Michael Cromer

Concentration fluctuations in polymer solutions under extensional flow

In this work, we extend the classical analysis of concentration fluctuations in polymer solutions under shear flow to consider the same phenomenology under extensional flow. Experimental work by van Egmond and Fuller [Macromolecules 26, 7182–7188 (1993)] revealed a four-lobe scattering pattern for a polystyrene solution in a planar extensional flow field. Similar to earlier results found in shear, they find the existence of finite-wavelength peak intensity locations. To investigate this phenomenon, we couple stress and concentration using a two-fluid model with fluctuations driven by thermal noise incorporated through a canonical Langevin approach. The polymer stress is governed by the Rolie–Poly model augmented with finite extensibility to account for large stretching of chains at high Weissenberg numbers. Perturbing the equations about homogeneous planar extensional flow for weak amplitude inhomogeneities, but arbitrary flow strength, we solve for the steady correlations. The resulting structure factor ...

Shear banding predictions for the two-fluid Rolie-Poly model

This paper continues our recent studies of the flow behavior of a two-fluid Rolie-Poly approximation for entangled polymer solutions. The model studied is similar to that used in our previous work, but now incorporates isotropic elastic contributions to the stress, as required for thermodynamic consistency. These contributions play no role in the dynamics of a single phase, incompressible polymer fluid. However, in the two-fluid model, each fluid phase, the polymer and the solvent, is compressible, and the isotropic elastic contribution to the stress cannot be neglected. We show that this change in the model leads to the prediction of a high Weissenberg number linear instability in simple shear flow, in addition to the linear instability at lower Weissenberg numbers that was identified in our preceding study. We then consider the dynamics of the full nonlinear system for both linear shear flow and the Taylor–Couette geometry. The linear shear flow solutions are used primarily to explore the details of the flow that develops from the high Weissenberg number instability. The Taylor–Couette geometry is studied for gap widths that are wide enough to allow detailed experimental measurements, and one primary focus is then to follow the transient evolution of the flow. We show that the time to reach steady state becomes very long as the gap width increases. However, the velocity and concentration distributions both show significant changes and banded structures for much smaller times, thus suggesting that the Taylor–Couette geometry is a practical system for experimental studies and can be directly compared with the present theory.

A study of pressure-driven flow of wormlike micellar solutions through a converging/diverging channel

The pressure-driven flow of a macroscale network model, which was derived from mescoscale network considerations and was developed to describe wormlike micellar mixtures, the Vasquez, Cook, McKinley (VCM) model [Vasquez et al., J. Non-Newtonian Fluid Mech. 144, 122–139 (2007)], is investigated in a periodically varying, converging/diverging channel. The VCM model consists of a coupled system of nonlinear partial differential equations describing two elastically active breaking and reforming micellar species. Previous analysis of the VCM model has shown that it captures both purely shear and purely extensional flow behaviors of wormlike micellar solutions. In this paper, the combined effects of shear and extension are probed by simulating the flow in a converging/diverging channel. Using domain perturbation analysis, the model equations are perturbed about the straight channel geometry thereby describing the flow through the perturbed, wavy wall geometry. The resulting first order perturbed system of (line...

Concentration fluctuations in polymer solutions under mixed flow

In this work, we extend the classical analysis of concentration fluctuations in polymer solutions under shear flow to consider the same phenomenology under mixed (shear + extensional) flows. To investigate this phenomenon, we couple stress and concentration using a two-fluid model with fluctuations driven by thermal noise incorporated through a canonical Langevin approach. The polymer stress is governed by the Rolie-Poly model augmented with finite extensibility to account for large stretching of chains at high Weissenberg numbers. Perturbing the equations about homogeneous flow for weak amplitude inhomogeneities, but arbitrary flow strength, we solve for the steady state structure factor (Fourier transformed pair correlation function) under general linear flows using a unique method of characteristics solver. Under shear flow, the model predicts butterfly patterns in accord with previous experimental and theoretical work, including a full rotation of peaks past the flow axis. In addition, the magnitude o...