^{1}, Donald L. Koch

^{1}and Ganesh Subramanian

^{2}

### Abstract

Many-fiber simulations and a linear stability analysis are used to explore the structure and dynamics that arise in a dilute suspension of sedimenting slender fibers with finite particle Reynolds numbers. Dynamic simulations based on a slender-body treatment of the fibers coupled with a pseudospectral solution of the Navier–Stokes equations reveal an inhomogeneous structure dominated by the largest wavelength that fits in the periodic simulation cell. This structure becomes weaker with increasing fiber Reynolds number and fiber concentration. A linear stability analysis shows that the stability of the homogeneous state of the suspension is determined by the direction of the horizontal migration of a fiber in a weak shear field with vertical streamlines produced by a perturbation to the fiber number density. The lift force on a settling fiber, in a plane transverse to gravity, has two contributions. The first contribution results from a broken symmetry in the presence of shear at finite Reynolds number, and involves the coupled effects of the shear and translational inertial terms. The second contribution is related to the sedimentation-driven drift of an inclined fiber, and is present even in the Stokes limit. Both contributions act to push fibers toward downward flowing, high density regions, and the net transverse drift is therefore of a destabilizing nature. The drift calculation and its implications for the instability of a homogeneous suspension are explored up to a Reynolds number of 10.6. The structure factor, fluid velocity fluctuations, and deviations of the fiber orientation away from the horizontal plane are found to generally decrease with increasing Reynolds number as a result of the increasing dominance of the inertial torque acting to rotate settling fibers toward the horizontal plane.

I. INTRODUCTION

II. SIMULATION METHOD

A. Slender-body theory

B. Pseudospectral solution of the outer velocity field

III. CROSS-STREAM MIGRATION OF A SETTLING FIBER IN A VERTICAL SHEAR FLOW

IV. LINEAR STABILITY ANALYSIS

V. SIMULATION RESULTS AND DISCUSSION

A. Mean settling velocity

B. Inhomogeneous spatial structure

C. Orientation distribution

D. Particle and fluid velocity fluctuations

VI. CONCLUSIONS

### Key Topics

- Suspensions
- 101.0
- Reynolds stress modeling
- 100.0
- Torque
- 55.0
- Sedimentation
- 45.0
- Shear flows
- 42.0

## Figures

Schematic of a suspension of fibers in a periodic unit cell. The superscript indicates the index of the among fibers.

Schematic of a suspension of fibers in a periodic unit cell. The superscript indicates the index of the among fibers.

The figure shows the variation in the zeroth and first moment coefficients of the fiber force distribution as a function of for a horizontal fiber. The scale for appears on the left, while that for is on the right.

The figure shows the variation in the zeroth and first moment coefficients of the fiber force distribution as a function of for a horizontal fiber. The scale for appears on the left, while that for is on the right.

The figure shows the variation in , a measure of the fiber projection along gravity, with and for ; the limiting values and , respectively, denote gradient-aligned and vorticity-aligned fibers. The analytical low asymptote (heavy line) has also been included for , and is seen to be nearly coincident with the numerical results for .

The figure shows the variation in , a measure of the fiber projection along gravity, with and for ; the limiting values and , respectively, denote gradient-aligned and vorticity-aligned fibers. The analytical low asymptote (heavy line) has also been included for , and is seen to be nearly coincident with the numerical results for .

The figure shows the variation in the transverse drift , measured in units of , as a function of and . The limiting values and , respectively, denote gradient-aligned and vorticity-aligned fibers.

The figure shows the variation in the transverse drift , measured in units of , as a function of and . The limiting values and , respectively, denote gradient-aligned and vorticity-aligned fibers.

Schematic of the role of inertia in the suspension instability. Solid curved arrows indicate the rotation by shear. Dotted curved arrows indicate the inertial rotation by sedimentation. Fibers 1 and 2 are oriented in the extensional and compressional quadrants of the local shear flow, respectively. Fiber 1 will evolve toward a fixed orientation while fiber 2 will flip into the extensional quadrant.

Schematic of the role of inertia in the suspension instability. Solid curved arrows indicate the rotation by shear. Dotted curved arrows indicate the inertial rotation by sedimentation. Fibers 1 and 2 are oriented in the extensional and compressional quadrants of the local shear flow, respectively. Fiber 1 will evolve toward a fixed orientation while fiber 2 will flip into the extensional quadrant.

The normalized drift velocity as a function of for three different aspect ratios. The solid line indicates , the full dashed line , and the alternate dashed line corresponding to .

The normalized drift velocity as a function of for three different aspect ratios. The solid line indicates , the full dashed line , and the alternate dashed line corresponding to .

The growth rate for , normalized by , as a function of for three different aspect ratios. The solid line indicates , the full dashed line , and the alternate dashed line corresponding to .

