Mathematical solutions for coupled nonlinear equations based on bioconvection in MHD Casson nanofluid flow

1.   Introduction

Nanofluids are generated when simple nanoparticles are dispersed in base fluids like ethylene glycol, oil, and water. These fluids have higher thermal conductivity than regular liquids. In 1995, at the Argonne National Laboratory, Choi and Eastman ^[1] were the first to show that solid nanoparticles dispersed in a base fluid make up a nanofluid. Nanofluids have applications in hybrid-powered engines, microelectronics, chemical catalytic reactors, particle storage facilities, pharmaceutical procedures, medicine diffusion, and electronic equipment cooling. To improve heat and mass transport properties, researchers have concentrated on nanofluids. Valipour et al. ^[2] numerically examined the CNT-polyethylene nanofluid around a stretching porous cylinder using the Runge–Kutta method and the impact of the Reynolds number, injection parameter, magnetic number, and volume fraction of CNT on flow style. They concluded that the CNT volume fraction and injection parameter had a direct relation with the velocity profile; on the other hand, the magnetic and Reynolds numbers had a reverse relation with the velocity profile. Mishra et al. ^[3] examined the heat characteristics of a silver-water nanofluid around a magnetized stretching cylinder. The investigation included the impacts of Joule heating, viscous dissipation, suction/injection, and slip boundary conditions. Two different models of thermal conductivity and dynamic viscosity were considered, corresponding to nanoparticles with different shapes: cylindrical and spherical (nanotubes). To solve the nonlinear equations, researchers employed a numerical approach, specifically the RKF45 order via the shooting algorithm, and found that when the nanofluid contains spherical nanoparticles, the fluid temperature is lower than when it contains cylindrical nanoparticles. Ramzan et al. ^[4] examined magnetohydrodynamic dusty Casson nanofluid flow over a deformable cylinder. They included the effects of Newtonian heating, heat sink/source, and binary chemical reactions together with activation energy to emphasize the novelty of their research. Numerically, they solved governing equations using the bvp4c technique. They found that the velocity field for the fluid flow decayed for high values of momentum dust particles, while a reverse pattern appeared for the dust phase. Rashed et al. ^[5] researched the nonhomogeneous Newtonian nanofluid flow over heated permeable cylinders and addressed Brownian and thermophoresis effects. The system of equations was computed numerically. Velocity, concentration, and temperature profiles of the nanofluid were boosted by increasing thermophoresis parameters. Recently, several studies were reported on nanofluid flow in cylinders ^[6,7,8].

The process by which fluid viscosity converts kinetic energy into internal energy is known as viscous dissipation. This process affects thermal distribution by acting like an energy source and changing heat transfer rates. Surface temperature influences viscous dissipation processes. Additionally, various factors, such as thermal radiation, Brownian diffusion, suction/injection, and viscous dissipation, may influence the enhancement of heat transfer. This characteristic has several real-world applications, such as in the production of electronic chips and plastics, food processing, electric power generation systems, cooling nuclear equipment, and groundwater contamination. Hsiao ^[9] analyzed the combined nanofluid and micropolar flow impacted by viscous dissipation and magnetohydrodynamic (MHD) through a stretching sheet. The finite difference technique was applied to discretize the resulting equations. The impact of important nondimensional parameters on temperature, velocity, and concentration was demonstrated. Temperature increased as the Eckect number increased, while the opposite was observed with the Prandtl number. Shahzad et al. ^[10] investigated the MHD Jeffery nanofluid flow over a stretching sheet with viscous dissipation aspects. They utilized copper nanoparticles dispersed with three various base fluids: polyvinyl alcohol (PVA) water, kerosene, and ethylene glycol. They found that the temperature profile improved as they increased the nanoparticle volume fraction, Eckert number, and magnetic parameter. Furthermore, when compared to the temperature profiles of Cu-kerosene nanofluids, the temperature distribution in the Cu-PVA-based Jeffrey nanofluid was higher. Zokri et al. ^[11] investigated the effect of viscous dissipation on Jeffrey flow over a horizontal circular cylinder, using the Keller-box approach to resolve numerically the nondimensional variables. The researchers discussed two specific cases: one where the cylinder was heated and another where it was cooled. They discovered that by increasing the Eckert number, there was a small increase in the skin friction coefficient (drag indicator) and a decrease in the Nusselt number (heat transfer indicator). Recent investigations on magnetic fields and viscous dissipation of nanofluid flow across surfaces may be found in ^{[12,13,14,15]}.

