Nonlinear Rheology of FENE Dumbbell with Friction-Reduction: Analysis of Brownian Force Intensity through Comparison of Extensional and Shear Viscosities

Abstract

Normalized extensional and shear viscosities, ηE/ηE0 and η/η0 with the subscript “0” standing for quantities in the linear viscoelastic (LVE) regime, are theoretically analyzed for the FENE-dumbbell in the presence of isotropic friction-reduction under flow. The basic parameters in the analysis are the friction reduction factor $.$ FENE factor rK = Kf Keq, and the Brownian force intensity factor rB = Bsf/Beq, with (Formula presented) and k being the bead friction coefficient and spring strength of the dumbbell, B denoting the intensity of the Brownian force acting on the bead, and the subscripts “sf” and “eq” standing for the quantities under steady flow and at equilibrium, respectively: (Formula presented) with kBT being the thermal energy, according to the fluctuation-dissipation theorem valid at equilibrium. The normalized viscosities were found to be analytically expressed as (Formula presented) with Wi being the Weissenberg number defined with respect to the LVE relaxation time. Both nE/nE0 and n/n0 are proportional to the rB factor, so that comparison of these quantities allows us to characterize the intensity of the Brownian force Bsf under flow. For example, for a test case of power-law decay of r( with Wi, the above expressions of nE/nE0 and n/n0 suggested a relationship (Formula presented) failure of the fluctuation-dissipation theorem) under fast flow given that nE/nE0 and n/n0 at high Wi are proportional to each other and exhibit the same power-law thinning behavior,(Formula presented) Thus, the analysis presented in this work offers an experimental route for specifying the Brownian force intensity under flow through direct comparison of the nE/nE0 and n/n0 data.