Document Type
Article
Publication Date
3-1-2016
Publication Title
Journal of Non-Newtonian Fluid Mechanics
Abstract
An analytic, asymptotic approximation of the nonlinear steady-state equations for viscoelastic creeping flow, modeled by the Oldroyd-B equations with polymer stress diffusion, is derived. Near the extensional stagnation point the flow stretches and aligns polymers along the outgoing streamlines of the stagnation point resulting in a stress-island, or birefringent strand. The polymer stress diffusion coefficient is used, both as an asymptotic parameter and a regularization parameter. The structure of the singular part of the polymer stress tensor is a Gaussian aligned with the incoming streamline of the stagnation point a smoothed δ-distribution whose width is proportional to the square-root of the diffusion coefficient. The amplitude of the stress island scales with the Wiessenberg number, and although singular in the limit of vanishing diffusion, it is integrable in the cross stream direction due to its vanishing width; this yields a convergent secondary flow. The leading order velocity response to this stress island is constructed and shown to be independent of the diffusion coefficient in the limit. The secondary circulation counteracts the forced flow and has a vorticity jump at the location of the stress islands, essentially expelling the background vorticity from the location of the birefringent strands. The analytic solutions are shown to be in excellent quantitative agreement with full numerical simulations, and therefore, the analytic solutions elucidate the salient mechanisms of the flow response to viscoelasticity and the mechanism for instability.
Keywords
Asymptotic analysis, Extensional flow, Stress diffusion, Viscoelastic creeping flow
Volume
229
First Page
101
Last Page
111
DOI
10.1016/j.jnnfm.2016.01.008
ISSN
03770257
Recommended Citation
A. Biello, Joseph and Thomases, Becca, "Equilibrium Circulation and Stress Distribution in Viscoelastic Creeping Flow" (2016). Mathematics Sciences: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/167
Comments
Peer reviewed accepted manuscript.