## 2 Steady-State Flow in a Circular Cylindrical Pipe at an Arbitrary Value of the \(n\) Power  
A circular pipe is the most common hydrodynamic object used for tests of theoretical solutions, since a huge amount of experimental material has been accumulated for the flow within this object, which makes it possible to evaluate the quality of the formulas obtained for resistance coefficients and velocity profiles. For the steady-state flow in a straight circular cylindrical pipe with radius \(R\), \(y\) is the coordinate measured from the wall (\(0\leq y\leq R\)). Let us introduce non-dimensional variables \(\eta\) (non-dimensional coordinate) and \(V\  
where \(Re_{*}=V_{*}R/\
u\) is the Reynolds number, calculated from the friction velocity. Then:  
\[\frac{dV^{2n-1}}{d\eta}=\frac{Re_{*}}{\chi_{n}^{n}}\left(1-\eta\right)^{n} \tag{7}\]  
The boundary condition for this equation is the no-slip condition: \(\eta=0\), \(V=0\).  
The integration of Equation (7) with this boundary condition leads to the following expression:  
\[V^{2n-1}=\frac{Re_{*}}{\chi_{n}^{n}(n+1)}\left[1-(1-\eta)^{n+1}\right]\]  
from which the non-dimensional velocity profile takes the form:  
\[V=\left(\frac{Re_{*}}{(n+1)\chi_{n}^{n}}\right)^{\frac{1}{2n-1}}\left[1-(1- \eta)^{n+1}\right]^{\frac{1}{2n-1}} \tag{8}\]  
The non-dimensional velocity averaged over the pipe's cross section is calculated using the formula [9; 10]:  
\[V_{av}=2\int\limits_{0}^{1}V(\eta)(1-\eta)d\eta\]  
Substitution of expression (8) for the velocity profile into this formula leads to the expression:  
\[V_{av}=2\bigg{(}\frac{Re_{*}}{(n+1)\chi_{n}^{n}}\bigg{)}^{\frac{1}{2n-1}}Y(n) \tag{9}\]  
where \(Y(n)\) has the following form for a flow in the pipe:  
\[Y(n)=\int\limits_{0}^{1}\left[1-(1-\eta)^{n+1}\right]^{\frac{1}{2n-1}}(1-\eta )d\eta \tag{10}\]  
This parameter is expressed through hyper-geometric functions; its values for different power values are shown in Table 1.  
For other \(n\) values, \(Y(n)\) can be found in reference materials. It is useful to note that when \(n\rightarrow\infty\), \(Y(n)\to 0.5\). The true Reynolds number \(Re=2 RV_{av}/\
u\), according to the average velocity \(V_{av}\), can be calculated [2] in terms of the \(Re_{*}\) number using friction velocity:  
\[Re=2 Re_{*}V_{av}\]  
Then, considering Formula (9), the following can be written:  
\[Re=4Y(n)\bigg{(}\frac{1}{(n+1)\chi_{n}^{n}}\bigg{)}^{\frac{2n}{2n-1}}Re_{*}^{ \frac{2n}{2n-1}} \tag{11}\]  
Hence, if expressing \(Re_{*}\) inversely in terms of \(Re\), one obtains  
\[Re_{*}=\frac{\left((n+1)\chi_{n}^{n}\right)^{\frac{1}{2n}}}{2n-1}Re^{\frac{2n -1}{2n}} \tag{12}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(n\) & **1** & **2** & **3** & **4** & **5** & **20** \\ \hline \(Y(n)\) & 0.25 & 0.403067 & 0.447761 & 0.467138 & 0.477358 & 0.498156 \\ \hline \hline \end{tabular}
\end{table}
Table 1: \(Y(n)\) values.  
The resistance coefficient \(\lambda=8|\tau_{w}|/\left(\rho V_{aw}^{2}\right)\) can be represented in terms of the square of the non-dimensional average velocity as \(\lambda=8/V_{aw}^{2}\), which, when considering Equation (9), leads to a formula for this coefficient at an arbitrary \(n\) value:  
\[\lambda=2^{\frac{n+2}{n}}\ \frac{(n+1)^{\frac{1}{n}}}{Y(n)^{\frac{2n-1}{n}}}\ \frac{\chi_{n}}{Re^{\frac{1}{n}}} \tag{13}\]  
Formulas (9)-(13) make it possible to describe the velocity field and resistance coefficient for a flow in the pipe for any values of the \(n\)th power.