In 1889, the scientist Svante Arrhenius defined the term activation energy as the smallest energy to start a chemical reaction. The definition of activation energy is derived from the Arrhenius equation, which describes how rate constants change with temperature. Mass equations that combine activation energy with binary chemical reactions have applications in geothermal and chemical engineering, food processing, oil reservoirs, and water and oil suspensions, among other fields. Bestman ^[16] first proposed this concept, and Azam et al. ^[17] modeled transient bioconvection Cross nanofluid flow over a contracting/expanding cylinder. The highly nonlinear system of ODEs was solved using the shooting Fehlberg approach to obtain numerical solutions for both when the cylinder was stretching and when it was shrinking. Accordingly, nanoparticle concentration increases with increasing values of the activation energy parameter for a stretching and contracting cylinder, but reverse outcomes are obtained for the chemical reaction parameter. For recent research on activation energy impact on various nanofluids under numerous conditions, see ^{[18,19,20,21,22]}.

Bioconvection is a naturally occurring phenomenon caused by the erratic movement of microorganisms, either as single cells or in colony-like arrangements. For example, gyrotactic bacteria may move in the reverse direction of gravity when there is no movement. In the context of bioconvection, the migration of microorganisms is influenced by the asymmetric mass distribution balance. Bioconvection processes have various applications in many medical and industry sectors; they have been included in a variety of micro-biosystems encompassing enzyme biosensors, ethanol, biotechnology, hydrogen gas manufacturing, fertilizer synthesis, biofuel generation, and biodiesel processing, as well as the use of subpopulations, purified cultures, and swimming classes. The first theoretical bioconvection model including a wide range of mobile microorganisms was developed by Plesset and Winet ^[23]. Abdelmalek et al. ^[24] evaluated the Williamson bioconvection nanofluid flow toward a stretched cylinder with second-order slip and variable thermal conductivity. The numerical solution was computed with the shooting scheme. Researchers noticed that the activation energy, thermophoresis, and slip parameters had a significant influence on the variation of nanoparticle concentration. Furthermore, they observed an increased gyrotactic microorganism distribution for the Weissenberg number. Imran et al. ^[25] studied bioconvection in Cross nanofluid flow over a cylinder with motile microbes and convective conditions. The system of ODEs was numerically solved using the bvp4c technique. They determined that higher Peclet and Lewis values reduce the microorganism profile. Nabwey et al. ^[26] examined a magneto-Carreau nanofluid over an inclined cylinder with gyrotactic microorganism effects. They concluded that the concentration of nanoparticles in the nanofluid improves as the activation energy limitation increases. Recently, some scholars performed numerical analyses of bioconvection swimming microorganisms ^[27,28,29].

It is believed that the thermal conductivity of fluids can be enhanced using nanoparticles. This is essential for applications like energy systems, industrial processes, and cooling systems that need effective heat transfer. Copper and silver show high thermal conductivity and are well-suited for increasing heat transfer rates. Further, the fluid can be controlled by an external magnetic field with the assistance of the magnetic properties of ferrous ferric oxide. These properties, combined in a Casson fluid, may aid in drug delivery systems, magnetic cooling, and microfluidic devices. This will optimize the fluid behavior in specific industrial, medical, and environmental settings. Therefore, the novelty of this paper is to perform a comparative analysis of ($F{e_3}{O_4}/Cu/Ag - C{H_3}OH$) nanofluids regarding concentration, thermal, momentum, and motile microorganism density along a stretching cylinder. The dimensionless equations are computed using the bvp4c approach in MATLAB software, and an investigation is carried out on the impact of representative factors. Furthermore, three ANN models are built to predict SFC at the cylinder surface. The results of this study have numerous applications on bioconvection in magneto-Casson nanofluids and could improve current technologies in several domains, particularly those pertaining to fluid flow, heat transfer, and biological processes. According to this study, a ferrous ferric oxide nanofluid raises the Nusselt number, which is an indicator of heat transfer efficiency. Cooling systems can use this to improve heat dissipation and boost system performance.

2.   Problem statement

The two-dimensional, steady, and incompressible laminar boundary layer flow for a non-Newtonian Casson nanofluid flowing through a permeable linearly stretching cylinder is considered here. Three different types of nanoparticles are considered: ferrous ferric oxide $\left({F{e_3}{O_4}} \right)$, copper $\left({Cu} \right)$, and silver $\left({Ag} \right)$, together with the base fluid methanol $\left({C{H_3}OH} \right)$. The cylindrical polar $\left({x, r} \right)$ coordinate system is considered, where x indicates the axial direction and r specifies the radial direction. At the boundary of a linearly stretched cylinder velocity is ${U_w}(x) = cx$. The constant strength ${B_0}$ of the magnetic field is inserted in a radial direction with inclination $\psi$. Moreover, motile gyrotactic microorganisms are included in the flow, and their movement is caused by bioconvective phenomena. Convective-type boundary conditions are imposed at the cylinder surface, also known as Newton boundary conditions. The fluid motion is initiated in the axial direction of the stretching cylinder. The geometry and coordinate system of the model is labeled as in Figure 1.

