Assessment of Analytical Orientation Prediction Models for Suspensions Containing Fibers and Spheres

Analytical orientation models like the Folgar Tucker (FT) model are widely applied to predict the orientation of suspended non-spherical particles. The accuracy of these models depends on empirical model parameters. In this work, we assess how well analytical orientation models can predict the orientation of suspensions not only consisting of fibers but also of an additional second particle type in the shape of disks, which are varied in size and filling fraction. We mainly focus on the FT model, and we also compare its accuracy to more complex models like Reduced-Strain Closure model (RSC), Moldflow Rotational Diffusion model (MRD), and Anisotropic Rotary Diffusion model (ARD). In our work, we address the following questions. First, can the FT model predict the orientation of suspensions despite the additional particle phase affecting the rotation of the fibers? Second, is it possible to formulate an expression for the sole Folgar Tucker model parameter that is based on the suspension composition? Third, is there an advantage to choose more complex orientation prediction models that require the adjustment of additional model parameters?

Fiber Orientation Predictions—A Review of Existing Models

Fiber reinforced polymers are key materials across different industries. The manufacturing processes of those materials have typically strong impact on their final microstructure, which at the same time controls the mechanical performance of the part. A reliable virtual engineering design of fiber-reinforced polymers requires therefore considering the simulation of the process-induced microstructure. One relevant microstructure descriptor in fiber-reinforced polymers is the fiber orientation. This work focuses on the modeling of the fiber orientation phenomenon and presents a historical review of the different modelling approaches. In this context, the article describes different macroscopic fiber orientation models such as the Folgar-Tucker, nematic, reduced strain closure (RSC), retarding principal rate (RPR), anisotropic rotary diffusion (ARD), principal anisotropic rotary diffusion (pARD), and Moldflow rotary diffusion (MRD) model. We discuss briefly about closure approximations, which are a common mathematical element of those macroscopic fiber orientation models. In the last section, we introduce some micro-scale numerical methods for simulating the fiber orientation phenomenon, such as the discrete element method (DEM), the smoothed particle hydrodynamics (SPH) method and the moving particle semi-implicit (MPS) method.

Stress relaxation in a dilute bacterial suspension: the active–passive transition

This paper follows a recent article of Nambiar et al. (J. Fluid Mech., vol. 812, 2017, pp. 41–64) on the linear rheological response of a dilute bacterial suspension (e.g. E. coli) to impulsive starting and stopping of simple shear flow. Here, we analyse the time dependent nonlinear rheology for a pair of linear flows – simple shear (a canonical weak flow) and uniaxial extension (a canonical strong flow), again in response to impulsive initiation and cessation. The rheology is governed by the bacterium orientation distribution which satisfies a kinetic equation that includes rotation by the imposed flow, and relaxation to isotropy via rotary diffusion and tumbling. The relevant dimensionless parameters are the Péclet number $Pe\equiv \dot{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D70F}$, which dictates the importance of flow-induced orientation anisotropy, and $\unicode[STIX]{x1D70F}D_{r}$, which quantifies the relative importance of the two intrinsic orientation decorrelation mechanisms (tumbling and rotary diffusion). Here, $\unicode[STIX]{x1D70F}$ is the mean run duration of a bacterium that exhibits a run-and-tumble dynamics, $D_{r}$ is the intrinsic rotary diffusivity of the bacterium and $\dot{\unicode[STIX]{x1D6FE}}$ is the characteristic magnitude of the imposed velocity gradient. The solution of the kinetic equation is obtained numerically using a spectral Galerkin method, that yields the rheological properties (the shear viscosity, the first and second normal stress differences for simple shear, and the extensional viscosity for uniaxial extension) over the entire range of $Pe$. For simple shear, we find that the stress relaxation predicted by our analysis at small $Pe$ is in good agreement with the experimental observations of Lopez et al. (Phys. Rev. Lett., vol. 115, 2015, 028301). However, the analysis at large $Pe$ yields relaxations that are qualitatively different. Upon step initiation of shear, the rheological response in the experiments corresponds to a transition from a nearly isotropic suspension of active swimmers at small $Pe$, to an apparently (nearly) isotropic suspension of passive rods at large $Pe$. In contrast, the computations yield the expected transition to a nearly flow-aligned suspension of passive rigid rods at high $Pe$. We probe this active–passive transition systematically, complementing the numerical solution with analytical solutions obtained from perturbation expansions about appropriate base states. Our study suggests courses for future experimental and analytical studies that will help understand relaxation phenomena in active suspensions.

Stress relaxation in a dilute bacterial suspension

In this communication, we offer a theoretical explanation for the results of recent experiments that examine the stress response of a dilute suspension of bacteria (wild-type E. coli) subjected to step changes in the shear rate (Lopez et al., Phys. Rev. Lett., vol. 115, 2015, 028301). The observations include a regime of negative apparent shear viscosities. We start from a kinetic equation that describes the evolution of the single-bacterium orientation probability density under the competing effects of an induced anisotropy by the imposed shear, and a return to isotropy on account of stochastic relaxation mechanisms (run-and-tumble dynamics and rotary diffusion). We then obtain analytical predictions for the stress response, at leading order, of a dilute bacterial suspension subject to a weak but arbitrary time-dependent shear rate profile. While the predicted responses for a step-shear compare well with the experiments for typical choices of the microscopic parameters that characterize the swimming motion of a single bacterium, use of actual experimental values leads to significant discrepancies. The incorporation of a distribution of run times leads to a better agreement with observations.

