## 5 Results and Discussion
Brownian diffusion (\(Nb\)), however, enhances nanoparticle species volume fraction (Figure 7b), and this is related to molecular agitation of the nanoparticles. \(Nb\) quantifies the random motion of small colloidal particles suspended in a fluid, caused by the collision of the fluid molecules with the particles [60]. Figure 7c shows that increasing the Brownian motion parameter weakly increases the micro-organism density number, whereas increasing thermophoresis parameter strongly reduces it. Thicker micro-organism species boundary layers are produced with higher \(Nb\), whereas the converse effect is induced with increasing \(Nt\). The distribution of heat, nanoparticles and micro-organisms in the boundary layer regime can be successfully manipulated with both nanoscale effects, i.e., the Brownian motion mechanism (higher \(Nb\) values) and thermophoresis effect. The influence of Brownian motion on the velocity fields was found to be inconsequential, and these plots are therefore excluded here. Figure 6: Effects \(M\) and \(Me\) on the dimensionless (**a**) velocity, (**b**) temperature, (**c**) nanoparticle volume fraction and (**d**) micro-organism density number function. Figure 7: Effects of \(Nb\) and \(Nt\) on the dimensionless (**a**) temperature, (**b**) nanoparticle volume fraction and (**c**) micro-organism density number function. Figure 8 presents the impact of second-order slip \(\gamma\) and bioconvection Peclet number \(Pe\) on the motile micro-organism density number. Higher \(Pe\) and smaller \(\gamma\) generally result in a decrease in microorganism density number magnitudes. Bioconvection Peclet numbers relate the rate of advection of micro-organisms driven by the flow to the rate of diffusion of micro-organisms under gyrotaxis. This parameter is often used in mathematical models of micro-organism propulsion as an alternative to bioconvection Rayleigh number (\(Ra_{b}\)) and bioconvection Lewis number (\(Le_{b}\)). Ordinary Peclet number is associated with convective heat transfer processes. In bioconvection, the Peclet number when sufficiently high has been shown to significantly change patterns of the motile micro-organism flow. Bioconvection is due to the internal energy of the microorganisms. With greater swimming speed (higher Peclet number), the micro-organisms move faster, which reduces their concentrations. At lower Peclet numbers the effect is reversed, i.e., motility of the micro-organisms is inhibited and move slower. This then leads to higher and more homogenous concentrations. The second-order slip parameter also arises in the velocity wall boundary condition but induces a lesser effect than first-order slip. It weakly decreases motile micro-organism species boundary layer thickness whereas the bioconvection Peclet number generates a strong reduction. Figure 9 exhibits the variation of the local Nusselt number, \(-\theta^{\prime}(0)\) versus temperature dependent viscous parameter, \(h_{2}\) and melting parameter, \(Me\) for a different value of Eckert number, \(Ec\). The results show that \(-\theta^{\prime}(0)\) decreases with increasing of \(h_{2}\), \(Ec\) and \(Me\). Greater nanomaterial viscosity, viscous dissipation and melting heat transfer therefore all _impede_ diffusion of heat from the nanofluid to the sheet surface. Figure 10 depicts the variation of the local Sherwood number, \(-\phi^{\prime}(0)\), versus Schmidt number, \(Sc\), and Brownian motion, \(Nb\), for a different value of thermophoresis parameter, Nt. Results show that \(-\phi^{\prime}(0)\) increases with increasing of \(Nt\), but decreases with \(Nb\). Nanoparticles species diffusion to the sheet (wall) is therefore stifled with greater Brownian motion whereas it is encouraged with stronger thermophoretic body force effect. The values of Schmidt number are selected between 1 and 1.4, and these correspond to common metallic nanoparticles (e.g., titanium, copper, silver) diffusing in water as the base fluid [15].