## 5 Results and Discussion  
Selected computations based on the BVP4C code are presented in Figures 3-10. Figure 3 illustrates the collective impact of the temperature-dependent thermal conductive parameter \(h_{4}\) and first-order slip parameter \(\delta\). The former parameter \(h_{4}\), based on the original definition in Equation (7) arises in two key terms in the energy conservation Equation (11), via in the augmented thermal diffusion term, \([1+h_{4}\theta]\theta^{\alpha}\), and the thermophoretic term, \(\left(h_{4}+Nt\right)\left(\theta^{\prime}\right)^{2}\). It also features in the melting interface wall boundary condition (14). It therefore plays a prominent role in heat diffusion in the regime. The latter parameter \(\delta\), only features in the wall velocity boundary condition. A strong decrease in temperature is associated with an increase in both \(h_{4}\) and \(\delta\). Hydrodynamic slip increases momentum diffusion near the wall, and this impedes thermal diffusion leading to a reduction in thermal boundary layer thickness. The non-inclusion of slip effects will clearly lead to erroneous computations which will diverge from results of relevance to practical applications. With increasing thermal conductivity there will be a faster diffusion of heat from the nanofluid to the wall which will effectively reduce temperatures in the medium. In other words, greater thermal conductivity will serve to cool the medium more efficiently. Asymptotically smooth solutions are achieved in the free stream indicating (as in Figure 2a-d) that enough infinity boundary conditions have been imposed in the MATLAB BVP4C code. Figure 4a,b depict the impact of nanoparticle mass diffusivity parameter, \(h_{6}\), and Schmidt number, \(Sc\), on nanoparticle volume fraction and microorganism density number function. A strong enhancement in both nanoparticle volume fraction and motile microorganism density is induced with  
Figure 3: Effects \(h_{4}\) and \(\delta\) on the dimensionless temperature. greater \(h_{6}\). The converse response is generated with greater Schmidt number. Both nanoparticle species and motile microorganism species boundary layer thicknesses are enhanced with accentuation in mass diffusivity. Increasing Schmidt number, however, corresponds to a decrease in nanoparticle species diffusivity which results in a reduction in both nanoparticle species and motile microorganism species boundary layer thicknesses. Monotonic decays from the wall (sheet surface) to the free stream are observed in both plots, although the trend is more consistent for the micro-organism density function (Figure 4b). Figure 5 visualizes the response in motile microorganism density number with a variation in both microorganism diffusivity parameter, \(h_{8}\), and bioconvection Schmidt number, \(Sb\). A rise in value of \(h_{8}\) clearly shows that micro-organism profile increases with greater \(Sb\) but decreases when the value of \(Sb\) increases. \(Sb=\frac{v_{\infty}}{D_{\infty,\infty}}\) is an important parameter controlling the bioconvection propulsion phenomenon. When \(Sb=1\), both viscous diffusivity and micro-organism diffusivity are equivalent, i.e., viscous and micro-organisms diffuse at the same rate and both velocity (hydrodynamic) and micro-organism boundary layer thicknesses are the same. When \(Sb=10\) clearly the viscous diffusivity exceeds the micro-organism diffusivity by a factor of ten and this results in a substantial reduction in diffusion rate of micro-organisms relative to the viscous diffusion. Micro-organism species concentration boundary layer thickness will therefore be considerably reduced. Figure 6a-d displays the influence of the magnetic body force parameter (\(M\)) and melting parameter (\(Me\)) on velocity, temperature, nanoparticle volume fraction and motile microorganism density number. Figure 4: Effects \(h_{6}\) and \(Sc\) on the dimensionless (**a**) nanoparticle volume fraction and (**b**) micro-organism density number function.