^{1,a)}and Tetsu Uesaka

^{1}

### Abstract

Particle-level simulations are performed to study semidilute suspensions of monodispersed non-Brownian fibers in shear flow, with a Newtonian fluid medium. The incompressible three-dimensional Navier-Stokes equations are used to describe the motion of the medium, while fibers are modeled as chains of fiber segments, interacting with the fluid through viscous drag forces. The two-way coupling between the solids and the fluid phase is taken into account by enforcing momentum conservation. The model includes long-range and short-range hydrodynamic fiber-fiber interactions, as well as mechanical interactions. The simulations rendered the time-dependent fiber orientation distribution, whose time average was found to agree with experimental data in the literature. The viscosity and first normal stress difference was calculated from the orientation distribution using the slender body theory of Batchelor [J. Fluid Mech.46, 813 (1971)], with corrections for the finite fiber aspect ratios. The viscosity was also obtained from direct computation of the shear stresses of the suspension for comparison. These two types of predictions compared well in the semidilute regime. At higher concentrations, however, a discrepancy was seen, most likely due to mechanical interactions, which are only accounted for in the direct computation method. The simulated viscosity determined directly from shear stresses was in fair agreement with experimental data found in the literature. The first normal stress difference was found to be proportional to the square of the volume concentration of fibers in the semidilute regime. As concentrations approached the concentrated regime, the first normal stress difference became proportional to volume concentration. It was also found that the coefficient of friction has a strong influence on the tendency for flocculation as well as the apparent viscosity of the suspension in the semidilute regime.

The reviewer is acknowledged for his/her important suggestion to examine the effects of the coefficient of friction and the contact force multiplier.

I. INTRODUCTION

II. COMPUTATIONAL METHOD

A. Coordinate system and domain of computations

B. Fluid model

C. Fiber model

D. Fiber-fluid interactions

E. Fiber-fiber interactions

III. SIMULATIONS

A. Initial condition and boundary conditions

B. Fiber orientation distribution

C. Rheological properties

IV. EFFECTS OF CONTACT FORCE PARAMETERS

V. CONCLUSIONS

### Key Topics

- Suspensions
- 58.0
- Shear rate dependent viscosity
- 41.0
- Viscosity
- 30.0
- Hydrodynamics
- 19.0
- Hydrological modeling
- 17.0

## Figures

Fibers are modeled as chains of fiber segments. Each segment, indexed , has a diameter , a length , a position vector , a direction vector , and an up-direction vector , all given with respect to .

Fibers are modeled as chains of fiber segments. Each segment, indexed , has a diameter , a length , a position vector , a direction vector , and an up-direction vector , all given with respect to .

Fiber-fluid interactions. A fiber segment ① disturbs the velocity field of a region of fluid ② at length scales equal to or smaller than the fiber segment length, by first-order approximation. The cell size of the three-dimensional spatial discretization ③ of the fluid flow solver is equal to the segment length, in order to capture larger length scales of the flow.

Fiber-fluid interactions. A fiber segment ① disturbs the velocity field of a region of fluid ② at length scales equal to or smaller than the fiber segment length, by first-order approximation. The cell size of the three-dimensional spatial discretization ③ of the fluid flow solver is equal to the segment length, in order to capture larger length scales of the flow.

Definition of coordinate system and orientation angles and . The orientation vector of a fiber traces a path on the unit sphere, characterized by the orbit constant , when the fiber is subjected to shear flow.

Definition of coordinate system and orientation angles and . The orientation vector of a fiber traces a path on the unit sphere, characterized by the orbit constant , when the fiber is subjected to shear flow.

The distribution of fibers with aspect ratio in shear flow. The solid line represents the theoretical prediction of Jeffery (Ref. 1) for isolated fibers. The bars indicate the computed distribution for . Hydrodynamic interactions are responsible for the discrepancy between Jeffery’s prediction and simulation results. Note the small asymmetry of the simulated distribution, whose mean value is slightly shifted below .

The distribution of fibers with aspect ratio in shear flow. The solid line represents the theoretical prediction of Jeffery (Ref. 1) for isolated fibers. The bars indicate the computed distribution for . Hydrodynamic interactions are responsible for the discrepancy between Jeffery’s prediction and simulation results. Note the small asymmetry of the simulated distribution, whose mean value is slightly shifted below .

The mean angle of the distribution as a function of dimensionless concentration . The × signs represent simulation results for . The triangles and squares denote experimental values obtained by Stover *et al.* (Ref. 24) for and , respectively. The error bars indicate an 80% confidence interval.

