## 3 Results
(**a**) frequency at \(A_{b}=0.0015\) m and \(Re=1000\) for both heat sinks; (**b**) amplitude at \(f=100\) Hz and \(Re=1000\) for both heat sinks; (**c**) amplitude at 0.0015 m under different Reynolds numbers for the thin plate-fin heat sink. results indicate that at a frequency and amplitude of 100 Hz and 0.0015 m, the Nusselt number of the thin plate-fin heat sink increased to a maximum of 16%, 9% and 6% at Reynolds numbers of 1000, 1500 and 2000, respectively. In order to estimate the possible reduction of the size of the fin through the utilization of vibration, it is necessary to calculate the enhancement of the Nusselt number in terms of the Nusselt ratio, which is calculated based on Equation (7). \[Nusselt\ Ratio=\frac{Nusselt\ number\ under\ vibrating\ conditions}{Nusselt\ number\ not\ -\ vibrating\ conditions} \tag{7}\]  
Figure 10 shows the variation in the Nusselt ratio with respect to the vibrational characteristics and Reynolds numbers. Since an enhancement in the Nusselt number is found to be insignificant with a vibrational frequency below 30 Hz, the enhancement of the Nusselt ratio is also found to be insignificant at low vibrational frequencies (f < 30 Hz). As shown in Figure 10a, at the vibrational frequency of 100 Hz, the Nusselt ratio is increased up to a maximum of 16% and 12% with the conventional heat sink and thin plate-fin heat sink, respectively. Moreover, as shown in Figure 10b, at the vibrational amplitude of 0.003 m, the thin plate-fin heat sink achieved a maximum of 20% enhancement in the Nusselt ratio, while the conventional heat sink achieved a 15% maximum increase in the Nusselt ratio in comparison to static conditions. Figure 10c shows the variation in the Nusselt ratio of the thin plate-fin heat sink for frequencies at different Reynolds numbers. Below the critical frequency of 30 Hz, the effect of vibration on the thermal performance of the thin plate-fin heat sink is insignificant for  
Figure 10: Nusselt ratio variation with vibrational characteristics. (**a**) frequency at \(A_{b}\) = 0.0015 m and Re = 1000 for both heat sinks; (**b**) amplitude at \(f\) = 100 Hz and \(Re\) = 1000 for both heat sinks; (**c**) amplitude at 0.0015 m under different Reynolds numbers for the thin plate-fin heat sink. all considered Reynolds values, resulting in an insignificant increase in the Nusselt ratio. However, beyond the 30 Hz critical frequency, an increase in vibrational frequency leads to an increase in the Nusselt ratio for each Reynolds value. It can be seen that higher Nusselt number ratio values were recorded at lower Reynolds numbers. This shows that vibration is less effective in increasing the Nusselt number (compared to Nusselt at 0 frequency) when the Reynolds number increases. At a heat-sink base frequency and amplitude of 100 Hz and 0.0015 m, the maximum enhancement in the Nusselt ratio is found to be 16%, 9% and 6% at Reynolds numbers of 1000, 1500 and 2000, respectively. The findings of this study demonstrate that an increase in vibrational frequency and amplitude enhances the thermal performance of both heat sink models. It is also demonstrated that at lower Reynolds number values, the effect of vibration on increasing the Nusselt number values from no-vibration values is more significant at lower Reynolds numbers. This heat-transfer enhancement under sinusoidal vibration can be utilized to reduce the size of the cooling system or fin, eventually leading to a compact cooling system. Thus, to estimate the possible reduction of the cooling system, it is necessary to present the percentage reduction in Reynolds number due to the application of vibration. Since the thermal performance of the thin plate-fin heat sink assisted by vibration diminishes with higher Reynolds numbers, the percentage reduction in Reynolds value is calculated and presented at a lower Reynolds value of 1000 for both heat sinks.