# Transient Friction Analysis of Pressure Waves Propagating in Power-Law Non-Newtonian Fluids
## 3 Results and Discussion
### Startup Flow  
In Section 2.4.1, the motion equation is solved based on the assumption of steady viscosity. In this section, the verification is conducted by comparing the analytical results with the numerical results provided in Section 2.4.2. The time domain solution for the startup flow in a circular pipe can be obtained through the inverse Laplace transform of Equation (13) using the partial fraction expansion method. The pressure gradient for the startup flow can be written as  
\[\frac{\partial p}{\partial x}=\rho P_{c}\to F(s)=\int_{0}^{\infty}\rho P_{c}e^ {-st}\text{d}t=\frac{\rho P_{c}}{s}. \tag{23}\]  
By substituting Equation (23) into Equation (13) and performing an inverse Laplace transform, the following result can be obtained:  
\[u(r,t)=P_{c}\frac{T^{1+\frac{1}{n}}-R^{1+\frac{1}{n}}}{4h\
u}+\sum_{k=1}^{ \infty}2P_{c}\frac{R^{2h}}{vh^{2}}\Big{(}\frac{r}{R}\Big{)}^{h-1}\frac{J_{- \sigma}\Big{(}\lambda_{k}\big{(}\frac{r}{R}\big{)}^{h}\Big{)}}{J_{1-\sigma}( \lambda_{k})\lambda_{k}^{3}}\text{exp}\left(-\frac{\lambda_{k}^{2}}{R^{2h}} vh^{2}t\right), \tag{24}\]  
where \(P_{c}\) is the constant pressure gradient, \(\lambda_{k}\), \(k=1,2,\ldots\) are the roots of the Bessel function, \(t\) is time, \(J\) is the Bessel function of the first kind, and \(h=\frac{u+1}{2n}\), \(\
u=\frac{\mu_{0}}{\rho}\). The first term on the right side of Equation (24) represents the steady-state term, while the second term represents the transient term. For the startup flow, when \(n=1\), we can obtain the results for Newtonian fluids [30], and the results are shown in Figure 2. As shown in the figure, the numerical method proposed in this article shows good agreement with the theoretical results. Moreover, we can compare the results of Equation (24) to the numerical results provided in Section 2.4.2, and the results are shown in Figure 3. It can be seen from the figures that the profiles of the analytical and numerical results are very similar for different values of \(n\). However, for the same flow profile, between the numerical and analytical results there is a significant difference in the time required. This difference increases as the deviation between \(n\) and \(1\) increases. For \(n<1\), the analytical results indicate the shorter time \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Density of drilling fluids, \(\rho\), \((\text{kg}/\text{m}^{3})\) & 1220 \\ Inner radius of drill string, \(R\), \((\text{m})\) & 0.05431 \\ Consistency coefficient of drilling fluids, \(K\), \((\text{Pa}\cdot\text{s}^{n})\) & 1.0 \\ Power law index, \(n\) & 0.25\(\sim\)1.75 \\ Flow rate, \(Q\), \((\text{m}^{3}/\text{s})\) & 0.03 \\ Amplitude of pressure waves, \(P_{p}\), \((\text{MPa})\) & 0.5\(\sim\)2.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the system being analyzed. required to establish a stable flow compared to the numerical results. On the contrary, for \(n>1\), the analytical results show that a longer time is required to reach a stable flow compared to the numerical results. The reason for this difference is due to the assumption of the stable apparent viscosity made in the analytical method. The shear rate and apparent viscosity obtained with numerical methods at different times are shown in Table 2.  
As shown in the table, the stable flow shear rate used in the analytical methods to calculate the apparent viscosity is the largest among all of the transient flows. Therefore, when \(n<1\), the analytical method employs the minimum apparent viscosity, whereas when \(n>1\), the analytical method utilizes the maximum apparent viscosity. For the startup flow, we also investigated the influence of the number of collocation points on the convergence of the numerical method. Table 3 presents the relative errors between numerical and theoretical solutions when the startup flow of power-law non-Newtonian fluids tends to stabilize. As shown in the table, the relative error decreases with the increase in collocation points.