Author Correlation Remarks
[98]
\(\frac{Q_{c}}{Q_{s}}=1-\left(\frac{1}{576}L\right)^{2}\left(0.03058-0.06426m\right)\)
\(m=0.5\alpha_{3}^{1}\) \(Q_{s}=\frac{W_{0}\pi a^{2}}{2}\) Low \(N_{\text{Dn}}\)
[83] \(\frac{Q_{c}}{Q_{s}}=1-\left(\frac{L}{576}\right)^{2}\left(0.0306+0.4607\alpha_{2}^{{}^{\prime}}+0.1982\alpha_{3}^{{}^{\prime}}+112.8\alpha_{5}^{{}^{\prime}}+257.83\beta_{1}^{{}^{\prime}}\right)\) \(\frac{Q_{c}}{Q_{s}}=1-\left(\frac{L}{576}\right)^{2}\left(0.0306+0.4607\alpha_{2}^{{}^{\prime}}+0.1982\alpha_{3}^{{}^{\prime}}+112.8\alpha_{5}^{{}^{\prime}}+257.83\beta_{1}^{{}^{\prime}}\right)\)
[100]
\(\frac{Q_{s}}{Q_{c}}=1+\left(N-1\right)6.3\times 10^{-4}\frac{R}{a}\left[0.14\left(\frac{R}{a}\right)^{0.5}+4.5\right]^{2}\left(\frac{Q_{c_{\text{Newtonian}}}}{Q_{s}}-1\right)^{2}\)
\(N\) is the number of bends or curves \(\frac{Q_{c_{\text{Newtonian}}}}{Q_{s}}\) is taken from the master curve \(6.3\times 10^{-4}\) is adjustable parameter that is dependent on geometry
[90]
\(\frac{Q_{s}}{Q_{c}}=1+\frac{1}{48}\left(\frac{a}{R}\right)^{2}\left[1-N_{\text{Re}}^{2}\left(\frac{11}{360}+N_{\text{Re}}^{2}\frac{1541}{87091200}\right)N_{\text{We}}^{2}\left(\frac{\mu_{p}}{\mu}\right)^{2}\frac{8}{3}\left(1-\frac{1}{15}\right)N_{\text{We}}^{2}\frac{\mu_{p}}{\mu}\left(3-2\frac{\mu_{p}}{\mu}\right)+N_{\text{We}}\text{Re}\frac{\mu_{p}}{\mu}\frac{1}{26880}\left(N_{\text{Re}}^{2}+5376\right)-N_{\text{We}}^{2}N_{\text{Re}}^{2}\frac{\mu_{p}}{\mu}\frac{1}{60480}\left(792-691\frac{\mu_{p}}{\mu}\right)-N_{\text{We}}^{3}N_{\text{Re}}\left(\frac{\mu_{p}}{\mu}\right)^{2}\frac{1}{90}\left(15-11\frac{\mu_{p}}{\mu}\right)\right]\)
\(N_{\text{Re}}=\frac{\rho W_{o}a}{\mu}\), \(N_{\text{We}}=\frac{\lambda W_{o}}{a}\) \(N_{\text{We}}\) Weissenberg number \(\lambda\) is the fluid relaxation \(W_{o}\) is the maximum axial velocity for flow in straight pipe \(\mu_{p}\) is the polymeric viscosity \(\mu\) is the total shear viscosity