The mean angle of the distribution as a function of dimensionless concentration . The × signs represent simulation results for . The triangles and squares denote experimental values obtained by Stover *et al.* (Ref. 24) for and , respectively. The error bars indicate an 80% confidence interval.

The standard deviation of the distribution plotted as a function of volume concentration. When the concentration increases, the distribution becomes narrower. The standard deviation predicted by Eq. (20) for isolated fibers is 0.302. Due to the relatively small sample size ( fibers), stochastic events may cause anomalies, as observed for .

The standard deviation of the distribution plotted as a function of volume concentration. When the concentration increases, the distribution becomes narrower. The standard deviation predicted by Eq. (20) for isolated fibers is 0.302. Due to the relatively small sample size ( fibers), stochastic events may cause anomalies, as observed for .

The distribution measured by Stover *et al.* (Ref. 24), for and (triangles), and the distribution obtained from simulations with the same parameters (solid line).

The distribution measured by Stover *et al.* (Ref. 24), for and (triangles), and the distribution obtained from simulations with the same parameters (solid line).

The mean orbit constant as a function of volume concentration. Each symbol denotes an experiment or simulation with a constant fiber aspect ratio : Anczurowski and Mason (Ref. 27), , circles; Stover *et al.* (Ref. 24), , up triangles; Stover *et al.*, , down triangles; Petrich *et al.* (Ref. 13), , squares; Petrich *et al.*, , diamonds; simulations, , × signs; simulations, , + signs. Rahnama *et al.* (Ref. 28) criticized the anomalous results of Anczurowski and Mason, providing experimental and theoretical evidences that they may be wrong.

The mean orbit constant as a function of volume concentration. Each symbol denotes an experiment or simulation with a constant fiber aspect ratio : Anczurowski and Mason (Ref. 27), , circles; Stover *et al.* (Ref. 24), , up triangles; Stover *et al.*, , down triangles; Petrich *et al.* (Ref. 13), , squares; Petrich *et al.*, , diamonds; simulations, , × signs; simulations, , + signs. Rahnama *et al.* (Ref. 28) criticized the anomalous results of Anczurowski and Mason, providing experimental and theoretical evidences that they may be wrong.

The mean orbit constant as a function of fiber aspect ratio. Each data point is an average taken over several experiments performed at different volume concentrations in the semidilute and the concentrated regimes. The symbols denote, in turn: Stover *et al.* (Ref. 24), triangles; Petrich *et al.* (Ref. 13), squares; simulations, × signs; isotropic distribution valid for spheres, + signs. A second degree polynomial fit has been included for guidance (solid line).

The mean orbit constant as a function of fiber aspect ratio. Each data point is an average taken over several experiments performed at different volume concentrations in the semidilute and the concentrated regimes. The symbols denote, in turn: Stover *et al.* (Ref. 24), triangles; Petrich *et al.* (Ref. 13), squares; simulations, × signs; isotropic distribution valid for spheres, + signs. A second degree polynomial fit has been included for guidance (solid line).

By summing all forces acting in the region , on the fibers intersecting , it is possible to obtain the fiber contribution to the shear stress in that section. Both fiber-fluid and fiber-fiber interaction forces are taken into account. We might as well have chosen to sum over the complement half-space by virtue of the force balance.

By summing all forces acting in the region , on the fibers intersecting , it is possible to obtain the fiber contribution to the shear stress in that section. Both fiber-fluid and fiber-fiber interaction forces are taken into account. We might as well have chosen to sum over the complement half-space by virtue of the force balance.

(a) Geometry of the viscometer used by Blakeney (Ref. 15). The outer cylinder was stationary, while the inner cylinder was kept at a constant angular velocity by applying torque on the attached cord. The radius of the outer cylinder was varied, and the geometry of the inner cylinder was fixed (; ). (b) Geometry of the viscometer used by Bibbo (Ref. 12). All geometry parameters were varied. A force transducer attached to the bottom rod was used to measure the shear forces.

(a) Geometry of the viscometer used by Blakeney (Ref. 15). The outer cylinder was stationary, while the inner cylinder was kept at a constant angular velocity by applying torque on the attached cord. The radius of the outer cylinder was varied, and the geometry of the inner cylinder was fixed (; ). (b) Geometry of the viscometer used by Bibbo (Ref. 12). All geometry parameters were varied. A force transducer attached to the bottom rod was used to measure the shear forces.

