## 5 Results and Discussion
Figure 5: Effects \(h_{8}\) and \(Sb\) on the dimensionless micro-organism density number function. As both magnetic field and melting heat transfer parameter increase, it is evident from Figure 6a, that the flow is accelerated strongly. Although in conventional stationary wall magnetohydrodynamics, the Lorentz body force decelerates flow, the presence of a stretching wall reverses this effect, as observed in various studies including [PERSON] et al. [59]. The motion of the wall drags the magnetic field with the free stream, and this results in flow acceleration. Momentum boundary layer thickness is therefore effectively decreased. The intensification in melting effect also imparts significant momentum to the nanofluid which results in flow acceleration and thinning of the momentum boundary layer. The electrically non-conducting scenario (\(M=0\)) therefore produces deceleration compared with the magnetic case. Conversely, Figure 6b shows that temperatures are reduced with increasing melting parameter, Me. Again, this is associated with the boost in momentum with greater melting effect which is enhanced relative to the heat diffusion rate. Thermal boundary layer thickness is therefore decreased. However, although the magnetic field accelerates the flow, it generates a similar effect to that encountered in stationary wall magnetohydrodynamics, namely an elevation in temperature. This is associated with the dissipation in supplementary work expended in dragging the nanofluid against the action of the magnetic field which manifests as thermal energy. The boundary layer regime is therefore heated, and thermal boundary layer thickness is enhanced. Figure 6c indicates that the nanoparticle species is elevated with increasing melting effect whereas it is strongly suppressed with increasing magnetic body force effect. Nanoparticle species diffusion is therefore assisted with phase change whereas it is impeded with a stronger magnetic field. The implication is that melting encourages better distribution of nanoparticles in the nanomaterial whereas a transverse magnetic field stifles dispersion of nanoparticles leading to less homogenous distributions. Figure 6d shows that micro-organism species concentration (number density function) is affected in a similar fashion to the nanoparticle volume fraction (concentration). It is consistently elevated by larger melting parameter values and reduced with greater magnetic field strengths. Micro-organism species boundary layer thickness is therefore respectively increased and decreased with a rise in melting heat transfer and magnetic body force parameter. Figure 7a-c presents the evolution in temperature, nanoparticle volume fraction and microorganism density number, with variation in Brownian motion parameter (\(Nb\)) and thermophoresis parameter (\(Nt\)). The impact of Brownian motion is strongly related to the size of nanoparticles. Physically smaller nanoparticles yield higher \(Nb\) values, which assist in thermal diffusion in the boundary layer via increased thermal conduction. On the contrary, larger nanoparticles show lower \(Nb\) values, and this depresses thermal conduction to the fluid and encourages heat diffusion to the wall. Higher \(Nb\) values will conversely stifle the diffusion of nanoparticles towards the sheet surface and enhance thermal diffusion into the nanofluid. This results in an escalation in temperatures as observed in Figure 7a. Increasing values of \(Nt\) indicate a stronger thermophoretic force (due to the temperature gradient) which draws the nanoparticles from the nanofluid towards the sheet surface and produces a depletion in nanoparticle volume fraction (concentration) and an associated decrease in nanoparticle species boundary layer thickness (Figure 7b). Increasing thermophoresis effect also elevates temperatures significantly as witnessed in Figure 7a and this trend is sustained throughout the boundary layer. Thermal boundary layer thickness is therefore enhanced consistently with greater thermophoresis.