## 3 Fluid Flow Development in the Pipe with Increasing Reynolds Number
For \(n=4\) and \(\chi_{n}=0.019746\), the Blasius formula \(\lambda=0.3164/Re^{0.25}\) takes place, for \(n=6\) and \(\chi_{n}=0.00910904\) the resistance curve, according to the (13), corresponds to:  
\[\lambda=\frac{0.1156}{Re^{1/6}} \tag{20}\]  
The intersection point of these curves corresponds to the Reynolds number, which can be found according to the relation:  
\[\frac{0.3164}{Re_{2}{}^{1/4}}=\frac{0.1156}{Re_{2}  
Figure 1c shows resistance curves for the flow in rough pipes with different values of the roughness parameter \(m=R/k\). For a flow in rough pipes, one can take \(n=1000\), which gives an almost horizontal line in coordinates as follows: (Logarithm of Reynolds number - logarithm of resistance coefficient), i.e., as \((log_{10}(Re)-log_{10}(100\lambda))\).The rheological power relation (2) can be presented as follows for \(n=1000\):  
\[\tau=\rho\chi_{n}\Big{[}\
u\frac{du^{1999}}{dy}\Big{]}^{\frac{1}{1000}},\quad \chi_{n}=\frac{0.02}{m^{1/3}} \tag{22}\]  
Hence, the velocity profile in non-dimensional coordinates, according to (8), can be presented in the form:  
\[V=\Big{(}\frac{Re_{*}}{101\chi_{n}{}^{1000}}\Big{)}^{\frac{1}{1999}}\Big{[}1- (1-\eta)^{1005}\Big{]}^{\frac{1}{1999}}\approx 1/\sqrt{\chi_{n}}\Big{[}1-(1- \eta)^{1001}\Big{]}^{\frac{1}{1999}} \tag{23}\]  
The average non-dimensional velocity value is  
\[V_{av}\approx\frac{2\ Y(100)}{\sqrt{\chi_{n}}}=\frac{2}{\sqrt{\chi_{n}}}*0.5= 1/\sqrt{\chi_{n}}\]  
taking into account the expression for \(\chi_{n}\), it takes the following form:  
\[V_{av}=\frac{m^{1/6}}{\sqrt{0.02}} \tag{24}\]  
For \(m=15;60;507\) values of \(V_{av}\) equal to 11.6, 14.4, and 20.2 can be obtained, which is in good agreement with experimental data [10]. The connection between Reynolds numbers for average and dynamic velocities for flow in rough pipes is as follows:  
\[Re=2 Re_{*}V_{av}=\frac{2 Re_{*}}{\sqrt{\chi_{n}}} \tag{25}\]  
The resistance coefficient, according to (13), is equal to \(\lambda=8\chi_{n}/Re^{0.001}\), and can be rounded to the approximate value in this case:  
\[\lambda\cong\frac{0.16}{m^{1/3}} \tag{26}\]  
which is in satisfactory agreement with the experimental data. The region of a hydraulically smooth pipe in Figure 1c transitions to the area with a constant value \(\lambda=Const\), depending on the value of the roughness parameter \(m\). The corresponding Reynolds number \(Re_{3}\) is found as a result of the intersection of the corresponding resistance curves:  
\[\frac{0.3164}{Re_{3}^{1/2}}=\frac{0.16}{m^{1/3}},\]  
which leads to  
\[Re_{3}=15.292\ m^{4/3} \tag{27}\]  
The Reynolds number \(Re_{3}\) values are presented in \[\lambda=\frac{0.1156}{Re^{1/6}} \tag{20}\]0 \begin{table}
\begin{tabular}{c c c c} \hline \(m\) & 60 & 100 & 507 \\ \hline \(Re_{3}\) & 3592.0 & 7097.9 & 61,822.0 \\ \(log_{10}(Re_{3})\) & 3.56 & 3.85 & 5.79 \\ \hline \end{tabular}
\end{table} 
Table 2: \(Re_{3}\) values. Resistance curves \(\lambda=Const=0.16/m^{1/3}\), when the quadratic law of resistance is satisfied, may have different positions according to each other. As a result, as the flow develops in rough pipes, and the \(n\)th power and \(\chi_{n}\) coefficient change:  
\[ \begin{array}{l}if\ Re<Re_{1},\ then\ n=1000\ \ and\ \ \chi_{n}=0.02/m^{1/3}\\ if\ Re_{1}\leq Re\leq Re_{3},\ then\ \ n=1000\ \ and\ \ \chi_{n}=0.02/m^{1/3}\\ if\ Re>Re_{3},\ then\ n=1000\ \ and\ \ \chi_{n}=0.02/m^{1/3}\\ \end{array} \]  
Thus, with an increase in the values of the Reynolds number, the \(n\)th power changes, which leads to changes in the velocity profiles and resistance coefficients according to formulas (8) and (13), respectively.