## 4 The Flow of Weakly Concentrated Aqueous Solutions of Polymers

Table 3: The values of the “\(A\)” coefficient depending on the pipe diameter for a WSR-301 polymer solution concentration \(C=15\times 10^{-6}\) according to Figure 3.  
\[Re_{2}=\left(\frac{A}{0.1364}\right)^{\frac{4}{3}};\]
3. In the \"laminar-type\" 2-3 region, where \(Re_{2}\leq Re\leq Re_{3}\), there is a deviation from the resistance curve of a hydraulically smooth pipe to the resistance curve for a polymer solution flow with a resistance law \(A/Re\). The value of the \"\(A\)\" constant determines the type of polymer, its concentration in a solution and the pipe's diameter [4; 5]. The \(Re_{3}\) value can be found according to the intersection conditions of the \"laminar-type\" curve \(A/Re\) with the Virk limit curve \(0.87/\sqrt{Re}\), i.e., \(\frac{A}{Re}=\frac{0.87}{\sqrt{Re}_{3}}\), and then \[Re_{3}=\left(\frac{A}{0.87}\right)^{2},\] for different \"\(A\)\" parameter values; the values of \(Re_{2}\) and \(Re_{3}\) are given in Table 4;
4. With a further increase in the Reynolds number when \(Re>Re_{3}\), there is a transition to the Virk resistance curve, complying with the law \(0.87/\sqrt{Re}\). Thus, with the development of the flow of weakly concentrated aqueous solutions of polymers, each region has its own values of the \(n\)th power, non-dimensional parameters \(\chi_{n}\) and \(Y(n)\):  
1. In region \(a\)-1, when \(Re<Re_{1}\), there are \(n=1\), \(\chi_{n}=1\), \(Y(1)=0.25\)
2. In region 1-2, when \(Re_{1}\leq Re<Re_{2}\), there are \(n=4\), \(\chi_{n}=0.019746\), \(Y(4)=0.467138\)
3. In region 2-3, when \(Re_{2}\leq Re\leq Re_{3}\), there are \(n=1\), \(\chi_{n}=A/64\), \(Y(n)=0.25\)
4. In region 3-\(b\), when \(Re>Re_{3}\), there are \(n=2\), \(\chi_{n}=0.032146\), \(Y(n)=0.403067\)  
In accordance with these values, for each region expressions for the resistance coefficients can be written according to (13), velocity profiles can be written in terms of universal coordinates according to (3), and Reynolds numbers can be written in terms of friction velocity. These expressions for each region are as follows:  
Figure 4: Resistance change scheme for fluid flow in a pipe. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(A\) & **130** & **400** & **400** \\ \hline \(Re_{2}\) & 3054.5 & 13,669.9 & 29,378.5 \\ \(Re_{3}\) & 22,311 & 211,230 & 665,508 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Values of \(Re_{2}\) and \(Re_{3}\) for different \(A\) parameters. 1. Region \(a\)-1: \[V=\frac{Re_{*}}{2}\Big{[}1-(1-\eta)^{2}\Big{]}\ ;\ \ Re_{*}=1.41 Re^{1/2},\ \ \lambda=64/Re\] (29) 2. Region 1-2: \[ \begin{array}{c}V=\Big{(}\frac{Re_{*}}{5\chi_{*}a}\Big{)}^{\frac{1}{2}} \Big{[}1-(1-\eta)^{5}\Big{]}^{\frac{1}{2}}=7.484\ Re_{*}^{\frac{1}{2}}\ \Big{[}1-(1-\eta)^{5}\Big{]}^{\frac{1}{2}}\\ Re_{*}=0.099436\ Re^{7/8};\ \ \ \ \lambda=0.3164/Re^{1/4}\end{array} \] (30) 3. Region 2-3: \[V=32\ \frac{Re_{*}}{A}\Big{[}1-(1-\eta)^{2}\Big{]}\ ;\ \ \ \ Re_{*}=3.535534\ Re^{1/2};\ \ \ \ \lambda=A/Re\] (31) 4. Region 3-\(b\): \[ \begin{array}{c}V=\Big{(}\frac{Re_{*}}{3\chi_{*}a}\Big{)}^{\frac{1}{2}} \Big{[}1-(1-\eta)^{3}\Big{]}^{\frac{1}{2}}=\ 6.858\ Re_{*}^{\frac{1}{2}}\ \Big{[}1-(1-\eta)^{3}\Big{]}^{\frac{1}{ 2}};\\ Re_{*}=0.164917\ Re^{3/4};\ \ \ \ \lambda=0.87/Re^{1/2}\end{array} \] (32)  
For the flow of polymer solutions in rough pipes with a roughness parameter \(m\), the Reynolds number \(Re_{4}=15.292m^{4/3}\) determines the beginning of the transition from the Blasius curve to the region with a constant resistance coefficient, according to (27). If \(Re_{4}\leq Re_{2}\), then there will be no transition to the flow regime of the \"laminar-type\" region and polymer additives will have no effect on the flow (Figure 4). If it turns out that \(Re_{2}<Re_{4}\), then the roughness does not have time to influence the flow, and the resistance curve during the flow of polymer solutions will contain the same four regions (29)-(32) as a hydrauically smooth pipe. For the resistance coefficients at points 1, 2, and 3 in Figure 4, there is a transition from one flow regime to another.