Journal ArticleExtensional rheology of dilute suspensions of spheres in polymeric liquidsSharma, Arjun (ORCID:0000000326635656); Koch, Donald L (ORCID:000000025474879X)<p>The extensional rheology of dilute suspensions of spheres in viscoelastic/polymeric liquids is studied computationally. At low polymer concentration<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline1.png' mime-subtype='png'/><tex-math>$c$</tex-math></alternatives></inline-formula>and Deborah number<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline2.png' mime-subtype='png'/><tex-math>$\textit{De}$</tex-math></alternatives></inline-formula>(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<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline3.png' mime-subtype='png'/><tex-math>$\Delta \textit{De}_{\textit{local}}\gt 0$</tex-math></alternatives></inline-formula>. This increases the suspension’s extensional viscosity with time and<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline4.png' mime-subtype='png'/><tex-math>$\textit{De}$</tex-math></alternatives></inline-formula>for<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline5.png' mime-subtype='png'/><tex-math>$De \lt 0.5$</tex-math></alternatives></inline-formula>. When<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline6.png' mime-subtype='png'/><tex-math>$\textit{De}$</tex-math></alternatives></inline-formula>exceeds 0.5, the coil-stretch transition value, the fully stretched polymers from the far-field collapse in regions with<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline7.png' mime-subtype='png'/><tex-math>$\Delta \textit{De}_{\textit{local}} \lt 0$</tex-math></alternatives></inline-formula>(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<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline8.png' mime-subtype='png'/><tex-math>$c$</tex-math></alternatives></inline-formula>. Highly stretched polymers impede local flow, reducing<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline9.png' mime-subtype='png'/><tex-math>$\Delta \textit{De}_{\textit{local}}$</tex-math></alternatives></inline-formula>, while<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline10.png' mime-subtype='png'/><tex-math>$\Delta \textit{De}_{\textit{local}}$</tex-math></alternatives></inline-formula>increases in regions with collapsed polymers. Initially, increasing<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline11.png' mime-subtype='png'/><tex-math>$c$</tex-math></alternatives></inline-formula>aligns<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline12.png' mime-subtype='png'/><tex-math>$\Delta \textit{De}_{\textit{local}}$</tex-math></alternatives></inline-formula>and local polymer stretch with far-field values, diminishing particle–polymer interaction effects. However, beyond a certain<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline13.png' mime-subtype='png'/><tex-math>$c$</tex-math></alternatives></inline-formula>, a new mechanism emerges. At low<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline14.png' mime-subtype='png'/><tex-math>$c$</tex-math></alternatives></inline-formula>, fluid three particle radii upstream exhibits<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline15.png' mime-subtype='png'/><tex-math>$\Delta \textit{De}_{\textit{local}} \gt 0$</tex-math></alternatives></inline-formula>, stretching polymers beyond their undisturbed state. As<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline16.png' mime-subtype='png'/><tex-math>$c$</tex-math></alternatives></inline-formula>increases, however,<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline17.png' mime-subtype='png'/><tex-math>$\Delta \textit{De}_{\textit{local}}$</tex-math></alternatives></inline-formula>in this region becomes negative, collapsing polymers and resulting in increasingly negative stress from particle–polymer interactions at large<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline18.png' mime-subtype='png'/><tex-math>$\textit{De}$</tex-math></alternatives></inline-formula>and time. At high<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline19.png' mime-subtype='png'/><tex-math>$c$</tex-math></alternatives></inline-formula>, this negative interaction stress scales as<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline20.png' mime-subtype='png'/><tex-math>$c^2$</tex-math></alternatives></inline-formula>, surpassing the linear increase of particle-free polymer stress, making dilute sphere concentrations more effective at reducing the viscosity of viscoelastic liquids at larger<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline21.png' mime-subtype='png'/><tex-math>$\textit{De}$</tex-math></alternatives></inline-formula>and<inline-formula><alternatives><inline-graphic href='S0022112025105302_inline22.png' mime-subtype='png'/><tex-math>$c$</tex-math></alternatives></inline-formula>.</p>Cambridge University Press2025-09-1010660094Journal of Fluid Mechanics10180022-1120https://doi.org/10.1017/jfm.2025.105302206851National Science Foundation