## 2 Simulation Model and Methods  
### Model Development  
The aluminum heat sink considered in the current study is presented in Figure 2 and the size of the rectangular fluid domain is 0.305 \(\times\) 0.310 \(\times\) 0.037 m\({}^{3}\). A uniformly distributed power input of 9.5 W is supplied to the bottom of the heat sink, which corresponds to 6250 W/m\({}^{2}\), which is usually generated in data centers [47; 48]. The uniform velocity is employed at the air inlet with a temperature of 25 \({}^{\circ}\)C, and the airflow is laminar at a Reynolds number of 1000, which corresponds to an averaged air inlet velocity of 1.46 m/s. Figure 1: Vibration wave plot of square and sinusoidal wave-shaped vibration. Mass, momentum, and energy conservation laws form the governing equations for the current study. The flow is assumed to be a laminar incompressible ideal gas with constant thermo-physical properties. The body forces are neglected, and a no-slip condition is applied on the interface between the heat sink and fluid. Thus, the governing equations can be presented as:  
Continuity equation:  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{ \partial z}=0, \tag{1}\]  
Momentum equations:  
\[\frac{\partial u}{\partial t}+\rho u\frac{\partial u}{\partial x}+\rho v\frac{ \partial u}{\partial y}+\rho w\frac{\partial w}{\partial z}=\mu\Big{(}\
abla^{ 2}u\Big{)}-\frac{\partial P}{\partial x} \tag{2}\]  
\[\rho\frac{\partial v}{\partial t}+\rho u\frac{\partial v}{\partial x}+\rho v \frac{\partial v}{\partial y}+\rho w\frac{\partial w}{\partial z}=\mu\Big{(} \
abla^{2}v\Big{)}-\frac{\partial P}{\partial y} \tag{3}\]  
\[\rho\frac{\partial w}{\partial t}+\rho u\frac{\partial w}{\partial x}+\rho v \frac{\partial w}{\partial y}+\rho w\frac{\partial w}{\partial z}=\mu\Big{(} \
abla^{2}w\Big{)}-\frac{\partial P}{\partial z} \tag{4}\]  
Energy conservation equation:  
\[\rho c_{p}\frac{\partial T}{\partial t}+\rho c_{p}\frac{\partial}{\partial x} (uT)+\rho c_{p}\frac{\partial}{\partial y}(vT)+\rho c_{p}\frac{\partial}{ \partial z}(wT)=k_{fluid}\Big{(}\
abla^{2}T\Big{)} \tag{5}\]  
Energy equation for heat sink:  
\[\
abla^{2}T=\frac{1}{\alpha}\,\frac{\partial T}{\partial t} \tag{6}\]  
where \(u\), \(v\), and \(w\) are the velocities along the \(x\), \(y\), and \(z\) directions, respectively, \(\rho\) is the fluid density, \(P\) is the hydrostatic pressure, and \(\mu\) is the fluid viscosity, which is assumed to be constant. \(k_{fluid}\) is the thermal conductivity of the fluid, \(c_{p}\) is the constant specific heat, \(T\) is the temperature, and \(\alpha\) is the thermal diffusivity of the fin material (\(k_{fin}/\rho c_{p}\)). Past  
Figure 2: Physical model. (**a**) Isometric view, (**b**) side view, (**c**) top view. research has found that a higher heat transfer enhancement is obtained with horizontally vibrating heated objects [49; 50]. Hence, the sinusoidal and square wave-shaped vibration is applied to the heat sink in the Y direction (Figure 2). To simulate vibration in FLUENT, User Define Function codes (UDFs) are developed, which include the displacement equations of sinusoidal and square wave-shaped vibration, given in Equations (7) and (8), respectively. \[Y=A_{v}sin(2\pi ft) \tag{7}\]  
\[Y=\frac{4A_{v}}{\pi}\sum_{n=1,3,5..}^{N}\frac{1}{n}sin(2\pi nft). \tag{8}\]  
where \(Y\) is the displacement of the heat sink at any time,\(f\) is the vibrational frequency, \(A_{v}\) is the vibrational amplitude, and \(t\) is time. The investigation is conducted over a range of vibrating frequencies 0-100 Hz with peak-to-peak amplitudes ranging from 0 m to 0.005 m. A Reynolds number of 1000 is used for the vibrating case. Theoretical Reynolds numbers under nonvibrating conditions have been calculated using Equations (9) and (10) [51; 52], which are used for comparative analyses with the vibrating heat sink.