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Structure and dynamics of dilute suspensions of finite-Reynolds-number settling fibers
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10.1063/1.3274612
/content/aip/journal/pof2/21/12/10.1063/1.3274612
http://aip.metastore.ingenta.com/content/aip/journal/pof2/21/12/10.1063/1.3274612

Figures

Image of FIG. 1.
FIG. 1.

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

Image of FIG. 2.
FIG. 2.

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.

Image of FIG. 3.
FIG. 3.

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 .

Image of FIG. 4.
FIG. 4.

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.

Image of FIG. 5.
FIG. 5.

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.

Image of FIG. 6.
FIG. 6.

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 .

Image of FIG. 7.
FIG. 7.

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 .

Image of FIG. 8.
FIG. 8.

Time evolution of sedimentation speed at and .

Image of FIG. 9.
FIG. 9.

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.

Image of FIG. 10.
FIG. 10.

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

Image of FIG. 11.
FIG. 11.

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

Image of FIG. 12.
FIG. 12.

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

Image of FIG. 13.
FIG. 13.

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

Image of FIG. 14.
FIG. 14.

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).

Image of FIG. 15.
FIG. 15.

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 .

Image of FIG. 16.
FIG. 16.

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

Image of FIG. 17.
FIG. 17.

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 (◻).

Image of FIG. 18.
FIG. 18.

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

Image of FIG. 19.
FIG. 19.

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.

Image of FIG. 20.
FIG. 20.

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

Image of FIG. 21.
FIG. 21.

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

Tables

Generic image for table
Table I.

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.

Generic image for table
Table II.

Steady-state results for the fiber orientation.

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/content/aip/journal/pof2/21/12/10.1063/1.3274612
2009-12-23
2014-04-17
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Structure and dynamics of dilute suspensions of finite-Reynolds-number settling fibers
http://aip.metastore.ingenta.com/content/aip/journal/pof2/21/12/10.1063/1.3274612
10.1063/1.3274612
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