## 3 Results
To investigate the effect of varying the amplitude at a constant frequency on heat transfer enhancement, a 100 Hz frequency is selected to show the effect on the Nusselt number. As seen in Figure 10b above, an increase in peak-to-peak amplitude at a frequency of 100 Hz leads to an increase in time-averaged Nusselt number in both types of excitations. The increase in the peak-to-peak amplitude at higher frequency imparts a higher fin displacement, resulting in a greater recirculation within the flow field and improved heat transfer. Figure 10: Variation in time-averaged Nusselt number with vibrational parameters and corresponding Reynolds number under nonvibrating conditions: (**a**) frequency at constant \(Amp=0.003\) m and \(Re=1000\); (**b**) peak-to-peak amplitude at constant \(f=100\) Hz and \(Re=1000\). However, the heat transfer enhancement is found to be higher with square wave excitations. Once the amplitude of the square-shaped vibration is increased, the impulse motion of the vibrating fins combined with the higher amplitude causes a higher air perturbation due to the impulsive nature of this vibration, resulting in a rapid change in airflow field direction near the fin surface, further enhancing the heat transfer. When the amplitude is increased from 0 to 0.005 m at 100 Hz, a 14.2% increase in Nusselt number values is recorded when the sinusoidal vibration is used. However, at 100 Hz frequency and 0.005 m amplitude, the Nusselt number magnitude increases by up to 25% with square wave-shaped vibration, providing the system with the highest thermal performance over the evaluated parameter range. From Figure 10, the results show that increasing amplitude and frequency lead to a higher thermal performance of the vibrating heat sink. Therefore, it is also necessary to present the heat transfer enhancement in terms of the ratio of the Nusselt number of vibrating and nonvibrating conditions at each vibration level, which could be used as an indicator to perform the comparative analysis of the increase in heat transfer enhancement under vibrating conditions:  
\[Nusselt\ Ratio=\left(\frac{Nusselt\ number\ with\ vibrating\ conditions}{Nusselt\ number\ with\ non\ - vibrating\ conditions}\right) \tag{11}\]  
Figure 11a,b show the variation in Nusselt ratio with vibrational frequency and amplitude under sinusoidal and square wave shapes. Due to the insignificant heat transfer enhancement with a lower frequency (\(f<30\) Hz), the Nusselt ratio is found to be one at lower vibrational frequencies. However, after exceeding the critical frequency, the Nusselt ratio is found to steadily increase with increasing frequency. As expected, a higher enhancement of the Nusselt ratio is obtained with square wave-shaped excitation as compared to the sinusoidal vibration at each vibration frequency. As compared to the nonvibrating heat sink, 12% and 20% enhancements in Nusselt number ratios are obtainable by applying sinusoidal and square wave-shaped excitation, respectively, at \(f=100\) Hz and \(Amp=0.003\) m.  
With the increase in peak-to-peak amplitude at constant frequency of 100 Hz, the Nusselt ratio also increases. As expected, an increase in Nusselt ratio is found to be higher with square wave-shaped vibration. As compared to the nonvibrating heat sink, 14.2 and 25% enhancements in Nusselt number ratios are obtained by utilizing the sinusoidal and  
Figure 11: Variation in Nusselt ratio with vibrational parameters at \(Re=1000\): (**a**) frequency at constant \(Amp=0.003\) m; (**b**) peak-to-peak amplitude at constant \(f=100\) Hz. square wave-shaped vibration, respectively, with the conventional heat sink at \(f\) = 100 Hz and \(Amp\) = 0.005 m.  
Given that the parametric study shows the possibility of heat transfer intensification of the vibrating heat sink under sinusoidal and square wave shapes, this method can be used to reduce the size of the conventional cooling system.