}{D_m}\sqrt{\frac{\upsilon l}{U_o}}$ is the concentration Biot number, $B{i}_t=\frac{h}{k}\sqrt{\frac{\upsilon l}{U_o}}$ is the thermal Biot number, $R=\frac{4\sigma T_{\mathrm{\infty }}^3}{k{k}^{*}}$ is the ratio factor, and $\sigma =n_{\mathrm{\infty }}/\left(n_w-n_{\mathrm{\infty }}\right)$ is the motile parameter.

The engineering-interested quantities $\left(C_f,N{u}_x,S{h}_x,N{n}_x\right)$ are stated as:

$$C_f=\frac{2{\tau }_w}{\rho U_w^2},N{u}_x=\frac{x{q}_w}{k\mathsf{\Delta }T},N{n}_x=\frac{x{q}_n}{D_n\mathsf{\Delta }n},S{h}_x=\frac{x{q}_m}{D_B\mathsf{\Delta }C}.$$

(13)

The converted form of Equation (13) is:

$$\frac{1}{2}{C}_f\sqrt{R{e}_x}=-f^{″}\left(0\right),\frac{S{h}_x}{\sqrt{R{e}_x}}=-{\phi }^{\prime }\left(0\right),\frac{N{n}_x}{\sqrt{R{e}_x}}=-{\mathrm{\hslash }}^{\prime }\left(0\right),\frac{N{u}_x}{\sqrt{R{e}_x}}=-{\theta }^{\prime }\left(0\right).$$

(14)

where $R{e}_x=\frac{U_0{x}^2}{vl}$ is the local Reynold’s number.

3. Numerical Solution

The numerical Matlab package bvp4c, which is specially designed for boundary value problems, is used to solve the dimensionless set of ODEs as:

$$\left.\begin{array}{c}{\mathrm{\wp }}_1=f\left(\eta \right),{\mathrm{\wp }}_3=f^{″}\left(\eta \right),{\mathrm{\wp }}_5={\theta }^{\prime }\left(\eta \right),{\mathrm{\wp }}_7={\varphi }^{\prime }\left(\eta \right),{\mathrm{\wp }}_9={\mathrm{\hslash }}^{\prime }\left(\eta \right),\\ {\mathrm{\wp }}_2=f^{\prime }\left(\eta \right),{\mathrm{\wp }}_4=\theta \left(\eta \right),{\mathrm{\wp }}_6=\varphi \left(\eta \right),{\mathrm{\wp }}_8=\hslash \left(\eta \right).\hfill \end{array}\right\}$$

(15)

By putting Equation (15) in Equations (8)–(12), we obtain:

$$\left(2\gamma \eta +1\right){\mathrm{\wp }}_3^{\prime }+\left(2\gamma +{\mathrm{\wp }}_1\right){\mathrm{\wp }}_3-{{\mathrm{\wp }}_2}^2+Ri\left({\mathrm{\wp }}_4-N_r{\mathrm{\wp }}_6-R_b{\mathrm{\wp }}_8\right)\mathit{cos}{\alpha }_1=0$$

(16)

$$\left(1+\frac{4}{3}R\right)\left(2\gamma \eta +1\right){\mathrm{\wp }}_5^{\prime }+\left(2\gamma +Pr{\mathrm{\wp }}_1\right){\mathrm{\wp }}_5+\left(2\gamma \eta +1\right)Pr\left(N_B{\mathrm{\wp }}_7+N_T{\mathrm{\wp }}_7\right){\mathrm{\wp }}_7+Pr\zeta {\mathrm{\wp }}_4=0$$

(17)

$$\left(2\gamma \eta +1\right)N_B{\mathrm{\wp }}_7^{\prime }+N_B\left(2\gamma +Sc{\mathrm{\wp }}_1\right){\mathrm{\wp }}_7+N_T\left(\left(2\gamma \eta +1\right){\mathrm{\wp }}_5^{\prime }+2\gamma {\mathrm{\wp }}_7\right)-Kr{\left(1+\epsilon \delta \right)}^n{\mathrm{\wp }}_6\mathit{exp}\left(-\frac{E}{1+\epsilon \delta }\right)=0$$

(18)

$$\left(2\gamma \eta +1\right){\mathrm{\wp }}_9^{\prime }+2\gamma {\mathrm{\wp }}_9+LbPr{\mathrm{\wp }}_9-Pe\left[\begin{array}{c}\sigma \gamma {\mathrm{\wp }}_7+\sigma \left(2\gamma \eta +1\right){\mathrm{\wp }}_7^{\prime }+\gamma {\mathrm{\wp }}_7{\mathrm{\wp }}_8+\\ \left(2\gamma \eta +1\right){\mathrm{\wp }}_7^{\prime }{\mathrm{\wp }}_8+\left(2\gamma \eta +1\right){\mathrm{\wp }}_7{\mathrm{\wp }}_9\end{array}\right]=0.$$