The growth rate for , normalized by , as a function of for three different aspect ratios. The solid line indicates , the full dashed line , and the alternate dashed line corresponding to .

Time evolution of sedimentation speed at and .

Time evolution of sedimentation speed at and .

Time-averaged sedimentation speed. Present simulations with : ◼, ; ●, ; ▽, Herzhaft and Guazzelli’s measurements (Ref. 6) at ; △, Butler and Shaqfeh’s simulations (Ref. 7) at (, ). Simulations of Kuusela *et al.* (Ref. 15) for prolate spheroids with different aspect ratios at : ——, ; – – –, 5; , 7.

Time-averaged sedimentation speed. Present simulations with : ◼, ; ●, ; ▽, Herzhaft and Guazzelli’s measurements (Ref. 6) at ; △, Butler and Shaqfeh’s simulations (Ref. 7) at (, ). Simulations of Kuusela *et al.* (Ref. 15) for prolate spheroids with different aspect ratios at : ——, ; – – –, 5; , 7.

Time-averaged sedimentation speed with varying at and the effect of different fiber length to the box size: ◼, ; ◻, .

Time-averaged sedimentation speed with varying at and the effect of different fiber length to the box size: ◼, ; ◻, .

Suspensions of settling fibers with (, , and ) at different dimensionless times, . Periodic cell has and .

Suspensions of settling fibers with (, , and ) at different dimensionless times, . Periodic cell has and .

Suspensions of settling fibers with at different fiber concentrations for (top) and 4 (bottom).

Suspensions of settling fibers with at different fiber concentrations for (top) and 4 (bottom).

Time evolution of the structure factors and at the smallest wave number for and : ——, ; – – –, , where and .

Time evolution of the structure factors and at the smallest wave number for and : ——, ; – – –, , where and .

Steady-state values of the structure factors (squares) and (circles) at the smallest wave number are plotted as a function of fiber concentration for (filled symbols) and 4 (open symbols).

Steady-state values of the structure factors (squares) and (circles) at the smallest wave number are plotted as a function of fiber concentration for (filled symbols) and 4 (open symbols).

Steady-state values of the structure factors (squares) and (circles) at the smallest wave number as a function of at . Filled symbols are for and open symbols for .

Steady-state values of the structure factors (squares) and (circles) at the smallest wave number as a function of at . Filled symbols are for and open symbols for .

Time variation in a single fiber’s polar angle at (a) and (b) : ——, ; – – –, 0.045; , 0.091.

Time variation in a single fiber’s polar angle at (a) and (b) : ——, ; – – –, 0.045; , 0.091.

Probability density function of the angle . Filled symbols indicate : ◼, ; ●, . Lines indicate : ——, ; – – –, 0.045; , 0.091. Herzhaft and Guazzelli’s measurements (Ref. 6) at and for a zero-Reynolds-number suspension are indicated by the open squares (◻).

Probability density function of the angle . Filled symbols indicate : ◼, ; ●, . Lines indicate : ——, ; – – –, 0.045; , 0.091. Herzhaft and Guazzelli’s measurements (Ref. 6) at and for a zero-Reynolds-number suspension are indicated by the open squares (◻).

The order parameter as a function of . Present simulations with : ◼, ; ●, . Herzhaft and Guazzelli’s data (Ref. 6) (◻) for settling fibers in Stokes flow with .

The order parameter as a function of . Present simulations with : ◼, ; ●, . Herzhaft and Guazzelli’s data (Ref. 6) (◻) for settling fibers in Stokes flow with .

The order parameter as a function of . Solid symbols are the present simulation results for : ◼, and ; ●, and . The open circle is for and . Simulations of Kuusela *et al.* (Ref. 15) for prolate spheroids with different aspect ratios at : ——, ; – – –, 5; , 7.

The order parameter as a function of . Solid symbols are the present simulation results for : ◼, and ; ●, and . The open circle is for and . Simulations of Kuusela *et al.* (Ref. 15) for prolate spheroids with different aspect ratios at : ——, ; – – –, 5; , 7.

The order parameter as a function of at : ●, .

The order parameter as a function of at : ●, .

Energy spectrum for and : ◻, dynamic simulation of settling fibers; ○, translating random array of fibers with fixed orientations and relative positions.

Energy spectrum for and : ◻, dynamic simulation of settling fibers; ○, translating random array of fibers with fixed orientations and relative positions.

## Tables

Steady-state values of the mean sedimentation velocity of the fibers and the root-mean-square vertical and horizontal velocities of the fibers and fluid. The asterisk indicates normalization with and and indicates a time average.

Steady-state values of the mean sedimentation velocity of the fibers and the root-mean-square vertical and horizontal velocities of the fibers and fluid. The asterisk indicates normalization with and and indicates a time average.

Steady-state results for the fiber orientation.

Steady-state results for the fiber orientation.

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