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The extensional rheology of dilute suspensions of spheres in viscoelastic/polymeric liquids is studied computationally. At low polymer concentration$$c$$and Deborah number$$\textit{De}$$(imposed extension rate times polymer relaxation time), a wake of highly stretched polymers forms downstream of the particles due to larger local velocity gradients than the imposed flow, indicated by$$\Delta \textit{De}_{\textit{local}}\gt 0$$. This increases the suspension’s extensional viscosity with time and$$\textit{De}$$for$$De \lt 0.5$$. When$$\textit{De}$$exceeds 0.5, the coil-stretch transition value, the fully stretched polymers from the far-field collapse in regions with$$\Delta \textit{De}_{\textit{local}} \lt 0$$(lower velocity gradient) around the particle’s stagnation points, reducing suspension viscosity relative to the particle-free liquid. The interaction between local flow and polymers intensifies with increasing$$c$$. Highly stretched polymers impede local flow, reducing$$\Delta \textit{De}_{\textit{local}}$$, while$$\Delta \textit{De}_{\textit{local}}$$increases in regions with collapsed polymers. Initially, increasing$$c$$aligns$$\Delta \textit{De}_{\textit{local}}$$and local polymer stretch with far-field values, diminishing particle–polymer interaction effects. However, beyond a certain$$c$$, a new mechanism emerges. At low$$c$$, fluid three particle radii upstream exhibits$$\Delta \textit{De}_{\textit{local}} \gt 0$$, stretching polymers beyond their undisturbed state. As$$c$$increases, however,$$\Delta \textit{De}_{\textit{local}}$$in this region becomes negative, collapsing polymers and resulting in increasingly negative stress from particle–polymer interactions at large$$\textit{De}$$and time. At high$$c$$, this negative interaction stress scales as$$c^2$$, surpassing the linear increase of particle-free polymer stress, making dilute sphere concentrations more effective at reducing the viscosity of viscoelastic liquids at larger$$\textit{De}$$and$$c$$.
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On the non-self-adjoint and multiscale character of passive scalar mixing under laminar advection
Except in the trivial case of spatially uniform flow, the advection–diffusion operator of a passive scalar tracer is linear and non-self-adjoint. In this study, we exploit the linearity of the governing equation and present an analytical eigenfunction approach for computing solutions to the advection–diffusion equation in two dimensions given arbitrary initial conditions, and when the advecting flow field at any given time is a plane parallel shear flow. Our analysis illuminates the specific role that the non-self-adjointness of the linear operator plays in the solution behaviour, and highlights the multiscale nature of the scalar mixing problem given the explicit dependence of the eigenvalue–eigenfunction pairs on a multiscale parameter$$q=2{\rm i}k\,{\textit {Pe}}$$, where$$k$$is the non-dimensional wavenumber of the tracer in the streamwise direction, and$${\textit {Pe}}$$is the Péclet number. We complement our theoretical discussion on the spectra of the operator by computing solutions and analysing the effect of shear flow width on the scale-dependent scalar decay of tracer variance, and characterize the distinct self-similar dispersive processes that arise from the shear flow dispersion of an arbitrarily compact tracer concentration. Finally, we discuss limitations of the present approach and future directions.
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- Award ID(s):
- 1835640
- PAR ID:
- 10492409
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 973
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2023.748