The steady-state specific viscosity as a function of volume concentration for straight rigid fibers in shear flow. The white symbols denote the experimental data of Blakeney (Ref. 15), and the black symbols denote the experimental data of Bibbo (Ref. 12). The + signs denote orientation distribution viscosity, and the × signs denote the shear stress viscosity of the simulations. The latter has been plotted as a function of the *computed* volume concentration in the sampled region. See Table I for details about the experimental conditions in each case. The solid line represents the theoretical prediction for a suspension of fibers with , where all fibers have an orbit constant , and the distribution is given by Eq. (20).

The steady-state specific viscosity as a function of volume concentration for straight rigid fibers in shear flow. The white symbols denote the experimental data of Blakeney (Ref. 15), and the black symbols denote the experimental data of Bibbo (Ref. 12). The + signs denote orientation distribution viscosity, and the × signs denote the shear stress viscosity of the simulations. The latter has been plotted as a function of the *computed* volume concentration in the sampled region. See Table I for details about the experimental conditions in each case. The solid line represents the theoretical prediction for a suspension of fibers with , where all fibers have an orbit constant , and the distribution is given by Eq. (20).

The computed mean value as a function of for fibers of aspect ratio . The dashed line indicates the upper limit of the semidilute regime .

The computed mean value as a function of for fibers of aspect ratio . The dashed line indicates the upper limit of the semidilute regime .

The normalized steady-state first normal stress difference is plotted as a function of volume concentration and fiber aspect ratio. In a formula suggested by Carter,^{32} the expressions on the axes should be proportional. The slope of the solid line indicates proportionality and has been included for guidance. Experimental data from the literature are denoted by squares[, Carter (Ref. 32)] and triangles [, Petrich *et al.* (Ref. 13)]. Simulation results are denoted by × signs .

The normalized steady-state first normal stress difference is plotted as a function of volume concentration and fiber aspect ratio. In a formula suggested by Carter,^{32} the expressions on the axes should be proportional. The slope of the solid line indicates proportionality and has been included for guidance. Experimental data from the literature are denoted by squares[, Carter (Ref. 32)] and triangles [, Petrich *et al.* (Ref. 13)]. Simulation results are denoted by × signs .

The simulated specific viscosity as measured from the shear stresses as a function of dimensionless concentration for different coefficients of friction: , circles; , triangles; , squares; , diamonds. The fiber aspect ratio was . Each data point represents the mean of three simulations.

The simulated specific viscosity as measured from the shear stresses as a function of dimensionless concentration for different coefficients of friction: , circles; , triangles; , squares; , diamonds. The fiber aspect ratio was . Each data point represents the mean of three simulations.

A comparison of the evolution of the fiber configuration with the amount of shear , in two experimental instances for different coefficients of friction . The fiber aspect ratio was , the volume concentration was , and the initial conditions were the same in both instances. Friction is one cause of the development of fiber flocs in the semidilute regime.

A comparison of the evolution of the fiber configuration with the amount of shear , in two experimental instances for different coefficients of friction . The fiber aspect ratio was , the volume concentration was , and the initial conditions were the same in both instances. Friction is one cause of the development of fiber flocs in the semidilute regime.

The simulated specific viscosity as measured from the shear stresses as a function of dimensionless concentration for different contact force multiplier values: , circles; , triangles; , squares. Each data point represents the mean of three simulations.

The simulated specific viscosity as measured from the shear stresses as a function of dimensionless concentration for different contact force multiplier values: , circles; , triangles; , squares. Each data point represents the mean of three simulations.

## Tables

Parameter sets used in the viscosity measurements of Blakeney (Ref. 15) and Bibbo (Ref. 12) for straight fibers, and the parameters used in the computer simulations. The symbol for each set of parameters is used in Fig. 12 for plotting the results.

Parameter sets used in the viscosity measurements of Blakeney (Ref. 15) and Bibbo (Ref. 12) for straight fibers, and the parameters used in the computer simulations. The symbol for each set of parameters is used in Fig. 12 for plotting the results.

Parameter sets used in the first normal stress difference measurements of Carter (Ref. 32) and Petrich *et al.* (Ref. 13) for straight rigid fibers, and the parameters used in the computer simulations. The symbol for each set of parameters is used in Fig. 14 for plotting the results.

Parameter sets used in the first normal stress difference measurements of Carter (Ref. 32) and Petrich *et al.* (Ref. 13) for straight rigid fibers, and the parameters used in the computer simulations. The symbol for each set of parameters is used in Fig. 14 for plotting the results.

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