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The motion of a freely rotating prolate spheroid in a simple shear flow of a dilute polymeric solution is examined in the limit of large particle aspect ratio,$$\kappa$$. A regular perturbation expansion in the polymer concentration,$$c$$, a generalized reciprocal theorem, and slender body theory to represent the velocity field of a Newtonian fluid around the spheroid are used to obtain the$$O(c)$$correction to the particle's orientational dynamics. The resulting dynamical system predicts a range of orientational behaviours qualitatively dependent upon$$c\, De$$($$De$$is the imposed shear rate times the polymer relaxation time) and$$\kappa$$and quantitatively on$$c$$. At a small but finite$$c\, De$$, the particle spirals towards a limit cycle near the vorticity axis for all initial conditions. Upon increasing$$\kappa$$, the limit cycle becomes smaller. Thus, ultimately the particle undergoes a periodic motion around and at a small angle from the vorticity axis. At moderate$$c\, De$$, a particle starting near the flow–gradient plane departs it monotonically instead of spirally, as this plane (a limit cycle at smaller$$c\, De$$) obtains a saddle and an unstable node. The former is close to the flow direction. Upon further increasing$$c\, De$$, the saddle node changes to a stable node. Therefore, depending upon the initial condition, a particle may either approach a periodic orbit near the vorticity axis or obtain a stable orientation near the flow direction. Upon further increasing$$c\, De$$, the limit cycle near the vorticity axis vanishes, and the particle aligns with the flow direction for all starting orientations. 

A finite difference scheme is used to develop a numerical method to solve the flow of an unbounded viscoelastic fluid with zero to moderate inertia around a prolate spheroidal particle. The equations are written in prolate spheroidal coordinates, and the shape of the particle is exactly resolved as one of the coordinate surfaces representing the inner boundary of the computational domain. As the prolate spheroidal grid is naturally clustered near the particle surface, good resolution is obtained in the regions where the gradients of relevant flow variables are most significant. This coordinate system also allows large domain sizes with a reasonable number of mesh points to simulate unbounded fluid around a particle. Changing the aspect ratio of the inner computational boundary enables simulations of different particle shapes ranging from a sphere to a slender fiber. Numerical studies of the latter particle shape allow testing of slender body theories. The mass and momentum equations are solved with a Schur complement approach allowing us to solve the zero inertia case necessary to isolate the viscoelastic effects. The singularities associated with the coordinate system are overcome using L’Hopital’s rule. A straightforward imposition of conditions representing a time-varying combination of linear flows on the outer boundary allows us to study various flows with the same computational domain geometry. For the special but important case of zero fluid and particle inertia we obtain a novel formulation that satisfies the force- and torque-free constraint in an iteration-free manner. The numerical method is demonstrated for various flows of Newtonian and viscoelastic fluids around spheres and spheroids (including those with large aspect ratio). Good agreement is demonstrated with existing theoretical and numerical results. 

The rheology of suspensions of rings (tori) rotating in an unbounded low Reynolds number simple shear flow is calculated using numerical simulations at dilute particle number densities ( n ≪ 1 ). Suspensions of non-Brownian rings are studied by computing pair interactions that include hydrodynamic interactions modeled using slender body theory and particle collisions modeled using a short-range repulsive force. Particle contact and hydrodynamic interactions were found to have comparable influences on the steady-state Jeffery orbit distribution. The average tilt of the ring away from the flow-vorticity plane increased during pairwise interactions compared to the tilt associated with Jeffery rotation and the steady-state orbit distribution. Particle stresses associated with the increased tilt during the interaction were found to be comparable to the stresses induced directly by particle contact forces and the hydrodynamic velocity disturbances of other particles. The hydrodynamic diffusivity coefficients in the gradient and vorticity directions were also obtained and were found to be two orders of magnitude larger than the corresponding values in fiber suspensions at the same particle concentrations. Rotary Brownian dynamics simulations of isolated Brownian rings were used to understand the shear rate dependence of suspension rheology. The orbit distribution observed in the regime of weak Brownian motion, P e ≫ ϕ T − 3, was surprisingly similar to that obtained from pairwise interaction calculations of non-Brownian rings. Here, the Peclet number P e is the ratio of the shear rate and the rotary diffusivity of the particle and ϕ T is the effective inverse-aspect ratio of the particle (approximately equal to 2 π times the inverse of its non-dimensional Jeffery time period). Thus, the rheology results obtained from pairwise interactions should retain accuracy even for weakly Brownian rings ( n ≪ 1 and ϕ T − 3 ≪ P e ).