Virk’s maximum drag reduction asymptote

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... 

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