We investigate the steady-state extensional rheology of a dilute suspension of spherical particles in a dilute polymer solution modeled by the FENE-P constitutive relation. The ensemble-averaged suspension stress uses a recent construct of Koch et al. [Phys. Rev. Fluids 1, 013301 (2016)], based on perturbation for small polymer concentration and the generalized reciprocal theorem, to determine the polymers' influence on the particle stresslet and the particles' influence on the polymer stress. The extensional viscosity is defined as half of the constant of proportionality between the deviatoric stress and the imposed rate of strain tensor in uniaxial extensional flow. For a particle-free polymeric fluid, the extensional viscosity (nondimensionalized by the solvent viscosity) is $1+{\mu }^{\text{poly}}$, where ${\mu }^{\text{poly}}$ is the polymer contribution to the extensional viscosity. When a small volume fraction, $\varphi$, of spheres is added to a polymeric fluid, we find that the stress is altered by the Einstein viscosity of $2.5\varphi$ and two additional stress contributions: the polymer influence on the stresslet and the particle-induced polymer stress (PIPS). At lower Deborah numbers (defined as the product of extension rate and polymer relaxation time), $\mathrm{De}\lesssim 0.5$, the net interaction stress is positive, while it becomes negative at large De. Relative to undisturbed flow, the presence of spheres in uniaxial extensional flow creates regions of both larger and smaller local stretching. Below the coil-stretch transition, the polymers far from the particles are in a coiled state, while they are stretched more than their undisturbed state by large stretching regions around the particle. Due to their finite relaxation time, they also form a wake of stretched polymers downstream of the particle. This leads to a positive contribution to the suspension stress from both the stresslet (surface) and the PIPS (stretched wake). Beyond the coil-stretch transition, polymers far from the particle are highly stretched, but they collapse closer to the coiled state as they arrive at the low stretching regions near the particle surface. Therefore, a negative PIPS results from the regions of collapsed polymers. At sufficiently high Deborah numbers, $\mathrm{De}\gtrsim 1.5$, this region is very thin, and it becomes thinner and more intense upon further increasing De. For large maximum polymer extensibility, $L$, the particle-polymer contribution to the suspension rheology is independent of $L$ below the coil-stretch transition, whereas it scales as $L^2$ above the coil-stretch transition. When $\mathrm{De}\gtrsim 0.6$, the changes in extensional viscosity from the stresslet and the PIPS are $\varphi {\mu }^{\text{poly}}$ and approximately $-1.85\varphi {\mu }^{\text{poly}}$, respectively. At large De, the polymer extensional viscosity, ${\mu }^{\text{poly}}$, is orders of magnitude larger than that of Newtonian solvent. Hence, adding particles reduces the extensional viscosity of the suspension as $(2.5-0.85{\mu }^{\text{poly}})\varphi <0$.

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• Complex fluids
• Fluid-particle interactions
• Low Reynolds number flows
• Polymer conformation & topology
• Rheology
• Suspensions
• Viscoelasticity
• Perturbative methods
• Stokes equations