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The motion of a freely rotating prolate spheroid in a simple shear flow of a dilute polymeric solution is examined in the limit of large particle aspect ratio,$$\kappa$$. A regular perturbation expansion in the polymer concentration,$$c$$, a generalized reciprocal theorem, and slender body theory to represent the velocity field of a Newtonian fluid around the spheroid are used to obtain the$$O(c)$$correction to the particle's orientational dynamics. The resulting dynamical system predicts a range of orientational behaviours qualitatively dependent upon$$c\, De$$($$De$$is the imposed shear rate times the polymer relaxation time) and$$\kappa$$and quantitatively on$$c$$. At a small but finite$$c\, De$$, the particle spirals towards a limit cycle near the vorticity axis for all initial conditions. Upon increasing$$\kappa$$, the limit cycle becomes smaller. Thus, ultimately the particle undergoes a periodic motion around and at a small angle from the vorticity axis. At moderate$$c\, De$$, a particle starting near the flow–gradient plane departs it monotonically instead of spirally, as this plane (a limit cycle at smaller$$c\, De$$) obtains a saddle and an unstable node. The former is close to the flow direction. Upon further increasing$$c\, De$$, the saddle node changes to a stable node. Therefore, depending upon the initial condition, a particle may either approach a periodic orbit near the vorticity axis or obtain a stable orientation near the flow direction. Upon further increasing$$c\, De$$, the limit cycle near the vorticity axis vanishes, and the particle aligns with the flow direction for all starting orientations.
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This content will become publicly available on September 10, 2026
Extensional rheology of dilute suspensions of spheres in polymeric liquids
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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- Award ID(s):
- 2206851
- PAR ID:
- 10660094
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 1018
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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