(19)

with boundary conditions

$$\begin{array}{c}{\mathrm{\wp }}_1\left(0\right)=0,{\mathrm{\wp }}_2\left(0\right)=1,{\mathrm{\wp }}_5\left(0\right)=-B{i}_c\left(1-{\mathrm{\wp }}_4\left(0\right)\right),{\mathrm{\wp }}_7\left(0\right)=-B{i}_t\left(1-{\mathrm{\wp }}_6\left(0\right)\right),{\mathrm{\wp }}_8\left(0\right)=1\eta \to \mathrm{\infty }:{\mathrm{\wp }}_2\to 0,{\mathrm{\wp }}_4\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\to 0,{\mathrm{\wp }}_6\to 0,{\mathrm{\wp }}_8\to 0\hfill \end{array}$$

(20)

The obtained results (Equations (16)–(20)) are further solved through the bvp4c package by using Matlab (2021a) software.

4. Results and Discussion

In the current examination, we investigated the numerical analysis of the boundary layer flow containing the micro gyrotactic organism. Similarity transformation was employed on the system of PDEs to transform them into non-dimensional ODEs; we then obtained the solutions to the proposed problem by using the bvp4c numerical scheme. Some engineering quantities were calculated in terms of numerical values. The geometry of the current flow regime is presented in Figure 1. The velocity fluctuation against various parameters is given in Figure 2a–e, and the impact of temperature against pertinent parameters is provided in Figure 3a–j; similarly, the concentration profile is highlighted in Figure 4a–c, and the motile microorganism profile is highlighted in Figure 5a,b. During the present analysis, some engineering values were calculated and are presented in

Table 1. Numerical results for Nusselt number, skin friction, the density of motile microbes, and Sherwood number.

$\mathit{L}\mathit{b}$	$\mathit{N}\mathit{b}$	$\mathit{S}\mathit{c}$	$\mathit{cos}{\mathit{\alpha }}_1$	$-{\mathit{f}}^{″}$(0)	$-{\mathit{\theta }}^{\mathbf{\prime }}$(0)	$-{\mathit{\varphi }}^{\mathbf{\prime }}$(0)	$-{\mathbf{\hslash }}^{\mathbf{\prime }}$(0)
0.1	0.1	0.3	0.3	1.1330	0.1710	1.2255	1.4126
0.3				1.1330	0.1710	1.2255	1.5335
0.5				1.1330	0.1710	1.2255	1.6345
	0.2			1.1242	0.4595	0.0754	0.7615
	0.4			1.1232	0.4193	1.1162	0.7675
	0.6			1.0171	0.3458	1.1469	0.7757
		0.5		1.1440	0.6570	0.3123	1.3420
		0.7		1.1422	0.5310	0.4231	1.3161
		0.9		1.1332	0.4313	0.5859	1.3364
			0.3	1.1461	0.1692	1.2226	1.4896
			0.707	1.1240	0.1848	1.2475	1.4364
			0.860	1.1490	0.1797	1.1899	1.4365

against various pertinent values.

4.1. Flow Characteristics

The effect of the inclination angle $\mathit{cos}{\alpha }_1$, $\gamma$, $Nr$, $Rb$, and $Ri$ is emphasized in Figure 2a–e, respectively. The impact of $\mathit{cos}{\alpha }_1$ on the fluid velocity over a stretching cylinder is portrayed in Figure 2a. The enhancement in the angle of inclination provides resistance to the fluid flow within an inclined cylinder and slows down the fluid motion. This is because during the inclination, the fluid in the cylinder resists, and as a result, the viscous forces are dominant, which slows down the motion of the fluid and due to this, the velocity is reduced. The influence of the curvature term is portrayed in Figure 2b. It was found that fluid velocity shows an enhancing nature for rising values of $\gamma$. Here, larger values of $\gamma$ correspond to a diminution in cylinder radius, causing less fluid particle interaction with the cylindrical surface, which implies less resistance to fluid flow. As a result, fluid velocity significantly increases. The graphical sketch of the buoyancy ratio factor $Nr$ on the fluid velocity is highlighted in Figure 2c. The velocity profile declines with the effect of Nr. Physically, it represents how much the species’ buoyancy force contributes to the thermal buoyancy force. It should be noted that the flow velocities are significantly reduced when Nr > 0, or when buoyancy forces are aiding. Increased Nr values reduced the fluid flow. The influence of $Rb$ is portrayed in Figure 2d. From the figure, the decay in velocity is noticed, and it is found that the bioconvection parameter is responsible for controlling the flow of the fluid. The fluctuation in the fluid velocity is portrayed in Figure 2e against various values of $Ri$. The fluctuation shows that when escalating the values of $Ri$, the velocity of the fluid increases as shown in the figure.

