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

TL;DR

This study examines how adding spherical particles affects the steady-state extensional rheology of dilute polymer solutions, revealing complex stress interactions and viscosity changes depending on flow conditions and Deborah number.

Contribution

It provides a detailed analysis of particle-induced stress contributions and their dependence on flow parameters in dilute suspensions of spheres in polymer solutions.

Findings

Particle addition alters extensional viscosity through interaction stresslet and PIPS.

Stress contributions depend on Deborah number, switching from positive to negative.

Adding particles can reduce the extensional viscosity at high Deborah numbers.

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, $\phi$, of spheres is added to a polymeric fluid, the stress is altered by the Einstein viscosity of 2.5$\phi$ 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), $De\lesssim0.5$, 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 $De\gtrsim0.6$, the changes in extensional viscosity from the interaction stresslet and the PIPS are $\phi\mu^\text{poly}$ and approximately -1.85$\phi\mu^\text{poly}$, respectively. At large $De$, the polymer extensional viscosity, $\mu^\text{poly}$, is large. Therefore, adding particles reduces the extensional viscosity of the suspension ($(2.5-0.85\mu^\text{poly})\phi<0$).