Shape Dynamics and Rheology of Soft Elastic Particles in a Shear Flow

The shape dynamics of soft, elastic particles in an unbounded simple shear flow is investigated theoretically under Stokes flow conditions. Three types of motion—steady-state, trembling, and tumbling—are predicted, depending on the shear rate, elastic shear modulus, and initial particle shape. The steady-state motion is found to be always stable. In addition, the existence of a trembling regime is documented for the first time in nonvesicle systems, and a complete phase diagram is developed. The rheological properties of dilute suspensions of such soft particles generally exhibit shear-thinning behavior and can even display negative intrinsic viscosity for sufficiently soft particles.

Figure 1

Three types of particle motion under shear at $G=0.2$. The solid black lines represent the unsteady TR and TU motions for initially elliptical (2D) particles, which are computed at ${\omega }_0=0.8$ and ${\omega }_0=0.6$, respectively. The dashed black line represent the SS motion of an initially circular (${\omega }_0=1$) particle [11, 15]. The blue and green symbols are arbitrary Lagrangian-Eulerian finite-element method results (with $\mathrm{Re}=0.01$). Shape contours are shown sequentially at times indicated by black squares. The black dotted line is the particle’s major axis.

Figure 2

Typical time-dependent solutions for $\theta$ at $G=0.2$ in the (a) TR and (b) TU regimes.

Figure 3

(a) Snapshots of 2D shape contours at five sequential time instants for $G=0.2$, where the TR-TU transition happens at ${\omega }_0^{*}\sim 0.68$. The blue and red lines represent two material lines initially coinciding with the semimajor and semiminor axes of the particle, respectively, while the dashed black lines represent the “current” semimajor axis. (b) Bifurcation in $\theta$ near the TR-TU transition for a 2D particle. (c) Phase diagram for both a 2D and a 3D (initially prolate spheroidal) particle under shear. Insets in (c): Typical $\mathrm{D}\mathrm{\text{−}}\theta$ orbits for a 2D particle in the two regimes at $G=0.2$, $0.3\le {\omega }_0\le 0.9$. The arrows show the direction of decreasing ${\omega }_0$.

Figure 4

The intrinsic viscosity [$\eta$] of a dilute suspension of initially prolate spheroidal, neo-Hookean particles. (a) Time evolution of [$\eta$] in the three regimes (SS, TR, and TU) for $G=0.2$. The dotted black, dashed blue, and solid red lines denote results for ${\omega }_0=1$, 0.8, and 0.5, respectively. (b) Mean values of [$\eta$], as functions of $G$, for several values of ${\omega }_0$.