Drag reduction asymptote, maximum

Sreenivasan, K. White, C.M. The onset of drag reduction by dilute polymer additives, and the maximum drag reduction asymptote. J. Fluid Mech. 2000, 409, 149-164. 

Rheology of polymer and surfactant solutions Maximum drag reduction asymptotes Type A and B drag reduction Mean velocity profiles and limiting asymptotes Turbulence intensities and stress balance... 

Over the past 60 years, a great deal of applied and theoretical research has been carried out on both polymer and surfactant DRAs because of their potential useful applications and the influence of the additives on both turbulent structure and rheology. Important results include the identification of maximum drag reduction asymptotes (MDRAs) both Virk s MDRA for polymer solutions [Virk et al., 1970 see Eq. (2.5)] and Zakin et al. s MDRA for surfactant solutions [Zakin et al., 1996 see Eq. (2.6)], relating solution nanostructures and rheological properties to macroscopic DR phenomena hypotheses on the influence of DRAs on turbulent structures, mechanisms for turbulent drag reduction, developing heat transfer enhancement techniques, and so on. 

There is a maximum drag reduction limit for polymer drag reduction [Castro and Squire, 1967 Giles and Pettit, 1967 Virk et al., 1970 Virk, 1975]. The maximum drag reduction asymptote (MDRA) proposed by Virk is generally accepted for high-polymer drag reduction [Virk, 1975]. Poiseuille s law for laminar flow [Eq. (2.3)], the von Karman law for Newtonian turbulent flow [Eq. (2.4)], and Virk s maximum... 

However, Sreenivasan and White (89) point out that the connection between fluctuating strain rates and large extensional viscosity is circumstantial. Further polymer coils can only be partially stretched in a random field of strain rate. Sreenivasan and White (89) point out that the elastic theory proposed by de Gennes (84) is compatible with at least two experimental observations ie, the dependence of drag reduction onset on polymer concentration and maximum drag reduction asymptote. 

Virk (10) observed that the maximum possible drag reduction asymptote (MDRA) is given by the following equation, where the drag reduction is insensitive to polymer properties ... 

In the regime below the maximum drag reduction asymptote, the friction factor varies with polymer properties and flow variables ... 

In case of surfactant solutions, friction factors signiflcantly lower than the predicted by MDRA for high polymer solutions are predicted. Zakin and coworkers (93) provided an equation given below for maximum drag reduction asymptote for surfactant solutions... 

Sreenivasan and White (89) recently invoked the theory based on the elastic behavior of stretched polymers of de Gennes (84), who states that coil-stretch transition does not occur in turbulent flows in randomly fluctuating strain rates and that, if moderately stretched, the polymers produce no measurable change in viscosity. Sreenivasan and White (89) were able to explain the onset of drag reduction and maximum drag reduction asymptote on the basis of the elastic theory, and their results qualitatively explain the existing experiments. 

Smith, G. L. and C. L. McCormick (2001). Water-soluble polymers. 79. Interaction of microblocky twin-tailed acrylamido terpolymers with anionic, cationic, and nonionic surfactants. Langmuir 17(5) 1719—1725. Sreenivasan, K. R. and C. M. White (2000). The onset of drag reduetion by dilute polymer additives, and the maximum drag reduction asymptote. Journal of Fluid Mechanics 409 149—164. 

---