The following equation represents the rheological relationship between the Cauchy stress tensor and an isotropic, incompressible flow of a Casson fluid ^[30]:

where $\tilde \tau$ is shear stress, $\tilde \alpha$ is shear rate, ${p_r}$ is a yield stress of fluid, ${\tilde \tau _0}$ is a Casson yield stress, and ${\mu _B}$ is a plastic dynamic viscosity.

2.1.   Mathematical formulation of the problem

The governing mass, momentum, heat, and motile microorganism equations within the boundary layer are described as follows ^[31,32]:

The following boundary conditions are suitable:

In Eq (6), the term $K{r^2}{\left({T/{T_\infty }} \right)^{{n_o}}}exp\left({ - {E^ * }/{k^ * }T} \right)$ indicates a modified Arrhenius equation in which ${n_0}$ is the fitted rate constant, taking values $- 1 < {n_0} < 1$, and ${k^ * } = 8.61 \times {10^{ - 5}}eV/K$ is the Boltzman constant.

2.2.   Similarity transformations

The process to convert the PDEs into ODEs involves the utilization of a similarity transformation ^[31] in the following manner:

here, prime signifies differentiation with respect to $\eta$. The continuity equation in Eq (3) is fulfilled automatically, while Eqs (4)–(8) are simplified by using the similarity transformation given in Eq (9) as follows:

And the transform boundary conditions are expressed as:

Where

${\mu _{n\, f}}$, ${\sigma _{n\, f}}$, ${\rho _{n\, f}}$, ${k_{n\, f}}$, and ${\left({\rho {c_p}} \right)_{n\, f}}$ are the dynamic viscosity, electrical conductivity, density, thermal conductivity, and electrical conductivity, respectively. The thermophysical properties are presented in Table 1.

Numerous thermophysical correlations for nanofluids are given in Table 2, used to calculate the ${\rho _{n\, f}}$, ${\mu _{n\, f}}$, ${\left({\rho {c_p}} \right)_{n\, f}}$, ${k_{n\, f}}$, and ${\sigma _{n\, f}}$ for nanofluids. The solid volume fraction of nanofluids is represented by ${\phi _{\, 1}}$. The dimensionless quantities involved in nondimensional equations and boundary conditions are displayed in Table 3.

2.3.   Engineering quantities

For engineering, the skin friction coefficient ${C_{{f_x}}}$, density number ${D_{{n_{\, x}}}}$, Nusselt number $N{u_x}$, and Sherwood number $S{h_x}$ are crucial physical variables and can be measured by the surface shear stress, heat/mass transfer effects, and microorganism diffusion of the system, respectively. These are written as follows:

where ${\tau _w}$, ${q_w}$, $\, {i_w}$, and ${j_w}$ are expressed as follows:

using Eq (9), the dimensionless form of these quantities is written in Eq (17).

here, ${{Re} _x} = \frac{{{U_w}\, x}}{{{\nu _f}}}$ denotes the local Reynolds number.

3.   Numerical technique

Several techniques exist for solving equations in fluid flow problems ^{[34,35,36,37,38,39,40]}; here, the shooting method was selected because it is easy to use and effective at turning boundary-value problems into initial-value problems. For nonlinear differential equations that are frequently encountered in fluid mechanics, this method guarantees greater accuracy and quicker convergence. Eqs (10)−(14) are difficult to solve analytically since they are highly nonlinear ODEs. Because of this, we move toward numerical solutions using the bvp4c function in MATLAB software. The three-stage Lobatto IIIa finite difference technique is executed by Bvp4c with accuracy up to the fourth order. First, highly nonlinear ODEs are transformed into a system of first-order differential equations with the support of appropriate variables, as given in Eq (19).

by inserting Eq (19) into Eqs (10)−(14), we get

and

It is important to note that the mesh size $\Delta \eta = 0.01$ and the step size $h = 0.01$ are frequently used interchangeably in the shooting method depending on how frequently the solution is calculated throughout the domain. All numerical outcomes are obtained with an ${10^{ - 6}}$ error tolerance in this problem.