Collective motion in a suspension of micro-swimmers that run-and-tumble and rotary diffuse

Recent experiments have shown that suspensions of swimming micro-organisms are characterized by complex dynamics involving enhanced swimming speeds, large-scale correlated motions and enhanced diffusivities of embedded tracer particles. Understanding this dynamics is of fundamental interest and also has relevance to biological systems. The observed collective dynamics has been interpreted as the onset of a hydrodynamic instability, of the quiescent isotropic state of pushers, swimmers with extensile force dipoles, above a critical threshold proportional to the swimmer concentration. In this work, we develop a particle-based model to simulate a suspension of hydrodynamically interacting rod-like swimmers to estimate this threshold. Unlike earlier simulations, the velocity disturbance field due to each swimmer is specified in terms of the intrinsic swimmer stress alone, as per viscous slender-body theory. This allows for a computationally efficient kinematic simulation where the interaction law between swimmers is knowna priori. The neglect of induced stresses is of secondary importance since the aforementioned instability arises solely due to the intrinsic swimmer force dipoles.Our kinematic simulations include, for the first time, intrinsic decorrelation mechanisms found in bacteria, such as tumbling and rotary diffusion. To begin with, we simulate so-called straight swimmers that lack intrinsic orientation decorrelation mechanisms, and a comparison with earlier results serves as a proof of principle. Next, we simulate suspensions of swimmers that tumble and undergo rotary diffusion, as a function of the swimmer number density$(n)$, and the intrinsic decorrelation time (the average duration between tumbles,${\it\tau}$, for tumblers, and the inverse of the rotary diffusivity,$D_{r}^{-1}$, for rotary diffusers). The simulations, as a function of the decorrelation time, are carried out with hydrodynamic interactions (between swimmers) turned off and on, and for both pushers and pullers (swimmers with contractile force dipoles). The ‘interactions-off’ simulations allow for a validation based on analytical expressions for the tracer diffusivity in the stable regime, and reveal a non-trivial box size dependence that arises with varying strength of the hydrodynamic interactions. The ‘interactions-on’ simulations lead us to our main finding: the existence of a box-size-independent parameter that characterizes the onset of instability in a pusher suspension, and is given by$nUL^{2}{\it\tau}$for tumblers and$nUL^{2}/D_{r}$for rotary diffusers; here,$U$and$L$are the swimming speed and swimmer length, respectively. The instability manifests as a bifurcation of the tracer diffusivity curves, in pusher and puller suspensions, for values of the above dimensionless parameters exceeding a critical threshold.

Stochastic dynamics of active swimmers in linear flows

Most classical work on the hydrodynamics of low-Reynolds-number swimming addresses deterministic locomotion in quiescent environments. Thermal fluctuations in fluids are known to lead to a Brownian loss of the swimming direction, resulting in a transition from short-time ballistic dynamics to effective long-time diffusion. As most cells or synthetic swimmers are immersed in external flows, we consider theoretically in this paper the stochastic dynamics of a model active particle (a self-propelled sphere) in a steady general linear flow. The stochasticity arises both from translational diffusion in physical space, and from a combination of rotary diffusion and so-called run-and-tumble dynamics in orientation space. The latter process characterizes the manner in which the orientation of many bacteria decorrelates during their swimming motion. In contrast to rotary diffusion, the decorrelation occurs by means of large and impulsive jumps in orientation (tumbles) governed by a Poisson process. We begin by deriving a general formulation for all components of the long-time mean square displacement tensor for a swimmer with a time-dependent swimming velocity and whose orientation decorrelates due to rotary diffusion alone. This general framework is applied to obtain the convectively enhanced mean-squared displacements of a steadily swimming particle in three canonical linear flows (extension, simple shear and solid-body rotation). We then show how to extend our results to the case where the swimmer orientation also decorrelates on account of run-and-tumble dynamics. Self-propulsion in general leads to the same long-time temporal scalings as for passive particles in linear flows but with increased coefficients. In the particular case of solid-body rotation, the effective long-time diffusion is the same as that in a quiescent fluid, and we clarify the lack of flow dependence by briefly examining the dynamics in elliptic linear flows. By comparing the new active terms with those obtained for passive particles we see that swimming can lead to an enhancement of the mean-square displacements by orders of magnitude, and could be relevant for biological organisms or synthetic swimming devices in fluctuating environmental or biological flows.

Observations of fibre orientation in simple shear flow of semi-dilute suspensions

The orientations of fibres in a semi-dilute, index-of-refraction-matched suspension in a Newtonian fluid were observed in a cylindrical Couette device. Even at the highest concentration (nL3 = 45), the particles rotated around the vorticity axis, spending most of their time nearly aligned in the flow direction as they would do in a Jeffery orbit. The measured orbit-constant distributions were quite different from the dilute orbit-constant distributions measured by Anczurowski & Mason (1967b) and were described well by an anisotropic, weak rotary diffusion. The measured ϕ-distributions were found to be similar to Jeffery's solution. Here, ϕ is the meridian angle in the flow-gradient plane. The shear viscosities measured by Bibbo (1987) compared well with the values predicted by Shaqfeh & Fredrickson's theory (1990) using moments of the orientation distribution measured here.