**3. Numerical Solution and Code Validation**

Using MAPLE 22, Equations (8)–(11) subject to boundary conditions (12) were solved numerically. In order to solve boundary value issues numerically, this software, by default, employs a four-fifths order Runge–Kutta–Fehlberg approach. Its reliability and precision have been repeatedly demonstrated in numerous heat transfer articles. The unity coefficient of the term was changed to a continuation (101 − 100*λ*), or (10 − 9*λ*) was used in the dsolve command in order to speed convergence for all values of the governing parameters selected for this investigation. Without making this adjustment, MAPLE produces results that do not conform to the asymptotic values, but produces results that have a sharp angle at which

the axis intersects. The numerical solution to challenging ODE boundary value problems in Maple's help section contains more details on resolving the convergence challenges. Using a value of 8 for the similarity variable *η*max, the asymptotic boundary conditions from Equation (12a,b) were substituted as follows.

$$F'(8) = \varepsilon, \theta(8) = 0, \phi(8) = 0, \chi(8) = 0 \tag{16}$$

The selection of *η*max = 8 guaranteed that all numerical solutions appropriately approximated the asymptotic values. This is a crucial feature that is frequently missed in the literature on boundary layer fluxes.

To authenticate the model's validity, we have compared the skin friction values for several values *ε* in Tables 1–3, while Table 4 compares the heat transfer values with the existing literature. It exhibits good agreement for various parameters, indicating that our numerical solution is valid.


**Table 1.** Comparison of numerical values of *F*(0) at *λ* = 0, *n* = 1.

**Table 2.** Comparison of *F*(0) for various values of *Pr* at *n* = 1 and *λ* = *Nr* = 0.


**Table 3.** Comparison of *F*(0) for various values of *n* and *a/c* at *n* = 1 and *λ* = *Nr* = 0.


**Table 4.** Comparison of numerical values of −*θ* (0) at *λ* = *ε* = *Nb* = *Nt* = 0, *n* = 1.


## **4. Results and Discussion**

In this study, the mixed bioconvective stagnation-point flow of a power-law nanofluid over a stretchy sheet was computationally studied using the Runge–Kutta–Fehlberg method of the seventh order (RKF7) in conjunction with the shooting method. The influence of gyrotactic microorganisms at the surface is taken into account. Figure 2a shows the effects of the bioconvection Rayleigh number and buoyancy parameter on the dimensionless velocity, with all other parameters held constant. We notice that the dimensionless velocity increases significantly in the vicinity of the surface and then drops to the boundary layer edge with the Rayleigh number Rb. The dimensionless velocity overshoots at the region of the surface owing to the existence of buoyant forces. The Rayleigh number increases the buoyancy forces because of bioconvection, increasing the dimensionless velocity. The effects of mixed convection and velocity ratio parameters on the dimensionless velocity are presented in Figure 2b. The convergence rate depends upon the velocity ratio parameter. As the velocity ratio increases, the convergence rate increases. Within the hydrodynamic boundary layer, the mixed convection parameter also plays an important role. When the mixed convection parameter is increased, the dimensionless velocity increases.

**Figure 2.** Variation of dimensionless velocity with (**a**) the buoyancy ratio parameter and bioconvection Rayleigh number and (**b**) the dimensionless mixed convection parameter and velocity ratio parameter.

The impacts of nanofluid parameters *Nb* and *Nt* are explained in Figure 3a. The Brownian motion parameter *Nb* keeps particles moving in a fluid. This keeps particles from settling, resulting in colloidal solutions that are more stable. It helps in enhancing the dimensionless velocity within the boundary layer. On the other side, the thermophoresis parameter generates a force due to temperature difference. Nanoparticles are transported towards the lower temperature zone by this force.

Consequently, the dimensionless velocity increases inside the boundary layer. In heat transfer, the bioconvection Schmidt number is equivalent to the Prandtl number. With a Schmidt number of one, momentum and mass transfer by diffusion are alike, and the velocity and concentration boundary layers are almost identical. An increasing bioconvection Schmidt number reduces the dimensionless velocity and hence the hydrodynamic boundary layer thickness, as shown in Figure 3b.

The effects of the thermophoresis parameter on the dimensionless temperature are presented in Figure 4 for several fluids. Thermophoresis is more important in a mixed convection process, where the flow is generated by the buoyancy force caused by a temperature differential. The nanoparticles move in the direction of a temperature drop, and decreasing the bulk density improves the heat transfer process. It is worth noting that nanoparticles transport thermal energy from high-temperature areas to lower-temperature areas. As a result, the thickness of the thermal boundary layer thickens as the thermophoresis parameter *Nt* increases. This fact is explained in Figure 4. The Prandtl number determines the

thickness of the thermal boundary layer. As the Prandtl number increases, the momentum diffusivity dominates the behavior and, as a result, the thickness of the thermal boundary layer decreases.

**Figure 3.** Variation of dimensionless velocity with (**a**) nanofluid parameters and (**b**) bioconvection Schmidt number.

**Figure 4.** Variation in dimensionless temperature with the thermophoresis parameter for different fluids.

Figure 5a demonstrates the influence of nanofluid parameters on the dimensionless concentration at different thermophoresis parameters *Nt* = 0.4 and *Nt* = 0.5. The thermophoresis and Brownian processes cause the nanoparticle distribution to become nonuniform throughout the domain, as shown in Figure 5a. When the particle concentration is low and the Rayleigh number is low, the distribution of nanoparticles is more uniform.

Nanoparticle concentration rises when *Nt* increases because the thermophoresis parameter represents the movement of particles due to the temperature gradient.