4.   Code validation

The numerical results are compared under certain conditions with results from the literature with excellent agreement (Table 4). Outcomes agree with those of Khan and Pop ^[41], Ali et al. ^[42], and Ahmad et al. ^[43]. Excellent associations in the findings ensure the correctness of our numerical code and solutions.

5.   Results and discussion

In this study, three distinct forms of nanofluids ($F{e_3}{O_4} - C{H_3}OH$, $Cu - C{H_3}OH$, $Ag - C{H_3}OH$) have been considered for 2D steady laminar Casson fluid flow above a stretching cylinder with inclined MHD and motile microorganisms. The impact of physical parameters on the dimensionless velocity $f'\left(\eta \right)$, concentration $\phi \left(\eta \right)$, temperature $\theta \left(\eta \right)$, and motile microorganisms $g\left(\eta \right)$ distributions is displayed in Figures 2–19, respectively for the magnetic parameter $M$, the Casson fluid parameter $\beta$, the permeability parameter ${P_0}$, the magnetic field inclination $\psi$, the Darcy–Forchheimer number ${D_f}$, the Eckert number $Ec$, the parameter of chemical reaction $\hat \sigma$, the activation energy parameter $Ea$, the bioconvective Peclet number ${P_e}$, the bioconvective Schmidt number ${S_b}$, the microorganism parameter $\Omega$, the thermal Biot number $Bi$, the diffusion Biot number $Bj$, the suction parameter ${\alpha ^ * }$, and the Schmidt number $Sc$. The fixed default values for fluid flow used in the simulation are as follows: $\gamma = 0.25, \, {\phi _{\, 1}} = 0.05, \, \psi = \pi /4, \, {D_f} = {P_0} = 0.5, Pe = \Omega = 0.2, Ec = \,$$\, \delta = {n_0} = Ea = 0.5, \, Sc = \hat \sigma = 1, \,$$Bi = 0.3,$ and ${B_j} = 0.3.$ The range of the flow parameters is carried within a convergence band of the adopted numerical scheme.

Figure 2 shows the increasing impact of the Casson fluid parameter $\beta$ on $f'\left(\eta \right)$. The fluids yield stress and the Casson fluid parameter are inversely correlated. A lower yield stress, which indicates that less force is needed to start the flow, is correlated with higher Casson parameter values. The fluid can move more easily as a result of the decreased flow resistance, which raises velocity.

Figure 3 presents the decreasing function between the magnetic parameter $M$ and $f'\left(\eta \right)$. The large values of the magnetic field parameter result in a decrease in fluid velocity because of the Lorentz force acting on the Casson fluid. An electrically conducting fluid experiences resistance to motion when a magnetic field is applied to it. This resistance is known as the Lorentz force.

Figure 4 reflects the impact of the magnetic field inclination $\psi$ on dimensionless $f'\left(\eta \right)$. The strength of applied magnetic field increases by increasing $\psi$. Therefore, $f'\left(\eta \right)$ declines for rising values of $\psi$; when the angle increases, the magnetic field force improves, and thus velocity drops.

The impact of the permeability parameter ${P_0}$ on $f'\left(\eta \right)$ is exhibited in Figure 5. An inverse relation for velocity toward ${P_0}$ is found. The resistance offered by the porous medium is responsible for the decrease in fluid velocity that occurs when the permeability parameter increases. Higher values of the permeability parameter signify a more restrictive porous medium, being inversely correlated with the resistance of the porous structure.

Figure 6 portrays the decreasing behavior of $f'\left(\eta \right)$ for rising estimates of Darcy–Forchheimer ${D_f}$. Physically inertial forces take over and produce a stronger opposing drag that prevents the fluid motion as the Darcy–Forchheimer parameter rises. This resistance leads to a noticeable velocity decrease in addition to lowering the momentum of flow. The larger Forchheimer parameter thus introduces stronger inertial drag forces, which are directly responsible for the decrease in fluid velocity.

Increasing values of the suction parameter ${\alpha ^ * }$ reduce $f'\left(\eta \right)$ (Figure 7). Physically, suction controls the flow of nanofluid, reducing drag on the nanoparticles in an external flow, causing a reduction in velocity. The improved removal of fluid particles from the boundary layer by suction causes a decrease in fluid velocity as the suction parameter increases. By physically drawing fluid away from the flow close to the surface suction, the total momentum in the boundary layer is lowered.

