^{1,a)}

### Abstract

We simulated the freely falling motion of an inhomogeneous flexible filament immersed in an incompressible viscous fluid under the action of gravity by the immersed boundary method. Our simulations show that the falling motion of an inhomogeneous filament is stable with respect to disturbances of small magnitude irrespective of the mass and bending modulus distributions. However, sufficiently large disturbances may bring the filament motion into a significantly different state: the filament deforms, rotates, and drifts towards one of the side boundaries while falling in the flowing fluid under the action of gravity. In addition our results indicate unstable filament motion depends more strongly on the bending modulus than the mass density. Our simulations also show the existence of two similar states for a homogeneous filament. The motions of inhomogeneous and homogeneous filaments are compared, and both quantitative and qualitative differences in the unstable motions are found. This is a starting point to understand the role of inhomogeneous filament properties in deciding its motion.

The author would like to thank the referees for their suggestions and comments which has helped him for a better representation of his work. The author also thanks Raymond Chin and Charles Peskin for helpful discussions regarding the revision of the paper.

I. INTRODUCTION

II. MATHEMATICAL FORMULATION

III. NUMERICAL RESULTS

A. Two states of the motion of an inhomogeneous filament

1. Two states of motion: Stable motion

2. Two states of motion: Unstable motion

3. Comparison with a homogeneous filament

B. Filament sideways drift motion

IV. SUMMARY

### Key Topics

- Viscosity
- 28.0
- Elasticity
- 16.0
- Vortex dynamics
- 12.0
- Reynolds stress modeling
- 9.0
- Navier Stokes equations
- 8.0

## Figures

The typical trajectories of the filament with no or small external disturbances. The left figure corresponds to no external disturbances and the right figure corresponds to small magnitude disturbances. The filament falling motion is stable when no external disturbances or instantaneous external disturbances of small magnitude are imposed.

The typical trajectories of the filament with no or small external disturbances. The left figure corresponds to no external disturbances and the right figure corresponds to small magnitude disturbances. The filament falling motion is stable when no external disturbances or instantaneous external disturbances of small magnitude are imposed.

The trajectories of the filament subjected to large disturbances. Each figure corresponds to a different bending rigidity and/or a different mass density. See Figs. 3 and 4 for details of density and rigidity distributions. The disturbances are imposed on the center portion of the filament for all cases on the bottom panel. For the top panel, disturbances on the right portion for the first and third cases, and on the left portion for the second case. We see that the large disturbances bring the filament motion into a totally different state than the stable state.

The trajectories of the filament subjected to large disturbances. Each figure corresponds to a different bending rigidity and/or a different mass density. See Figs. 3 and 4 for details of density and rigidity distributions. The disturbances are imposed on the center portion of the filament for all cases on the bottom panel. For the top panel, disturbances on the right portion for the first and third cases, and on the left portion for the second case. We see that the large disturbances bring the filament motion into a totally different state than the stable state.

The filament bending rigidity as a function of filament length for each case in Fig. 2. Each figure corresponds to the case at the corresponding location in Fig. 2, respectively. The axis is the filament length (dimensionless), the axis is the dimensionless bending rigidity. The values of the dimensionless mean bending rigidity are from top left to bottom right, respectively.

The filament bending rigidity as a function of filament length for each case in Fig. 2. Each figure corresponds to the case at the corresponding location in Fig. 2, respectively. The axis is the filament length (dimensionless), the axis is the dimensionless bending rigidity. The values of the dimensionless mean bending rigidity are from top left to bottom right, respectively.

The filament mass density as a function of filament length for each case in Fig. 2. The axis is the filament length (dimensionless), the axis is the dimensionless mass density. The curved line corresponds to the second case on the top of Fig. 2. The straight line corresponds to all of the other cases. The values of the dimensionless mean mass density are 0.9778, 0.9756, 0.9778, 0.9778, 0.9778, 0.9778, from top left to bottom right, respectively.

The filament mass density as a function of filament length for each case in Fig. 2. The axis is the filament length (dimensionless), the axis is the dimensionless mass density. The curved line corresponds to the second case on the top of Fig. 2. The straight line corresponds to all of the other cases. The values of the dimensionless mean mass density are 0.9778, 0.9756, 0.9778, 0.9778, 0.9778, 0.9778, from top left to bottom right, respectively.

The visualization of the flow field for the third case on the upper panel of Fig. 2. The top group is the instantaneous fluid marker positions, and the bottom group is the vorticity contours. The dark line represents the filament. The four dimensionless instants from the left to the right are 2.0, 3.0, 4.0, 5.0, respectively.

The visualization of the flow field for the third case on the upper panel of Fig. 2. The top group is the instantaneous fluid marker positions, and the bottom group is the vorticity contours. The dark line represents the filament. The four dimensionless instants from the left to the right are 2.0, 3.0, 4.0, 5.0, respectively.

The visualization of the flow field for the first case on the lower panel of Fig. 2. The top group is the instantaneous fluid marker positions, and the bottom group is the vorticity contours. The dark line represents the filament. The four dimensionless instants from the left to the right are 2.0, 3.0, 4.0, 5.0, respectively.

The visualization of the flow field for the first case on the lower panel of Fig. 2. The top group is the instantaneous fluid marker positions, and the bottom group is the vorticity contours. The dark line represents the filament. The four dimensionless instants from the left to the right are 2.0, 3.0, 4.0, 5.0, respectively.

