Calculation of the effect of polymer additive in a converging flow

Abstract

The conical-channel flow of a dilute polymer solution is investigated theoretically. The stress field due to polymer additive is calculated using a new molecular model, based on the physical picture of the polymer molecules unravelling in strong flows and Batchelor's theory for the stress in a suspension of elongated particles. Good agreement is obtained with the experimental results of James & Saringer (1980). The absence of a significant polymer effect in a two-dimensional case (the wedge-channel flow), observed by the same authors (James & Saringer 1982a), is also explained. The fundamental differences between the proposed model and the elastic-dumbbell models are discussed.

Drag Reduction by Polymer Additives

“…Rg Pe AM; (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) here n is the number of nodes in a branched polymer or tethered surface, and a' is proportional to m 3 '/ where m is the average number of monomers per nodes (inn ; N). The exponent, v, is approximately, 0.5 for branched polymers [10], and 0.4 for polymerized surfaces [9].…”

“…Ryskin [14] describes a "yo-yo" model for affine polymer deformation in extensional flow. It appears to satisfactorily describe convergent flow in a cone.…”

“…This description is peculiar and of dubious applicability because cone flow is strongly convergent and has a large (negative) divergence at its apex, while incompressible pipe flow and turbulence are divergence-free. Also, a given molecule is simultaneously subjected to turbulent eddies of all sizes, as pointed out by Ryskin [14]. Having made these assumptions, they define a new inner scale t1* for the turbulent flow field, by equating the polymer elastic energy density to the Reynolds stress of the turbulence.…”

“…7 (3 -17) In addition to streaming, stretching and diffusive terms similar to those appearing in Equation (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15), there is a thermal forcing contribution (4kBT/7 )6Sj. This term balances against the Zimm relaxation term -2w.Qij in a uniform quiescent fluid to give the expected equilibrium solution of (3-17), namely Qi = ik R26 3 (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18) corresponding to an isotropic distribution of polymers with size R,.…”

Effect of initial conformation, flow strength, and hydrodynamic interaction on polymer molecules in extensional flows

“…Neglecting Brownian motion at high strain rates, Rallison and Hinch 14 observed segmental alignment and multiple fold formation to a highly kinked state, leading to viscous stresses. Ryskin 15 suggested preferential extension of the chain center in the initial stages. But Wiest, Wedgewood, and Bird 16 predicted nearly simultaneous orientation of all segments in the initial stages, which was followed by unfolding of the chains.…”