The nanofluid ($F{e_3}{O_4} - C{H_3}OH$) shows higher velocity than the other two nanofluids ($Cu - C{H_3}OH$, $Ag - C{H_3}OH$) in response to parameters $M$, $\psi$, ${P_0}$, ${D_f}$, and ${\alpha ^ * }$, while the opposite is observed for $\beta.$ The effect of several parameters on the temperature profile $\theta \left(\eta \right)$ for the three nanofluids is shown in Figures 8–12.

The impact of the Casson fluid parameter $\beta$ on the temperature profile $\theta \left(\eta \right)$ is clearly shown in Figure 8. The temperature profile decreases with increasing $\beta$. Larger values of the Casson fluid parameter result in a drop in temperature because 𝛽 is inversely proportional to the fluid yield stress. When fluid resistance to deformation decreases, there is a smoother flow with less energy dissipation in response to an increase in 𝛽.

Figure 9 shows the impact of the magnetic parameter $M$ on the temperature profile $\theta \left(\eta \right)$. The fluid generates more Joule heating when the magnetic field parameter increases; hence, the temperature rises. A magnetic field interaction with an electrically conducting fluid results in an induced Lorentz force that prevents the fluid from moving; this produces heat through electrical resistance.

Figure 10 illustrates the influence of a magnetic field inclination $\psi$ on the temperature profile $\theta \left(\eta \right)$. Higher $\psi$ increases $\theta \left(\eta \right)$.

The influence of the Eckert number $Ec$ on temperature distribution $\theta \left(\eta \right)$ is displayed in Figure 11. The work done against a viscous fluid is often expressed as the Eckert number, a dissipation parameter. As $Ec$ increases, the fluid temperature also increases. This phenomenon involves converting kinetic energy into internal energy through the work done against the forces within viscous fluids. Here, $Ec$ depends upon the kinetic energy; as the kinetic energy increases, so will the $Ec$.

Figure 12 shows the interaction between the temperature of the nanofluids and the thermal Biot number $Bi$. The fluids' temperature close to the surface increases as Bi increases because of the improved efficiency of the heat exchange between the fluid and surface. This results in a thicker thermal boundary layer and a higher temperature profile across the convective heat transfer–affected region.

From Figures 8−12, it is clear that the temperature is higher for the nanofluid ($Ag - C{H_3}OH$) than the other two nanofluids ($Cu - C{H_3}OH$, $F{e_3}{O_4} - C{H_3}OH$).

The response of concentration distribution to several manipulating parameters is given in Figures 13–16. Figure 13 shows the influence of the Schmidt number $Sc$ on the concentration profile $\phi \left(\eta \right)$ of nanofluids. The Schmidt number is the ratio of momentum to mass diffusivity; it $\phi \left(\eta \right)$ decreases as a function of $Sc$ since higher $Sc$ values correspond to lower mass diffusion and, as a consequence, a reduced concentration of the boundary layer.

The influence of the chemical reaction parameter $\hat \sigma$ on the concentration gradient $\phi \left(\eta \right)$ is demonstrated in Figure 14. $\phi \left(\eta \right)$ decreases with increasing $\hat \sigma$; it consumes the chemical species in the fluid, and the chemical reaction lowers the concentration. Species depletion becomes more pronounced with the increase in the reaction rate, which lowers the concentration of fluid.

Figure 15 shows the variation of the concentration profile $\phi \left(\eta \right)$ in response to the activation energy parameter $Ea$. Increasing values of $Ea$ lead to increased $\phi \left(\eta \right)$. Physically, the concentration profile will show an upward trend if a significant amount of energy is necessary to start the chemical reaction.

Figure 16 illustrates the influence of the Biot number $Bj$ (sway of diffusion) on the concentration profile $\phi \left(\eta \right)$. $\phi \left(\eta \right)$ increases with increasing values of $Bj$. According to the definition of the Biot number, the transmitted mass will be dispersed throughout the surface by convection and thus, concentration distribution increases.

There is a higher influence of the relevant parameters on the mass distribution of the nanofluid $Ag - C{H_3}OH$ than of the other two nanofluids $Cu/F{e_3}{O_4} - C{H_3}OH$.

