Steady state extensional rheology of a dilute suspension of spheres in a dilute polymer solution

Abstract

We investigate the steady-state extensional rheology of a dilute suspension of spherical particles in a dilute polymer solution. For a particle-free polymeric fluid, in addition to the solvent viscosity, the extensional viscosity due to the polymers, ${\mu }^{\text{poly}}$, contributes to the total non-dimensionalized extensional viscosity $1+{\mu }^{\text{poly}}$. When a small volume fraction, $\varphi$, of spheres is added to a polymeric fluid, the stress is altered by the Einstein viscosity of 2.5$\varphi$ and two additional stress contributions: the interaction stresslet and the particle-induced polymer stress (PIPS). The net interaction stress is positive at lower Deborah numbers (product of extension rate and polymer relaxation time), $De\lesssim 0.5$, and negative at large $De$. Relative to undisturbed flow, the presence of spheres in uniaxial extensional flow creates larger and smaller local stretching regions. Below the coil-stretch transition ($De<0.5$), the polymers far from the particles are in a coiled state, and a wake of stretched polymers forms downstream of the particle as they are stretched by the large stretching regions around the particle. This leads to a positive interaction 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. When $De\gtrsim 0.6$, the changes in extensional viscosity from the interaction 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 large. Therefore, adding particles reduces the extensional viscosity of the suspension ($(2.5-0.85{\mu }^{\text{poly}})\varphi <0$).