4.2. Heat Characteristics

We plotted the influence of the values of flow constraints such as $\mathit{cos}{\alpha }_1$,$\xi$, $\gamma$, $Nb$, $Nr$, $Pr$, $Rb$, $Ri$, $Rd$, and $B{i}_t$ on the energy curve in Figure 3a–j, respectively. The impact of $\mathit{cos}{\alpha }_1$ on the fluid temperature is portrayed in Figure 3a. The angle of inclination disturbs the flow within an inclined cylinder, and due to this disturbance, some inner forces are produced in the fluid. This increases the friction among the fluid particles, due to which the temperature becomes high within an inclined stretching cylinder. The upshot of the heat source is that it boosts the energy curve as demonstrated in Figure 3b. The fluctuation in temperature is calculated versus positive values of curvature factor $\gamma$ and is presented in Figure 3c. From the temperature variation, it can be observed that $\gamma$ is responsible for the increase in the fluid temperature passing through the inclined stretch cylinder. Figure 3d exhibits the consequences of $Nb$ on the energy field. It can be noticed that the rising values of $Nb$ augment the energy curve due to the vibration produced in the fluid.

Figure 3e highlights the effect of the buoyancy ratio factor $N_r$ on the energy profile. From the figure, temperature increases with the increase in $N_r$, which is physically precise. The temperature goes high because $N_r$ increases the inner forces, due to which some extra energy develops within the fluid. The impact of $Pr$ on the energy curve is portrayed in Figure 3f. From the figure, the increase in $Pr$ as a result of the temperature increase is due to the fact that $Pr$ is the main factor behind the increase in the thermal characteristics of the fluid within the inclined stretching cylinder. The effect of $Rb$ on the energy field is detected in Figure 3g. The graphical analysis shows that the effect of $Rb$ enhances the energy curve. The influence of Ricardson number $Ri$ on the temperature curve is detected in Figure 3h. The graphical analysis of the temperature shows a decrease versus the significance of Ri because $Ri$ controls the temperature of the system. The energy profile augments versus the growing trend of the $Rd$ factor (see Figure 3i). Figure 3j displays the effect of the Biot number on the energy curve. It can be noticed that the mounting values of $B{i}_t$ enhance the temperature curve, this is due to the fact that $B{i}_t$ raises the fluid temperature because the cumulative values of $B{i}_t$ yields a disturbance in the fluid.

4.3. Mass Characteristics

The effect of $Kr$, $Ea$, and $B{i}_c$ on the concentration field is revealed in Figure 4a–c. The impact of chemical reaction parameter $Kr$ is displayed in Figure 4a. The rising effect of Kr reduces the concentration profile. Physically, in the exothermic reaction, the heat is released from the system, so according to the mathematical relation E = Mc², the energy and mass is directly proportional, so that is why the effect of the chemical reaction, in this case, reduces the concentration profile. The consequence of $Ea$ is depicted in Figure 4b. The mass curve enhances with the positive variation in activation energy $Ea$. Physically, the activation energy is the minimum amount of energy/heat needed to boost the chemical reaction; thus, the action of Ea accelerates the fluid molecules so it is moving fast, which causes an elevation in the mass distribution. Figure 4c depicts the influence of mass Biot number $B{i}_c$ on the mass profile. The mass profile of the fluid grows higher for the escalating values of $B{i}_c$. According to the general mathematical expression of mass Biot number $B{i}_c=\frac{k_m}{D_m}L$, where L is the characteristic length, D_m is the mass diffusivity, and k_m is the mass transfer factor, this is why the rising effect of $B{i}_c$ improves the mass profile.

4.4. Motile Microorganism Interpretation

In this section, we focused on presenting the flow analysis of curvature parameter $\gamma$ and Lewis number $Lb$ on the microorganism profile shown in Figure 5a,b, respectively. The impact of curvature parameter $\gamma$ on the motile gyrotactic microorganism profile is depicted in Figure 5a. The effect of $Lb$ on the motile microorganism curve is shown in Figure 5b. The numerical outputs for the Sherwood number, motile microorganism, Nusselt number, and skin friction are shown in Table 1. It was observed that the velocity and energy transfer rate increases with the rising effect of Brownian motion, whereas the inclination angle lowers the mass and energy conveyance rate.

Table 2. Numerical results for Sherwood number versus physical parameters.