Comparison of trajectories for homogeneous (dashed curves) and inhomogeneous (solid curves) filaments. The three figures correspond to three different loading locations of external disturbances: on the left portion (left figure), on the right portion (middle figure), and on the middle portion (right figure). All the other parameters are the same for the three figures. For each figure, two simulation results were plotted together. The only difference between the two simulations is the mass density distribution: one is linearly distributed (see Fig. 8) the other is uniformly distributed (taken as the mean value of the linear distribution). The dimensionless bending modulus is a constant (0.00926).

Comparison of trajectories for homogeneous (dashed curves) and inhomogeneous (solid curves) filaments. The three figures correspond to three different loading locations of external disturbances: on the left portion (left figure), on the right portion (middle figure), and on the middle portion (right figure). All the other parameters are the same for the three figures. For each figure, two simulation results were plotted together. The only difference between the two simulations is the mass density distribution: one is linearly distributed (see Fig. 8) the other is uniformly distributed (taken as the mean value of the linear distribution). The dimensionless bending modulus is a constant (0.00926).

The mass distribution over the filament arc-length for the inhomogeneous filament in Fig. 7.

The mass distribution over the filament arc-length for the inhomogeneous filament in Fig. 7.

Comparison of trajectories for homogeneous (shadowed curves) and inhomogeneous (solid curves) filaments. The only difference for the two results in each figure is the bending modulus. See Fig. 11 for bending modulus distribution of the inhomogeneous filament. The bending modulus of the homogeneous one is a constant (the mean value of the inhomogeneous distribution). Three figures correspond to three different loading locations of external disturbances. Counting from the left to the right, the disturbance was loaded on the left, right and middle portion of the filament, respectively.

Comparison of trajectories for homogeneous (shadowed curves) and inhomogeneous (solid curves) filaments. The only difference for the two results in each figure is the bending modulus. See Fig. 11 for bending modulus distribution of the inhomogeneous filament. The bending modulus of the homogeneous one is a constant (the mean value of the inhomogeneous distribution). Three figures correspond to three different loading locations of external disturbances. Counting from the left to the right, the disturbance was loaded on the left, right and middle portion of the filament, respectively.

The trajectories of the filament subjected to initial large external disturbances on different locations. The disturbance is imposed on the left, right, and center portions of the filament from the left to the right, respectively. As a consequence the filament falls in significantly different ways. This indicates the complexity of the filament motion in a flowing viscous fluid because of the interaction of elasticity and hydrodynamics.

The trajectories of the filament subjected to initial large external disturbances on different locations. The disturbance is imposed on the left, right, and center portions of the filament from the left to the right, respectively. As a consequence the filament falls in significantly different ways. This indicates the complexity of the filament motion in a flowing viscous fluid because of the interaction of elasticity and hydrodynamics.

The bending rigidity distribution for the case in Fig. 10. The axis is the filament length (dimensionless), the axis is the dimensionless bending rigidity. The mean bending rigidity is 0.0046, and the dimensionless mass density is 0.9017.

The bending rigidity distribution for the case in Fig. 10. The axis is the filament length (dimensionless), the axis is the dimensionless bending rigidity. The mean bending rigidity is 0.0046, and the dimensionless mass density is 0.9017.

The flow and filament visualization for the first case in Fig. 10 for ten different time instants (continued in Fig. 13). The top figures are instantaneous positions of fluid markers and the lower figures are corresponding vorticity contours at the same time instant. The dimensionless time instants are 0.667, 1.200, 1, 733, 2.267, 2.800, from the left to the right, respectively.

The flow and filament visualization for the first case in Fig. 10 for ten different time instants (continued in Fig. 13). The top figures are instantaneous positions of fluid markers and the lower figures are corresponding vorticity contours at the same time instant. The dimensionless time instants are 0.667, 1.200, 1, 733, 2.267, 2.800, from the left to the right, respectively.

Continued from Fig. 12. The dimensionless time instants are 3.333, 3.867, 4.400, 4.933, 5.467, from the left to the right, respectively.

Continued from Fig. 12. The dimensionless time instants are 3.333, 3.867, 4.400, 4.933, 5.467, from the left to the right, respectively.

The flow and filament visualization for the third case in Fig. 10 for ten different time instants (continued in Fig. 15). The top figures are instantaneous positions of fluid markers and the lower figures are corresponding vorticity contours at the same time instant. The dimensionless time instants are 0.667, 1.200, 1.733, 2.267, 2.800, from the left to the right, respectively.

The flow and filament visualization for the third case in Fig. 10 for ten different time instants (continued in Fig. 15). The top figures are instantaneous positions of fluid markers and the lower figures are corresponding vorticity contours at the same time instant. The dimensionless time instants are 0.667, 1.200, 1.733, 2.267, 2.800, from the left to the right, respectively.

Continued from Fig. 14. The dimensionless time instants are 3.333, 3.867, 4.400, 4.933, 5.333, from the left to the right, respectively.

Continued from Fig. 14. The dimensionless time instants are 3.333, 3.867, 4.400, 4.933, 5.333, from the left to the right, respectively.

## Tables

Parameters used in the simulations.

Parameters used in the simulations.

Nondimensional parameters used in the simulations. For the meanings of the symbols used in the definition of the nondimensional parameters, see Table I.

Nondimensional parameters used in the simulations. For the meanings of the symbols used in the definition of the nondimensional parameters, see Table I.

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