## 4 Heat Transfer around a Stationary Cylinder with a Detached Filament  
A stationary cylinder with a detached flexible filament is considered in this section, as shown in Figure 7, where \(D\) is the diameter of the fixed cylinder, \(L\) is the length of the filament, and \(l\) is the distance from the origin of the cylinder to the fixed end of the filament. This model has been extensively used to study fish behaviors in a street wake [17; 59; 60; 61]. Simulations are performed at \(l/D=3.0\), \(m_{s}=0\), and Re = 100. The non-dimensional stretching coefficient defined by \(K_{s}^{*}~{}=~{}K_{s}/(\rho\mathrm{U}^{2})\) is 1000 (the filament is approximately inextensible), and the non-dimensional bending rigidity defined by \(K_{b}^{*}~{}=~{}K_{b}/(\rho\mathrm{U}^{2}D^{3})\) is \(10^{-4}\). The mesh size around the cylinder and filament is D/160. The dimensionless time step \(\Delta t^{*}=\Delta tU_{0}/D=6.25\times 10^{-4}\). Three power-law indices 0.6, 0.8, and 1.0, and two filament lengths \(2.5D\) and \(4.0D\) are considered.  
The averaged drag coefficient (\(C_{d}\)), the Strouhal number (\(St\)), the averaged Nusselt number (\(Nu\)) of the cylinder, and the vertical flapping amplitude (\(A/D\)) of the trailing end of the filament are presented in Table 4. The good agreements of the drag coefficient, Strouhal number, and flapping amplitude between the present results and the data from [17; 60] show the excellent accuracy of the present solver. The longer filament generates a larger flapping amplitude, but the filament length does not significantly affect other fluid and temperature characteristics, as shown in Table 4. The results from non-Newtonian fluid flow show that the shear-thickening fluid generates larger drag comparing with shear-thinning fluid, but \(Nu\) presents an opposite tendency, as analysed in Section 3.2. Additionally, the current results show that the power index does not have significant influence on the flapping amplitude. The instantaneous vortex and temperature contours are presented in Figure 8. The non-Newtonian results presented here can be used to extend the limited database involving FSI.  
Figure 8: Instantaneous vorticity (left column, ranging from \(-2U_{0}/D\) to \(2U_{0}/D\)) and temperature (right column, ranging from 0 to 1.0) contours at Re = 100, Pr = 1.0: **(a)**\(n=0.6\); **(b)**\(n=1.0\); **(c)**\(n=1.4\).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Sources** & \(n\) & \(L/D\) & \(C_{D}\) & \(St\) & \(Nu\) & \(A/D\) \\ \hline Present & 0.6 & 2.5 & 1.25 & 0.175 & 5.881 & 1.10 \\  & & 4.0 & 1.24 & 0.175 & 5.872 & 1.28 \\ \hline Present & & 2.5 & 1.38 & 0.156 & 5.407 & 1.11 \\ [PERSON] et al. [60] & 1.0 & 2.5 & 1.41 & 0.156 & – & 1.14 \\ Present & & 4.0 & 1.37 & 0.152 & 5.401 & 1.30 \\ [PERSON] et al. [17] & & 4.0 & 1.39 & 0.153 & – & 1.34 \\ \hline Present & 1.4 & 2.5 & 1.51 & 0.143 & 5.077 & 1.10 \\  & & 4.0 & 1.48 & 0.141 & 5.069 & 1.30 \\ \hline \end{tabular}
\end{table}
Table 4: The averaged drag coefficient (\(C_{d}\)), the Strouhal number (\(St\)), the averaged Nusselt number (\(Nu\)) of the cylinder, and the vertical flapping amplitude \(A/D\) of the trailing end of the filament.  
Figure 7: Schematic of heat transfer around a stationary cylinder with a detached filament in power-law fluid flow.