## 3 Validations
### Non-Newtonian Power-Law Fluid Flow and Heat Transfer around a Stationary Cylinder  
Non-Newtonian power-law fluid flow and heat transfer from a stationary cylinder is considered to validate the accuracy of the present numerical method in handling FSI and heat transfer. The two-dimensional power-law fluid inlets with a uniform flow \(U_{0}\) over a stationary circular cylinder (of diameter \(D\)), with the computational domain extending from \((-15D,-5D)\) to \((25D,5D)\), as shown in Figure 3.  
In order to improve the computational efficiency and validate the geometry-adaptive technique utilized in present in-house code, four blocks are used in the simulations, as shown in Figure 4, and the most refined mesh size of the fluid domain is \(D/80\). The dimensionless time step \(\Delta t^{*}=\Delta tU_{0}/D=6.25\times 10^{-4}\).  
The non-dimensional parameters that control this problem include the Reynolds number (Re) and the Prandtl number (Pr), which are defined as  
\[\text{Re}=\frac{\rho_{0}D^{n}U_{0}^{2-n}}{m},\quad\text{Pr}=\frac{c_{p}m(U_{0} /D)^{n-1}}{K}. \tag{26}\]  
Figure 3: Schematic of heat transfer around a stationary cylinder in power-law fluid flow.  
Figure 2: Velocity profiles of the fully developed flow in a channel predicted by the present numerical method (markers) and those calculated by Equation (25) (solid lines), at Re = 10 and \(n=0.3\) (+), 0.5 (o), 0.7 (\(\square\)), 1.0 (\(\Diamond\)), and 1.5 (\(\Delta\)).  
The drag coefficient, lift coefficient, and Strouhal number are defined as  
\[C_{D}=-\frac{\sum F_{x}}{0.5\rho U_{0}^{2}D},\quad C_{L}=\frac{\sum F_{y}}{0.5 \rho U_{0}^{2}D},\quad\text{St}=\frac{D}{U_{0}T} \tag{27}\]  
where \(T\) is the vortex shedding period, and \(F_{x}\) and \(F_{y}\) are the horizontal and vertical components of \(\mathbf{F}\) acting on the cylinder calculated by Equation (16).  
Firstly, we consider the unsteady flow of non-Newtonian power-law flow over a cylinder at \(\text{Re}=100\) (at which the flow patterns show von-Karman periodic vortex shedding behind the cylinder) to validate the geometry-adaptive fluid solver in terms of handling FSI problem. In order to achieve the early von-Karman vortex shedding, a vertical velocity perturbation is introduced into the initial inlet boundary. Four different indexes \(n=0.6,1.0,1.4\), and \(1.8\) are calculated. Table 2 shows the  
After the validation of the fluid solver in computing FSI problems involving non-Newtonian flow, we further consider the steady flow and heat transfer around a cylinder. The average Nusselt number (\(Nu\)) used for quantitative comparison with the available data from the literature are defined as  
\[Nu=\frac{1}{2\pi}\int_{0}^{2\pi}Nu(\theta)\mathrm{d}\theta \tag{28}\]  
where \(Nu(\theta)\) is the local Nusselt number on the surface of the cylinder, it can be evaluated using the temperature field according to  
\[Nu(\theta)=-\frac{\partial T}{\partial n_{s}} \tag{29}\]  
where \(n_{s}\) is the unit vector normal to the surface of the cylinder.  
The average Nusselt number from the current computation along with the available data from the literature of two Reynolds number (Re = 10 and 40) and three indexes (\(n=0.8\), \(1.0\), and \(1.4\)) are presented in Table 3. It is found that the present results are in good agreement with other published data. With the increase of \(n\) from 0.8 to 1.4, the average Nusselt number decreases slightly at both Reynolds numbers. Comparisons between the Reynolds numbers of 10 and 40 show that the average Nusselt number changes more remarkably with larger Reynolds numbers. It is because of the enhancement of the heat convection effects under the larger Reynolds number. The snapshots of the temperature field from current simulations are presented in Figure 6. As is readily seen from Figure 6, the length of the vortex attached behind the cylinder stretches significantly with \(n\) increasing from 0.8 to 1.4, which also qualitatively agrees with the analysis in [54].