The variation in density of motile microorganisms $g\left(\eta \right)$ in response to several parameters and nanofluids is shown in Figures 17–19. Figure 17 shows the effect of the bioconvective Schmidt number ${S_b}$ on the density profile of motile microorganisms $g\left(\eta \right)$. For higher values of ${S_b}$, the impact of this parameter on the rescaled $g\left(\eta \right)$ is found to be insignificant. The variation of bioconvection Schmidt numbers tend to lower the boundary layer thickness. Figure 18 portrays the impact of bioconvective Peclet number ${P_e}$ on the density distribution of motile microorganisms $g\left(\eta \right)$. The Peclet number is a key parameter to recognize microorganisms that are swimming in a fluid. This number represents the ratio of the product of the maximum cell swimming speed and the chemotaxis constant to the diffusion coefficient of the microbes. Diffusion is the process by which the material travels from an area of high concentration to an area of low concentration, describing the movement of substances in the fluid. It has been discovered that an increase in ${P_e}$ leads to an increase in the speed of fluid particles and a decrease in the diffusivity of microbes, thus reducing $g\left(\eta \right)$.

Figure 19 shows the effect of the microorganism parameter $\Omega$ on the density profile $g\left(\eta \right)$. Increasing $\Omega$ results in a decrease in the concentration of microorganisms in ambient fluid. The $Ag$ nanoparticle shows a higher motile microorganisms profile than the $Cu$ and $F{e_3}{O_4}$ nanofluids.

The dimensionless skin friction coefficient $0.5\, {C_{{f_{\, x}}}}\sqrt {{{{Re} }_x}}$, Nusselt number $N{u_x}/\sqrt {{{{Re} }_x}}$, density number ${D_n}_{_x}/\sqrt {{{{Re} }_x}}$, and Sherwood number $S{h_x}/\sqrt {{{{Re} }_x}}$ versus several parameters are calculated and presented in Tables 5–8. Table 5 highlights the impact of $\gamma$, $\beta$, $M$, ${P_0}$, $\psi$, ${D_f}$, and ${\alpha ^ * }$ on the coefficient of skin friction. The coefficient increases by increased $\gamma$, $M$, $\psi$, ${P_0}$, and ${D_f}$; an opposite trend is observed for $\beta$ and ${\alpha ^ * }$. Moreover, the $Ag - C{H_3}OH$ nanofluid has higher $0.5\, {C_{{f_{\, x}}}}\sqrt {{{{Re} }_x}}$ than $Cu - C{H_3}OH$ and $F{e_3}{O_4} - C{H_3}OH$ nanofluids. Table 6 depicts the influence of various physical parameters on the Nusselt number. The heat transfer rate (Nusselt number) decreases with increasing $\gamma$, $\psi$, $M$, and $Ec$; however, it increases with increasing $\beta$ and $Bi$. This is a significant finding, as a higher Nusselt number would ultimately indicate a larger heat transfer from the surface to its surroundings, which aids to cool the system. Finally, the heat transfer rate for the $F{e_3}{O_4} - C{H_3}OH$ nanofluid is higher than for the $Cu - C{H_3}OH$ and $Ag - C{H_3}OH$ nanofluids.

Table 7 represents the impact of involved parameters on mass transfer rate (Sherwood number). A direct relation is observed with $\gamma, \, \beta, \, Sc, \, \hat \sigma$ and $Bj$, while a reverse relation is found for $M$ and $Ea$. Moreover, the mass transfer rate is smaller for $Cu - C{H_3}OH$ and $Ag - C{H_3}OH$ than $F{e_3}{O_4} - C{H_3}OH$. The impact of important parameters on the mass rate of microorganisms (density number) is presented in Table 8. Density increases with increased $\gamma, \, \beta, \, {S_b}, \, {P_e}$, and $\Omega$ but decreases with increased $M$. Moreover, the mass rate of microorganisms for the $F{e_3}{O_4} - C{H_3}OH$ nanofluid is higher than for $Cu - C{H_3}OH$ and $Ag - C{H_3}OH$ nanofluids.

6.   Artificial neural networking (ANN) model

The solution of nonlinear partial differential equations (PDEs), which are computationally demanding, is frequently required for fluid flow problems. Frequently, the flow field is challenging to describe for a wide range of parameters because of singularities appearing in the solution scheme. Therefore, ANN can help us to predict the quantity of interest by learning and approximating these nonlinear systems. The mathematical modeling of Casson fluid flow over a horizontally stretched cylinder is studied here, and the resulting flow equations are solved using numerical methods. The skin friction coefficient (SFC) is computed using three distinct nanoparticles: $F{e_3}{O_4} - C{H_3}OH$, $Cu - C{H_3}OH$, and $Ag - C{H_3}OH$. In each scenario, ANN models are established to forecast the SFC values on the cylinder.

