## 3 Mathematical Modeling
### Analytical Steady-State Thermal Model
For the bottom surface of the bus bars, the vibration-assisted convection heat transfer coefficient \(\mathrm{h_{CB}}\) is estimated as follows [7; 22; 26; 27]:  
\[\rm I_{cp}=\frac{\left[(2\rm h_{c}S_{S}+\rm h_{c}F_{T}+\rm h_{c}B_{S}+\rm h_{r }S_{01})\big{(}T_{cp}-T_{a}\big{)}\right]^{1/2}S}{\left(\rm k_{s}k_{p}\rho_{dc} V\right)^{1/2}} \tag{5}\]7  
\[\rm I_{cp}=\frac{\left[(2\rm h_{c}S_{S}+\rm h_{c}F_{T}+\rm h_{c}B_{S}+\rm h_{r }S_{01})\big{(}T_{cp}-T_{a}\big{)}\right]^{1/2}S}{\left(\rm k_{s}k_{p}\rho_{dc} V\right)^{1/2}} \tag{5}\]8  
\[\rm I_{cp}=\frac{\left[(2\rm h_{c}S_{S}+\rm h_{c}F_{T}+\rm h_{c}B_{S}+\rm h_{r }S_{01})\big{(}T_{cp}-T_{a}\big{)}\right]^{1/2}S}{\left(\rm k_{s}k_{p}\rho_{dc} V\right)^{1/2}} \tag{5}\]9  
\[\rm T_{s}=\frac{Q_{bg,v}V}{2\rm h_{c}S_{S}+\rm h_{c}F_{T}+\rm h_{c}B_{S}+\rm h _{r}S_{01}}+T_{a} \tag{6}\]0  
\[\rm T_{s}=\frac{Q_{bg,v}V}{2\rm h_{c}S_{S}+\rm h_{c}F_{T}+\rm h_{c}B_{S}+\rm h _{r}S_{01}}+T_{a} \tag{6}\]1  
\[\rm T_{s}=\frac{Q_{bg,v}V}{2\rm h_{c}S_{S}+\rm h_{c}F_{T}+\rm h_{c}B_{S}+\rm h _{r}S_{01}}+T_{a} \tag{6}\]2  
\[\rm T_{s}=\frac{Q_{bg,v}V}{2\rm h_{c}S_{S}+\rm h_{c}F_{T}+\rm h_{c}B_{S}+\rm h _{r}S_{01}}+T_{a} \tag{6}\]3  
where \(\mathrm{Gr_{B}}\) is the Grashof number for a stationary horizontal plate, \(\mathrm{Nu_{B1}}\) is the Nusselt number for free convection from the bottom surface of a stationary horizontal plate calculated using Equation (22) for \(\mathrm{Gr_{B}Pr}\leq 8\cdot 10^{6}\) or \(\mathrm{Gr_{B}Pr}>8\cdot 10^{6}\), \(\mathrm{Re_{BV}}\) is the Reynolds number that is identical with \(\mathrm{Re_{TV}}\), \(\mathrm{Nu_{B2}}\) is the Nusselt number for fully developed turbulent forced convection from the bottom surface of a stationary horizontal plate, and \(\mathrm{Nu_{BV}}\) is the Nusselt number for vibration-assisted convection from the bottom surface of a horizontal plate. The idea that the effects of vibrations on free convection heat transfer from the surfaces of an isothermal plate can be modeled using pure forced convection is found in [7]. In practice, according to [26], a fully developed turbulent boundary layer (i.e., pure forced convection) over a stationary horizontal plate may be realized by \"tripping\" the existing boundary layer at the leading edge, using a fine wire or some other turbulence promoter. Based on the previous findings, it is assumed that the effect of horizontal transverse vibrations on free convection heat transfer from the top or bottom surface of an isothermal horizontal plate is the same as the effect of any turbulence promoter on forced convection heat transfer over a stationary horizontal surface. This assumption also agrees with the results of [6; 19]. Accordingly, Equation (17) is obtained from the following equations [26; 27]:  
\[\rm T_{s}=\frac{Q_{bg,v}V}{2\rm h_{c}S_{S}+\rm h_{c}F_{T}+\rm h_{c}B_{S}+\rm h _{r}S_{01}}+T_{a} \tag{6}\]4  
\[\rm T_{s}=\frac{Q_{bg,v}V}{2\rm h_{c}S_{S}+\rm h_{c}F_{T}+\rm h_{c}B_{S}+\rm h _{r}S_{01}}+T_{a} \tag{6}\]5  
which are valid for \(0.6\leq\mathrm{Pr}\leq 60\) and \(\mathrm{Re_{TV,cr}}\leq 8\mathrm{Re_{TV}}\leq 10^{8}\), considering that, for a fully developed turbulent boundary layer, \(\mathrm{Re_{TV,cr}}=0\) and \(A=0\). In Equations (28) and (29),\(\mathrm{Re}_{\mathrm{TV,cr}}\) represents the critical Reynolds number associated with the laminar to turbulent flow transition. In addition, for the transition, the critical Reynolds number and the constant A are equal to \(\mathrm{Re}_{\mathrm{TV,cr}}=5\cdot 10^{5}\) and A = 871, respectively. The same applies to the bottom surface of a rectangular bus bar. The above model can be applied to rectangular bus bars with surfaces whose characteristic lengths are large enough that a boundary layer can be formed along them. However, if these dimensions are small, as in the case of flat fins in [21], then the effects of convection and radiation from such surfaces can be ignored; i.e., those surfaces can be regarded as adiabatic. It is obvious that this was done in the experiments by [21].