## 3 Results
As seen in Figure 9b-e, increasing the amplitude leads to a widespread decrease in air temperature near the fin surfaces, tending to have a stronger cooling effect. However, as seen in Figure 9e, increasing the amplitude with square wave-shaped vibration leads to a higher decrease in air temperature near the fin surfaces due to stronger flow circulations, leading to a higher mixing effect within the flow field. The discussion shows that vibrational frequency, amplitude, and wave shape are three important parameters that induce air mixing and rupture the thermal boundary layer. To evaluate the thermal effectiveness and consequent cooling enhancement of the system by varying those parameters, time-averaged Nusselt numbers are used as an indicator. As a result, the parameterization of time-averaged Nusselt number with vibration frequency and amplitude for each wave shape is performed and presented in Figure 10. The parametrization helps to more simply estimate the heat transfer enhancement and consequent reduction in the cooling system based on the vibrational parameters under each wave shape. The parametrization also aids in determining the optimum scenario of parameter combinations that provide the best thermal performance of the system. Moreover, according to the Nusselt number value obtained from the modeling, Reynolds numbers are calculated from Equations (9) and (10), which gives the value of the Reynolds number under no vibrating conditions, and those values are presented at each data point in Figure 10.  
Figure 9: Temperature profiles on a plane passing across the tip of the heat sink at \(Re=1000\): (**a**) static heat sink; (**b**) sinusoidal wave shape, \(f=100\) Hz, \(Amp=0.003\) m; (**c**) square wave shape, \(f=100\) Hz, \(Amp=0.003\) m; (**d**) sinusoidal wave shape, \(f=100\) Hz, and \(Amp=0.005\) m; (**e**) square wave shape, \(f=100\) Hz, \(Amp=0.005\) m.  
As shown in Figure 10a below, when the vibrational frequency is below 30 Hz, the Nusselt number almost remains unchanged with each type of wave shape compared with the static heat sink. When the frequency is lower than the threshold value, the energy of vibration propagates through the solid interface, and the fluid may not be adequate to induce the strong recirculation zones or secondary flows within the flow field. Thus, applied vibration is insufficient to disturb the developed thermal resistance caused by the thermal boundary layer. Notably, this threshold is observed to be independent of the vibration wave shape. Thus, using sinusoidal or square wave-shaped vibration on the heat sink with a lower frequency (\(f<30\) Hz) does not influence the thermal performance of the system. However, after exceeding the threshold frequency, Nusselt values show a steady rise with increasing frequency at each vibrational wave shape. This result can be attributed to the vibration effect, where a lower vibration frequency may not induce sufficient air perturbation, which cannot disrupt the thermal boundary layer. However, when the frequency increases beyond the threshold, it causes a higher agitation and air perturbation, causing disruption to the thermal boundary layer and consequently enhancing heat transfer. As expected, the square wave excitation of the heat sink causes the Nusselt number to shift to an elevated value compared with the sinusoidal motion because square wave excitation of the heat sink attains the maximum amplitude instantly, which induces a strong impulsive motion, leading to a significant velocity change and stronger recirculation zones, thereby enhancing the cooling effect. Moreover, the relative Nusselt number enhancement difference in square and sinusoidal wave motion is improved with increased frequency. Compared with the static heat sink, the Nusselt number is improved up to 20% with square wave-shaped vibration, whereas a 12% increase in Nusselt number values is recorded when the sinusoidal frequency is used.