Researchers believe that the combination of the artificial neural networking models ^[44,45,46] and the multilayer perceptron (MLP) has outstanding learning capabilities, making it a popular choice for predicting various physical circumstances. The MLP network is composed of three separate layers. The first layer is responsible for accepting inputs; the hidden layer is centrally involved, and the last layer, known as the output layer, stores predicted data. Figure 20 illustrates the structural arrangement of this ANN model. Three separate ANN models were created and designated as ANN Model-Ⅰ, ANN Model-Ⅱ, and ANN Model-Ⅲ to forecast the SFC for the three nanofluids $F{e_3}{O_4} - C{H_3}OH$, $Cu - C{H_3}OH$, and $Ag - C{H_3}OH$, respectively. These ANN models utilize seven distinct flow parameters as inputs: Curvature parameter, Casson, suction, magnetic field inclination, permeability parameter, and Darcy–Forchheimer number.

Table 9 contains the symbolic details corresponding to the three different models. The desired output for all these models is the SFC. To train these models effectively, we have assembled a dataset comprising 70 different samples, each associated with its corresponding SFC value. The dataset is divided into 70%, 15%, and 15% for training, validation, and testing purposes, respectively. Complete information regarding this matter is demonstrated in Table 10.

The impact of seven flow parameters on SFC for all three nanofluids is presented in Tables 11–17. The architecture of these models incorporates a hidden layer housing ten neurons. The training network is facilitated by adopting the Levenberg–Marquardt (LM) method. Compared to traditional gradient descent algorithms, the LM algorithm converges more rapidly especially when dealing with small-to-medium-sized datasets. In fluid dynamics where precise profile and coefficient predictions are essential, LM is particularly beneficial. The Purelin and Tan-Sig transfer functions used in the hidden and output layers are mathematically defined as:

To calculate the effectiveness of the ANN models for predicting the SFC on a horizontal cylinder within three distinct nanofluids, we have considered specific performance indicators. These indicators include the mean squared error (MSE) and the coefficient of determination (R). The computation of these performance metrics involves the utilization of mathematical formulations, which can be expressed as:

The training process is a crucial stage in the development of ANN models. To perfect network training, the Levenberg–Marquardt algorithm was used. Within multi-layer perceptron (MLP) network structures, the training iteration continues until the error between target and prediction data generated at the output layer reaches a minimum level.

As the training advances through each epoch, the mean squared error (MSE) values, which were initially large, gradually drop, indicating that the network's performance is being refined. Figure 21(a)–(c) presents the training performances across all three ANN models. Notable observations occur throughout the training cycle (epoch) of an MLP network. The best validation performance for ANN model-Ⅰ is 0.0041131 at epoch 5. Similarly, the best validation performance for ANN model-Ⅱ is obtained at 0.013587 during epoch 4. Correspondingly, for ANN model-Ⅲ, the most favorable validation performance occurs at 0.012446 in epoch 6. Across all graphs, a consistent trend is apparent: the initial MSE values are relatively high, gradually decreasing as the training advances toward more refined stages.

The examination of error histograms is an additional stage for evaluating the training efficacy of ANN models. Throughout the training process, these histograms visually display the differences between predicted values and associated target values. In this regard, Figure 22(a)–(c) depicts the error histograms for the constructed ANN models intended to predict SFC values. Figure 22(a) provides the error histogram for ANN model-Ⅰ regarding the $F{e_3}{O_4} - C{H_3}OH$ nanofluid on a horizontal cylinder. Figure 22(b) shows the error histogram for the ANN model-Ⅱ, focusing on the $Cu - C{H_3}OH$ nanofluid on a horizontal cylinder. Figure 22(c) displays the error histogram for the ANN model-Ⅲ regarding the $Ag - C{H_3}OH$ nanofluid on a horizontal cylinder. Each of these histograms was divided into 20 bins. A close examination shows that the errors obtained from each ANN model tend to cluster around the zero-error line. Furthermore, the magnitudes of these differences are generally small, which is an important point.

Figure 23(a)–(c) exhibits the regression values corresponding to the SFC data across all three ANN models. It is essential to understand that the regression value represents the correlation between the predicted and target values; a regression value close to 1 suggests a strong correlation. For ANN model-Ⅰ, the regression values were determined as R = 0.99698 for training, R = 0.99649 for validation, and R = 0.99362 for testing. The cumulative regression value for this model was computed as R = 0.99559 (Figure 23(a)). Regarding the ANN model-Ⅱ, the regression values were R = 0.99867 (training), R = 0.99434 (validation), and R = 0.9998 (testing), resulting in an overall regression value of R = 0.99764 (Figure 23(b)). For the ANN model-Ⅲ, the regression values were found to be R = 0.99822 (training), R = 0.98961 (validation), and R = 0.99598 (testing), resulting in an overall regression value of R = 0.99612 (Figure 23(c)). Based on the comprehensive statistical analysis presented in Figure 23(a)–(c), it is reasonable to conclude that the formulated ANN models are effective at predicting the SFC on a horizontal cylinder.

