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.
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 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 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 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.
(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 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 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.
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.
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.
Article metrics loading...
Full text loading...