## 4 Results and Discussion
[PERSON] [3] also described that random Brownian motion results in the net movement of solute or suspended particles from regions of higher concentration to regions of lower concentration, a process called diffusion. Therefore, diffusion acts in opposition to centrifugal sedimentation, which tends to concentrate the particles. Brownian motion causes the arbitrary motion of nanoparticles within the working fluid. Therefore, the increase in the \(Nb\) causes a chaotic movement of the particles which affects the heat transfer positively. This mechanism is shown in Figure 5a, while this random movement of the nanoparticles reduces the solute field (see Figure 5b). [PERSON] and [PERSON] [4] also reported that the Brownian motion factor is favorable to the growth of the thermal boundary layer, while Brownian motion has a very limited effect on the Nusselt number. Our Brownian motion results are similar to those reported by [PERSON] and [PERSON] [6]. Another technique for improving thermal energy transfer is the thermophoresis mechanism, in which an increase in thermal energy is achieved by providing additional energy to the system through the thermal gradient. Figure 4: (**a**) \(F(\zeta)\), (**b**) \(H(\zeta)\), (**c**) \(\theta(\zeta)\), and (**d**) \(\phi(\zeta)\) for different values of \(R\). Therefore, the increase in \(Nt\) significantly increases the thermal and solute layer, as can be seen in Figure 6a,b. The effect of the thermophoresis number on the thermal field (Nusselt number) is also more evident than that of the Brownian motion parameter. [PERSON] and [PERSON] [7] report a similar observation, that is, the thermophoresis factor influences the Nusselt number with a rate of 0.1052 while the Brownian motion parameter influences the Nusselt number with a rate of 0.0001. The \(Le\) relates to both thermal and mass diffusivity. The physical mechanism of the Lewis number on heat and mass flow can be seen in Figure 7a,b. Figure 7a portrays the downturn in the thermal boundary layer and Figure 7b shows a significant reduction in the NVF layer thickness as \(Le\) increases i.e., the attenuation of mass diffusivity. Our Lewis number results are similar to those reported by [PERSON] and [PERSON] [6], and [PERSON] and [PERSON] [7]. Figure 5: (a) \(\theta(\xi)\) and (b) \(\phi(\xi)\) for different values of \(Nb\). The significance of two distinct thermal energy modulations (ESHS and THS) on \(\theta(\xi)\) and \(\phi(\xi)\) is explained pictorially in Figures 8 and 9. An increase in \(Q_{E}\) and \(Q_{T}\) yields an extra heat in the system, which leads to an increase in the thermal field, whereas increasing values of the \(Q_{E}\) and \(Q_{T}\) leads to a dual behavior in the \(\phi(\xi)\). Amongst two different thermal energy modulations, namely ESHS and THS, ESHS is more significant for solar thermal applications which encompass the heating process. The current results of ESHS and THS parameters are in agreement with our previous findings [19]. Three-Dimensional surface and columns plots are utilized to analyze the influence of key parameters on the Nusselt number (\(Nu_{t}\)) and Sherwood number (\(Sh_{t}\)) (see Figure 10a-e). It is observed that the higher the \(Nb\) and \(Nt\), the lower the heat transfer rate \(Nu_{t}\). This is because the thermal layer thickness is wider for larger values of both \(Nb\) and \(Nt\). Similarly, \(Nu_{t}\) inversely responses to \(Q_{E}\) and \(Q_{T}\). As ESHS and THS modulations improve the thickness of the thermal layer structure, the heat transfer rate is reduced for larger \(Q_{E}\) and \(Q_{T}\). Therefore, to achieve the maximum heat transfer rate, both \(Q_{E}\) and \(Q_{T}\) must be kept as low as possible. Higher values of Hartman number (\(H\alpha\)) and Lewis number (\(Le\)) lead to significant heat transfer. The maximum heat transfer rate \(Nu_{t}\) is found when both Marangoni number and Marangoni ratio parameters are kept at higher values. Therefore, the Marangoni convection process is vital in thermal energy systems.