## 1 Introduction  
Packed electronic components have increased heat flux generation, prompting the investigation of various cooling strategies [1]. The application of conventional heat sinks, extended surfaces used to increase the heat-transfer area in electronic components, is a popular cooling mechanism among electronic cooling systems [2]. Two types of conventional heat sinks are available: plate-fin and pin-fin heat sinks. The application of conventional plate-fin heat sink is popular due to their advantages of being simple to design and fabricate [3]. Recent studies have proposed conventional plate-fin designs, including cross-fin arrangements [4], wavy [5, 6] or Y-shaped plate fins [7], radial plate fins [8], combinations of plate and pin fins [9, 10], plate fins with fillets [11, 12], slots [13, 14, 15], dimples [16], and rods [17]. The current trend in electronic product design is focused on making them lighter, thinner, shorter, and smaller [18]. Thus, the challenging tasks lie not only in improving heat dissipation rates but also in reducing the size and weight of the cooling system. Consequently, numerous studies have been conducted on different conventional plate-fin heat sink modules to achieve high-performance heat removal while maintaining compactness. The utilization of thin plate fins in a heat sink allows for a higher number of fins within a given substrate size compared to conventional plate fins. This increase in the number of fins results in a larger heat-transfer area, ensuring improved heat removal. Furthermore,the application of thin plate-fin heat sinks enables the reduction of weight and material usage in the cooling system as compared to conventional plate-fin heat sinks. As a result, researchers have become increasingly interested in enhancing heat transfer in thin plate-fin heat sinks. Heat-transfer enhancement through modifications in heat sink geometry has its limitations due to the development of thermal boundary layers. To achieve further improved thermal performance, heat sinks have incorporated passive and active cooling methods. Researchers have explored roughened heat sinks [19; 20; 21], nanoscale structuration techniques [22], and the application of reverse or swirl flow devices [23] as passive cooling methods to obtain higher thermal performance in heat sinks. However, the increasing demand for heat transfer necessitates the use of active techniques that utilize external power. The implementation of active cooling methods such as fan-based air cooling systems, air jets [24; 25; 26], electro-hydrodynamics (EHDs), and magneto-hydrodynamics (MHDs) in heat sinks is common among researchers [27]. The combination of passive and active cooling technologies, such as the implementation of a jet system on heat sinks with micro-scale surface treatment [28] or heat-sink cooling with nanofluids, which is subjected to a magnetic field [29], has also been investigated by researchers. Additionally, vibration-assisted convection cooling mechanisms have shown significant heat-transfer enhancement. Nevertheless, the application of vibrations in the cooling system can lead to reliability and stability issues within the cooling system due to the formation of fatigue stress on the vibrating surface [30]. Vibrational energy provided to the heated surfaces is propagated through the interface of the solid and fluid, inducing secondary flows within the flow regime, disrupting thermal boundary layers, thus enhancing the heat transfer [31]. However, this induced disturbance depends on the vibration characteristics, flow parameters, and geometrical parameters of the heated body. Experimental and numerical investigations have been conducted to prove the concept of enhancing heat transfer via vibration using gauge wire [32], spheres [33], and cylinders [34; 35; 36]. The results of these studies have shown significant improvements in heat transfer due to induced secondary flows within the flow field.