## 3 Fluid Flow Development in the Pipe with Increasing Reynolds Number  
As the Reynolds number increases on the resistance curve, there is a transition in the values of resistance coefficient from one region to another due to a change in rheology. Figure 1 shows three schemes for the change in the resistance coefficient \(\lambda\) for three different cases of flow description implementations corresponding to different values of the \(n\)th power and the \(\chi_{n}\) non-dimensional coefficient, respectively. These diagrams indicate: 1--Hagen-Poiseuille curve \(n=1\), \(\chi_{n}=1\); 2--Blasius curve, \(n=4\), \(\chi_{n}=0.019746\); 3--curve for \(n=6\), \(\chi_{n}=0.00910904\), 4--curve for \(n=1000\), \(\chi_{n}=0.02/m^{1/3}\), where \(m\) is the roughness parameter, equal to the ratio of the pipe radius \(R\) to the reduced height of the roughness bumps \(k\) (\(m=R/k\)). Figure 1a shows the transition from a laminar flow regime to a turbulent one, Figure 1b shows this transition to the Prandl-Nikuradze curve, which is approximated by curves 2 and 3 with different values of the \(n\)th power; and Figure 1c shows the transition of curve 2 to curve 4 with a horizontal region with its own roughness value. The transition of the laminar flow regime with resistance \(\lambda=64/Re\) to turbulent one, when the Blasius formula takes place \(\lambda=0.3164/Re^{0.25}\) is shown in Figure 1a. From curve 1 corresponding to the \(n=1\) and \(\chi_{n}=1\), there is a transition to the Blasius curve 2, which corresponds to \(n=4\) and \(\chi_{n}=0.019746\). The velocity profile (9) takes the following form for \(n=1\) :  
\[V=\frac{Re_{*}}{2}\Big{[}1-(1-\eta)^{2}\Big{]} \tag{14}\]  
Figure 1: Resistance curves for different cases of flow of the \(n\)th power: (**a**) shows the transition from a laminar flow regime to a turbulent one; (**b**) shows this transition to the Prandl–Nikuradze curve; (**c**) shows the transition of curve 2 to curve 4 with a horizontal region. and it becomes a turbulent profile for \(n=4\) :  
\[V=\Big{(}\frac{Re_{*}}{5\chi_{n}{}^{4}}\Big{)}^{\frac{1}{7}}\Big{[}1-(1-\eta)^{5} \Big{]}^{\frac{1}{7}} \tag{15}\]  
In dimensional form, taking into account the notation (3) after the transition to cylindrical coordinates, when \(\eta=y/r=(R-r)/R=1-r/R\), Formula (14) takes the form:  
\[u=\frac{1}{4\mu}\Big{(}-\frac{dp}{dz}\Big{)}\Big{(}R^{2}-r^{2}\Big{)} \tag{16}\]  
and then (15) takes the form given below:  
\[u=0.93677\frac{1}{v^{1/7}}\bigg{(}\frac{1}{\rho}\frac{\Delta p}{l}\bigg{)}^{ \frac{4}{7}}\Big{(}R^{5}-r^{5}\Big{)}^{\frac{1}{7}} \tag{17}\]  
The Reynolds number (\(Re_{1}\) value) of the transition originating from the laminar regime to the turbulent one can be calculated as the result of the intersection of the Hagen-Poiseuille and Blasius curves:  
\[\frac{64}{Re_{1}}=\frac{0.3164}{Re_{1}{}^{0.25}}\]  
from which \(Re_{1}=1187.4\). This value turns out to be less than 2300, indicated in the references [1; 2], which is due to the disregard of the transition region from the laminar regime to the turbulent one. If we assume that this region exists, then it would be advisable to accept that \(Re_{1}=2300\) and understand that there is a gap in the resistance curve at this value of the Reynolds number. Thus, for the case of the flow shown in the diagram in Figure 1a, one obtains:  
\[ \begin{array}{l}\text{Re}<2300,n=1,\chi_{n}=1\\ \text{Re}>2300,n=4,\chi=0.019746\end{array}  \tag{18}\]  
We shall not consider this case further. Figure 1b shows a resistance curve, complying with the Prandl-Nikuradze formula:  
\[\lambda=0.0032+\frac{0.221}{Re^{0.237}} \tag{19}\]  
which is in good agreement with experimental data [9; 11], and is approximated by two curves corresponding to \(n=4\) and \(n=6\).