Viscoelastic laminar drag bounds in pipe flow

Abstract

The velocity and friction properties of laminar pipe flow of a viscoelastic solution are bounded by the corresponding values for two Newtonian fluids, namely, the solvent and a fluid with a viscosity identical to the total viscosity of the solution. The lower friction factor for the flow of the solution when compared to the latter is tracked to an increased strain rate needed to enhance viscous dissipation. Lastly, we show analytically that the effective viscosity varies similarly to the radial diagonal component of the conformation tensor as observed numerically in turbulent flows, and give a lucid interpretation of shear-thinning through a sequence of underlying constitutive physical phenomena.Elasticity effects have long been known to affect transition to turbulence and drag in fluid flows, whether these effects are induced by fluid-structure interaction 1-3 , or by viscoelastic rheological aspects associated with non-Newtonian fluids 4-8 . For instance, the addition of low concentrations of long-chain polymers generates an astounding 80% drag reduction (DR) in turbulent regimes 9,10 , which has significant implications for practical applications. At relatively low Reynolds numbers, elasto-inertial effects sets in 11,12 , which are also observed in the instabilities of flow bounded by compliant walls 1-3 .When considering turbulent dilute polymeric solutions, the exhibited DR has been attributed to the interplay between flow turbulence and elasticity of the polymers in the near-wall region 9,10,13,14 . DR is most prominent when the time scale of the polymer elastic dynamicsknown to be dependent on the number and length of monomers making the polymer-is of the same order or higher than that of the turbulent fluid flow 15 . In addition, such fluids exhibit a maximum drag asymptote with respect to the polymer concentration 15-20 , which is suggested to be associated with elasto-inertial instability 11,12 . A phenomenon of reverse transition has even been uncovered in such flow recently 21,22 . The energy cascade is also different from that of the Newtonian counterpart 23 since some energy is rerouted to the polymer stretching dynamics, thereby reducing the formation of the smallest eddies and the associated viscous energy dissipation.However, this DR phenomenology is absent in steady laminar flows with polymers 9,16 due to absence of small time scale in the flow dynamics. Here, we show analytically that the laminar drag of a FENE-P fluid (the solution) in a cylindrical pipe exhibits a set of lower and upper bounds. Specifically, the laminar drag is lower than that of a Newtonian fluid with a viscosity matching that of the total viscosity of the solution, while being higher than that of the pure solvent. This previously una) This letter accepted by Physics of Fluids could be found after publication at https://aip.scitation.org/journal/phf reported phenomenon is due to the effective viscosity of the solution being bounded by the limits of that of two Newtonian flows: (i) the flow of pure solvent and (ii) that of a ...