Figure 24(a)–(c) shows the mean square error (MSE) for each of the 70 data samples for the SFC predictions at the cylinder surface for the ANN Model-Ⅰ, ANN Model-Ⅱ, and ANN Model-Ⅲ, respectively. These figures collectively indicate the successful completion of the learning stages for the three models. A smaller MSE value, approaching zero, indicates fewer errors during the training of the ANN models to forecast SFC values. The average MSE is 0.000334 for ANN Model-Ⅰ, 0.012691 for ANN Model-Ⅱ, and 0.001038 for ANN Model-Ⅲ, all of which being notably low.

As a result, the training process of the neural network models is considered highly effective in accurately predicting SFC values at the horizontal cylinder for all three types of nanoparticles. Figure 25(a)–(c) shows a comprehensive comparison between the actual and predicted SFC values for each ANN model: ANN model-Ⅰ, ANN model-Ⅱ, and ANN model-Ⅲ, respectively. It is clear that the ANN model outputs closely align with the SFC target values. With such an overlap, we can conclude that the three constructed models can accurately predict SFC.

Table 18 shows the performance information needed to compute the MSE and R values across each of the ANN models during the training, validation, and testing stages. It is evident that the MSE values are minimal, and the R-values closely approach 1. This underscores the capability of the constructed models to predict SFC values on the surface of a horizontal cylinder with high accuracy.

7.   Final remarks

The objective of the current paper is to explore inclined magneto-Casson fluid flow over a stretching cylinder consisting of Darcy–Forchheimer medium and gyrotactic microorganisms. Additionally, the impact of joule heating, activation energy, viscous dissipation, binary chemical reaction rate, and convective boundary conditions are considered. The nanoparticles used in this research are $F{e_3}{O_4}$, $Cu$, and $Ag$ with $C{H_3}OH$ as the base fluid. These nanofluids are compared based on relevant parameters, and quantities are simulated numerically using tabular and graphical ways. The skin friction coefficient (SFC) values are calculated for all three types of nanoparticles using artificial neural network models along the cylinder's surface. The main outcomes are summarized as follows:

o Velocity decreases with magnetic, permeability, inclination, Darcy–Forchheimer number, and suction parameters and increases with the Casson fluid parameter.

o Temperature increases with magnetic field inclination, magnetic parameter, Eckert number, and thermal Biot number but decreases with the Casson fluid parameter.

o The concentration is enhanced by increasing diffusion Biot number and activation energy parameter and decreases with rising chemical reaction parameter and Schmidt number.

o The density distribution of motile microorganisms decreases with larger values of bioconvective Schmidt number, bioconvective Peclet number, and microorganism parameter.

o The skin friction coefficient increases with curvature parameter, magnetic parameter, permeability parameter, magnetic field inclination, and Darcy–Forchheimer number; an opposite behavior is observed with Casson fluid and suction parameters.

o The heat transfer rate decreases with the curvature parameter, magnetic parameter, Eckert number, and magnetic field inclination; an opposite influence can be noted with the Casson fluid parameter.

o The thermal Biot number has a positive impact on the heat transfer rate; the diffusion Biot number has a positive impact on mass transfer rate.

o The silver nanofluid has a dominant effect on the coefficient of skin friction, temperature, concentration, and motile microorganisms profiles; the ferrous ferric oxide nanofluid has a dominant impact on density number, Sherwood number, Nusselt number, and velocity profiles.

o The MSE values for all ANN Models Ⅰ, Ⅱ, and Ⅲ are 0.000334, 0.012691, and 0.001038, respectively, indicating that these possess the capability to accurately predict the SFC.

o This idea can be extended to more extensive interactions between nanofluids and microorganisms by refining flow systems inspired by biotechnology, verifying computational models using experimental data, and applying results to real-world biotechnology and engineering problems.

Author contributions

K. U. Rehman: Writing – original draft, validation, investigation, formal analysis; N. Fatima: Writing – review & editing, visualization, validation, investigation; W. Shatanawi: Visualization, supervision, investigation, conceptualization; N. Kousar: Investigation, data curation, conceptualization. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to thank Prince Sultan University, Saudi Arabia, for the technical support through the TAS research lab.

Conflict of interest

The authors declare no conflict of interest.