# Transient Friction Analysis of Pressure Waves Propagating in Power-Law Non-Newtonian Fluids
## 3 Results and Discussion
### Pressure Waves Transmitted inside the Drill String
In addition, the apparent viscosity adopts the maximum shear rate of the stable pipe flow, meaning that when \(n<1\), the analytical results for shear rate at the wall are greater than the actual value, as shown in Figure 5a,c, and when \(n>1\), the analytical results for shear rate at the wall are smaller than the actual value, as shown in Figure 6a,c. Numerical analyses show that the apparent viscosity is not only dependent on the shear rate, but also on Figure 4: Numerical and analytical results for the periodic flow of Newtonian fluids with different frequencies. The time of the curve shown in the figures is the last moment of a pulsating cycle. Figure 5: Comparison of analytical and numerical results for periodic flow with different frequency when power-law index \(n<1\); the time of the curve shown in the figures is the last moment of a pulsating cycle. (**a**) \(n=0.6\) without correction, (**b**) \(n=0.6\) after correction, (**c**) \(n=0.85\) without correction, (**d**) \(n=0.85\) after correction. The corrected velocity profile is shown in Figure 5b,d for \(n<1\) and Figure 6b,d for \(n>1\). It can be seen from the figures that the analytical results for the shear rate at the wall are in good agreement with the numerical results after correction. The closer \(n\) is to 1, the better the correction effect. However, there is a significant difference between the analytical and numerical results for the flow near the pipe axis, and this difference increases with the increase in frequency. Moreover, the correction effect for \(n>1\) is better than that for \(n<1\). It can also be seen from the Figures 5 and 6 that the shear rate at the wall increases with the increase in frequency for the same \(n\). Therefore, the friction loss of the periodic flow increases with the increase in frequency. Additionally, the shear rate at the wall decreases as \(n\) increases. In this study, the trend of \(C(n)\) changing with \(n\) based on the numerical results is shown in Figure 7, and the fitted expression for \(C(n)\) is  
\[C(n)=-0.422n^{3}+1.600n^{2}-1.831n+0.712,0.5\leq n\leq 1.5. \tag{28}\]  
The correction basis for Equations (27) and (28) is the maximum velocity profile in one cycle, and its adaptability to other moments needs further verification. Figure 8 shows the time history curve of the velocity profile within half a cycle for \(n=0.8\) and \(n=1.2\). As  
Figure 6: Comparison of analytical and numerical results for periodic flow with different frequency when power-law index \(n>1\). The time of the curve shown in the figures is the last moment of a pulsating cycle. (**a**) \(n=1.2\) without correction, (**b**) \(n=1.2\) after correction, (**c**) \(n=1.4\) without correction, (**d**) \(n=1.4\) after correction. shown in the figure, the corrected analytical velocity profiles match the numerical results very well in terms of the occurrence time and shape. In particular, the velocity distributions of the analytical results near the wall almost coincide with the numerical results, ensuring an accurate evaluation of wall friction losses.