## 4 Results and Discussion
### Outcomes of Wave Amplitude (\(A\))  
The 3-D plots and the cross-sectional view in the \(X-Y\) plane of the streamlines, isotherms, and the nanoparticle distribution for various wave amplitudes (\(A\)) are depicted in Figures 7 and 8, respectively. The numbers of waves (\(N\)), Rayleigh number (\(Ra\)) and nanoparticle volume fraction (\(\phi\)) are kept fixed at 4, 106, and 0.02, respectively. From the first columns of Figures 7 and 8, it can be seen that the stream lines consist of four pairs of strong recirculation cells at the upper part occupying most part of the domain and eight comparatively lower strength cell at the lower part of the domain for \(A=0.05\). For increasing wave length the vortices at the lower part become weaker and finally disappear at \(A=0.125\), on the other hand, the vortices at the upper portion become larger in size and strength and occupy the whole domain as the values of \(A\) rises. On the increase of wave amplitude, wavelength  
Figure 6: Variation of the streamlines (**left**), isotherms (**middle**), and nanoparticle distribution (**right**) evolution by number of oscillations (\(N\)) for \(A=0.1\), \(Ra=10^{6}\) and \(\phi=0.02\) in the plane (\(X,Y\)). increases which leads the strong circulation of flow. In conclusion, the number of oscillations and extent of wave amplitude play vital role in transferring heat from a wavy surface. For the present collector design, augmentation in oscillations number as well as wave amplitude reduce the fluid domain but enlarge the surface area which is capable to collect more solar irradiation and consequently contribute to enhance heat transfer. The effect of different wave amplitude on thermal field is presented in the middle columns of Figures 7 and 8. The isothermal lines are almost parallel to the nearest wall and the higher temperature gradient region remains in the vicinity of the concave portions of the upper wall which indicates the higher heat transfer region also because the space between the upper and lower surface is least on theses parts of the collector. When escalating the value of \(A\), the isotherms slightly distorted from the convex portion of the upper surface and became denser near the concave portions where the temperature gradient become higher because these regions getting closer to the base wall making more heat transfer. The third columns of Figures 7 and 8 display the concentration field of nanoparticle for different \(A\). The overall appearance concerning the figures determines that thermophoresis and Brownian forces cause the nanoparticles concentration to grow nonuniform everywhere within the region. For any case, nanoparticle concentration remains higher at the top middle of each wave. The isoconcentrations are more concentrated near the walls with an increase of wave amplitude. The solutal boundary layer becomes thicker by increasing by \(A\). Such performance designates the outline of the thermal carrier within nanofluid, and one requirement remains concerned concerning the near-wall section and hold lower or higher particle concentration which directs to higher or lower rates of heat transfer. Figure 7: Variation of the 3D streamlines (**left**), isotherms (**middle**), and nanoparticle distribution (**right**) evolution by amplitude (\(A\)) for \(N=4\), \(Ra=10^{6}\), and \(\phi=0.02\). Figure 9 shows the distribution of nanoparticles at the plane of \(Y=0.1\) along the \(X\)-direction. Figure 9a,b depicts the results for different \(N\) and \(A\), respectively. In both forms, a noticeable variation of particles concentration close to the horizontal surfaces can be recognized. A significant level of nanoparticles could be located at the lower surface. That is due to the thermophoresis force, which manages to move the nanoparticles of hot to cold zones. Close to the heated upper surface, the concentration of nanoparticles is low. This is again due to the thermophoresis effect, which sweeps the particles away from the hot surface.