$\mathit{S}\mathit{c}$	$\mathit{E}$	Kr	$-{\mathit{f}}^{″}\left(0\right)$	$-{\mathit{\theta }}^{\mathbf{\prime }}\left(0\right)$	$-{\mathit{\varphi }}^{\mathbf{\prime }}\left(0\right)$	$-{\mathbf{\hslash }}^{\mathbf{\prime }}\left(0\right)$
0.1	1.0	0.2	1.2441	0.5971	0.699480	1.2921
0.3			1.2423	0.5211	0.826871	1.3060
0.5			1.2398	0.4212	1.016631	1.4263
0.7			1.2333	0.1611	1.104621	1.4025
	1.0		1.1440	0.3458	0.858235	0.7675
	1.5		1.1422	0.6570	0.828462	0.7757
	2.0		1.1332	0.5310	0.795743	1.3420
	2.5		1.1230	0.4313	0.653740	1.3161
		0.2	1.1461	0.1692	0.793485	1.3364
		0.4	1.1240	0.1848	0.828462	1.4896
		0.6	1.1490	0.1797	0.861540	1.4364
		0.8	1.1521	0.1692	0.893523	1.4365

reveals the numerical results for the Sherwood number versus the variation in the chemical reaction, the Schmidt number, and activation energy.

Table 3. Validation of the results with the published study.

Parameters		Ref. [52]	Present Work	Error Analysis	Ref. [52]	Present Work	Error Analysis	Ref. [52]	Present Work	Error Analysis
$\mathit{L}\mathit{b}$	$\mathit{N}\mathit{b}$	$-{\mathit{f}}^{″}$(0)	$-{\mathit{f}}^{″}$(0)		$-{\mathit{\theta }}^{\mathbf{\prime }}$(0)	$-{\mathit{\theta }}^{\mathbf{\prime }}$(0)	$2.4728\times {10}^{-5}$	$-{\mathit{\varphi }}^{\mathbf{\prime }}$(0)	$-{\mathit{\varphi }}^{\mathbf{\prime }}$(0)
0.5	0.2	1.1330	1.133031	$1.1123\times {10}^{-5}$	0.2610	0.261085	$4.1742\times {10}^{-6}$	1.3155	1.315582	$3.4840\times {10}^{-5}$
0.6		1.1330	1.133035	$3.4840\times {10}^{-5}$	0.2610	0.261035	$3.7381\times {10}^{-7}$	1.3155	1.315523	$2.1238\times {10}^{-6}$
0.7		1.1330	1.133052	$2.1238\times {10}^{-6}$	0.2610	0.261036	$3.6371\times {10}^{-5}$	1.3155	1.315546	$3.5942\times {10}^{-6}$
0.8		1.1330	1.133063	$3.5942\times {10}^{-6}$	0.2610	0.261083	$4.8294\times {10}^{-8}$	1.3155	1.315543	$4.3718\times {10}^{-7}$
	0.2	1.3242	1.324275	$4.3718\times {10}^{-7}$	0.5495	0.549563	$5.8321\times {10}^{-7}$	1.1654	1.165474	$5.9372\times {10}^{-8}$
	0.3	1.3232	1.323247	$5.9383\times {10}^{-6}$	0.5095	0.509568	$7.6479\times {10}^{-8}$	1.2062	1.206232	$6.9378\times {10}^{-7}$
	0.5	1.3217	1.321794	$6.1242\times {10}^{-5}$	0.4358	0.435824	$4.9724\times {10}^{-9}$	1.2369	1.236953	$5.0338\times {10}^{-8}$
	1.0	1.3184	1.318457	$4.9724\times {10}^{-6}$	0.3246	0.324657	$1.8473\times {10}^{-8}$	0.3952	0.395264	$2.4125\times {10}^{-7}$

validates the present results with the published literature, and it is observed that the presented results are in a good agreement with the published data. Furthermore, Table 3 presents the error analysis of the current results with the published work.

5. Conclusions

This research investigates the numerical analysis of the boundary layer flow containing the micro gyrotactic organism. The flow is allowed to pass through an inclined stretching cylinder with the effects of heat generation/a heat source, and activation energy. The similarity transformation is employed on the system of PDEs to transform them into non-dimensional ODEs, then solutions are obtained to the proposed problem by using the bvp4c numerical scheme. The graphical results are plotted for various flow parameters in order to show their impact on the flow, temperature, concentration, and motile microorganism profiles. Some engineering quantities are calculated and presented in tabular form. During the present research, we obtained the following results:

• The angle of inclination disturbs the flow within an inclined cylinder and slows down the fluid motion while elevating the temperature of the fluid inside an inclined cylinder.
• The curvature effect $\gamma$ is also highlighted in the dynamics of the fluid velocity, temperature, and motile microorganism profile, and it is found that escalating values of $\gamma$ accelerate the temperature, velocity, and motile microbes’ profiles.
• Velocity is lowered for the rising values of $Nr$ and $Rb$.
• The temperature grows higher for the cumulative values of $\xi$, $Nb$, and $Rb$.