**Figure 5.** Variation in the dimensionless nanofluid concentration with (**a**) nanofluid parameters and (**b**) Schmidt number for different fluids.

Figure 5b depicts the influence of the Schmidt number Sc on the dimensionless concentration for several fluids. The dimensionless concentration overshoots near the surface and drops to zero, as observed. The reason is that the dimensionless velocity rises towards the surface before falling to zero. It is worth noting that, in the surface region, the dimensionless concentration rises with the Schmidt number. This is because the mass diffusivity diminishes, resulting in a lower concentration of nanoparticles. As a result, the concentration boundary layer thickness decreases as the Schmidt number Sc distance from the cone surface increases and increases as the Schmidt number *Nt* decreases. However, in the passively controlled model, it is assumed that there is no nanoparticle flux at the plate and that its particle fraction value adjusts accordingly. Thus, the passively controlled nanofluid model [26] can be used in practical applications. A numerical survey is then performed for all four profiles embodying the velocity, temperature, nanoparticle volumetric fraction, and density of motile microorganisms.

The effects of the bioconvection Schmidt number for different values of Peclet number are presented in Figure 6a. The large values of the Peclet number suggest an advectively dominated distribution, while a smaller value indicates a dispersed flow. As the Peclet number increases, the boundary layer thickness of motile microorganisms increases. Conversely, the bioconvection Schmidt number tends to suppress the boundary layer thickness. It is observed that the dimensionless motile microorganism density decreases with the bioconvection Schmidt number, but increases with the Peclet number. This can be attributed to the substantial decrease in mass diffusivity, which generates a lower concentration. Figure 6b illustrates the influence of the bioconvection constant for different values of the Brownian motion parameter. Nanoparticles are not self-propelled in a nanofluid. Brownian motion and the thermophoresis effect cause them to move. Motile microorganisms are mixed with a dilute suspension of nanoparticles to increase mass transfer and microscale mixing, as well as nanofluid stability in the flow. For this reason, the Brownian motion parameter tends to decrease the dimensionless microorganisms while the bioconvection constant tends to increase the microorganism's density.

**Figure 6.** Variation of dimensionless motile micro-organism density with (**a**) the bioconvection Schmidt number with different Peclet numbers and (**b**) the bioconvection constant and Brownian motion parameter.

For several values of the bioconvection Rayleigh number and dimensionless mixed convection parameter, the variation in skin friction with the buoyancy ratio parameter is shown in Figure 7a. The skin friction is observed to rise with the bioconvection Rayleigh number and dimensionless mixed convection parameter, but decreases slightly with the buoyancy parameter. The skin friction is exceptionally high in the absence of the buoyancy effect and subsequently tends to decrease along with the buoyancy parameter. However, compared with buoyancy force convection, increasing the bioconvection Rayleigh number and dimensionless mixed convection parameter enhanced the convection heat. These results are anticipated; in the interim, heat is produced as a result of increased skin friction, resulting in the formation of a layer of hot fluid close to the surface. It is interesting to note that, as long as skin friction values are positive, we can see that drag force is imparted to the surface via the fluid. Figure 7b explains the effects of nanofluid parameters on the dimensionless heat transfer for different fluids. It is important to note that the Nusselt number decreases with both nanofluid parameters, but increases with the Prandtl number. It is commonly known that the Nusselt number represents the ratio of convective to conductive heat transfer; as a result, with more significant Nusselt numbers, heat convection predominates and the Nusselt number decreases as *Nb* and *Nt* increase.

The ratio of momentum diffusivity to mass diffusivity is known as the Schmidt number. In heat transmission, this is like the Prandtl number. When there is simultaneous momentum and mass transmission, this is used to characterize flows. The Sherwood number measures the efficacy of mass transfer at the surface. The variation in the Sherwood number with the Schmidt number is depicted in Figure 8a for different values of the thermophoresis parameter and Brownian number. It is perceived that the Sherwood number increases significantly with the Schmidt number and Brownian motion parameter. However, the thermophoresis parameter tends to reduce the Sherwood number.

The variation in the dimensionless local density number of the motile microorganisms with bioconvection parameters is depicted in Figure 8b. The motile microorganism mass transfer rate is significantly increased when the bioconvection Schmidt number increases. The bioconvection Peclet number and bioconvection constant reduce the rate of motile microorganism mass transfer. This is because Pe is directly related to *Wc* (maximum cell

swimming speed) and inversely proportional to *Dm* (the diffusivity of microorganisms). As a result, larger Pe values diminish microorganism speed and reduce microorganism diffusivity. This will result in lower microorganism concentrations in the border layer and a higher rate of motile micro-organism mass transfer.

**Figure 7.** Variation in skin friction with (**a**) buoyancy ratio parameters for different values of bioconvection Rayleigh number and dimensionless mixed convection parameter and (**b**) variation in the Nusselt number with the Prandtl number for different values of nanofluid parameters.

**Figure 8.** (**a**) Variation in the Sherwood number with the Schmidt number for different nanofluid parameters and (**b**) variation in the density number of the motile microorganisms.

#### **5. Conclusions**

The mixed bioconvective flow of water-based nanofluid containing micro-organisms is investigated numerically using a Runge-Kutta–Fehlberg method of the seventh order (RKF7) coupled with a shooting method. The effects of bioconvection and nanofluid parameters on the dimensionless variables and quantities of interest are investigated numerically. The results are presented graphically and are compared with the existing data. In the RKF7 method, the following are the primary conclusions that can be deduced from the study:


**Author Contributions:** All authors contributed equally. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** Data are available upon request.

**Conflicts of Interest:** The authors declare no conflict of interest.
