# Dynamics of Swimmers in Fluids with Resistance

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Fluid Model

#### 2.2. Sperm Representation

#### 2.3. Numerical Methods

## 3. Results

#### 3.1. Dynamics of a Single Swimmer

#### 3.2. Pairs of Swimmers

#### 3.3. Wall Interactions

## 4. Discussions and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic of the internal structure of the flagellum. The circular cross-section contains nine sets of microtubule doublets close to the perimeter and the dynein (molecular force generators) are shown. (

**b**) Cartoon of the discretization of the cell body or head and how it is connected to the flagellum. Each line corresponds to a spring connecting points and the stars are nodes from the discretization (gray dashed lines and nodes correspond to the flagellum).

**Figure 2.**Flagellar positions are given over two beat periods, rotated and shifted to have the point at arc length $s=L$ co-located, for non-dimensional time $t=8-10$. The top two rows, (

**a**–

**d**), is for decreased curvature stiffness ${S}_{C}$ whereas the bottom row, (

**e**,

**f**), is for the baseline curvature stiffness. Similarly, the left column with (

**a**,

**c**,

**e**) is for resistance parameter $\alpha =0.01$ whereas the right column with (

**b**,

**d**,

**f**) is for $\alpha =8$. The filament cases in (

**a**,

**b**) correspond to head radius ${r}_{h}=0$ and the trace to the right of the swimmer is the right most point of the swimmer. When head radius is ${r}_{h}=0.1$, (

**c**–

**f**), the trace to the right of the tail is of the left most point of the head. Other parameters used are specified in Table 1.

**Figure 3.**Quantifying difference in flagellar beat forms for different values of resistance parameter $\alpha $ and curvature stiffness ${S}_{C}$ at non-dimensional time $t=15$. (

**a**) Difference between preferred and emergent simulation beat forms, superimposed in the case of ${r}_{h}=0.1$. (

**b**) Procrustes measure of dissimilarity between the emergent beat of the filament (no head or ${r}_{h}=0$) and head cell size of ${r}_{h}=0.075$, 0.1, and 0.125. For both (

**a**,

**b**), $\beta =0.1$; other parameters used are specified in Table 1.

**Figure 4.**Swimmers at $t=15$ (non-dimensional, after 30 beat periods) for different resistance parameter $\alpha $ and preferred beat form amplitude $\beta $: (

**a**) $\alpha =0.01$, $\beta =0.2$, and head radius ${r}_{h}=0.1$, (

**b**) $\alpha =8$, $\beta =0.2$, and head radius ${r}_{h}=0.1$, (

**c**) $\alpha =0.01$, $\beta =0.1$, and head radius ${r}_{h}=0.1$, (

**d**) $\alpha =8$, $\beta =0.1$, and head radius ${r}_{h}=0.1$, (

**e**) $\alpha =0.01$, $\beta =0.1$, and head radius ${r}_{h}=0.05$, (

**f**) $\alpha =8$, $\beta =0.1$, and head radius ${r}_{h}=0.05$. The flow field is shown with arrows and normalized pressure is shown with the background color with scale given on the colorbar. The last point on the flagellum is also traced out (corresponding to arc length parameter $s=0$). Other baseline parameters utilized are given in Table 1.

**Figure 5.**Swimming speeds vary as a function of the resistance parameter $\alpha $. Speeds for two different preferred beat amplitudes $\beta $ and curvature stiffness parameters ${S}_{C}$: (

**a**) swimmer with head radius ${r}_{h}=0.1$ and (

**b**) filament (F, head radius ${r}_{h}=0$). (

**c**) Speeds for four different head radii ${r}_{h}$ using baseline parameters. In (

**a**–

**c**), speeds are scaled by the swimmer using $\alpha =0.01$ and with baseline parameters ($L=1$, $\beta =0.2$, ${S}_{C}$, ${r}_{h}=0.1$; refer to Table 1). All speeds are calculated as an average over a beat period and are the total distance traveled by the center point of the head (or the first point on the filament with no head) over the time interval $\tau =0.025$.

**Figure 6.**Swimmers at $t=15$ (non-dimensional, after 30 beat periods) for different resistance parameter $\alpha $ and preferred beat form amplitude $\beta $ when initialized a vertical distance ${I}_{d}=0.5$ apart: (

**a**) $\alpha =0.01$, $\beta =0.1$, (

**b**) $\alpha =0.01$, $\beta =0.2$, (

**c**) $\alpha =4$, $\beta =0.1$, (

**d**) $\alpha =4$, $\beta =0.2$, (

**d**) $\alpha =8$, $\beta =0.1$, (

**e**) $\alpha =8$, $\beta =0.2$. The flow field is shown with arrows and normalized pressure is shown with the background color with scale given on the colorbar. The last point on the flagellum is also traced out (corresponding to arc length $s=0$). Other baseline parameters utilized are given in Table 1.

**Figure 7.**Distance between pairs of swimmers initialized with the same initial beat form and a vertical distance of ${I}_{d}=0.5$ apart. (

**a**,

**c**) Distance between the center of the cell body (or last point on the filament, $s=1$, when ${r}_{h}=0$) (

**b**,

**d**) Distance between the last point on the tail (arc length $s=0$). The amplitude of the preferred beat form is $\beta =0.1$ for the top row (

**a**,

**b**) and $\beta =0.2$ for the bottom row (

**c**,

**d**). The top row, (

**a**,

**b**), is distance between swimmers at time $t=15$, plotted as a function of resistance parameter $\alpha $ for different head radii ${r}_{h}$ (where filament is ${r}_{h}=0$). The bottom row, (

**c**,

**d**), shows the distance between swimmers plotted as a function of time for ${r}_{h}=0.1$. For parameters not specified, refer to Table 1 for baseline values used.

**Figure 8.**Swimming speeds at time $t=15$ as a function of resistance parameter $\alpha $ for a pair of swimmers initialized with the same initial beat form and a vertical distance of ${I}_{d}=0.5$ or ${I}_{d}=2$ apart. Speeds for different preferred beat amplitude $\beta $ for baseline parameters: (

**a**) filament (F, head radius ${r}_{h}=0$) and (

**b**) speed for head radius ${r}_{h}=0.1$. In (

**a**,

**b**), for each $\alpha $, speeds are scaled by the corresponding solo swimmer with the same parameters; the rest of the baseline parameters are given in Table 1.

**Figure 9.**Long-term dynamics of a pair of swimmers: (

**a**,

**b**) Tail traces from $t=0-30$ and final flagellar locations at $t=30$ for $\alpha =0$ with amplitude $\beta =0.1$ in (

**a**) and $\beta =0.2$ in (

**b**). The change in angle of swimming trajectories of the top swimmer S1, with respect to the solo swimmer, for $t=0-30$ for $\beta =0.2$ (calculated using Equation (10)). The rest of the baseline parameters are given in Table 1.

**Figure 10.**Solo swimmer in free-space in left hand column and swimmer initialized ${W}_{d}=0.5$ above an elastic wall in right hand column. End of the tail traces are shown for $t=0-13$ and the tail beat form over four beat periods ($t=13-15$) are overlayed and shown in gray. The black tail curve corresponds to the time point $t=13$. All simulations are utilizing the baseline parameters in Table 1 with amplitude $\beta =0.1$. The resistance parameter is varied: (

**a**,

**b**) are $\alpha =0$, (

**c**,

**d**) are $\alpha =4$, and (

**e**,

**f**) are $\alpha =8$.

**Figure 11.**(

**a**) y-locations of the swimmer tail at the end of each beat period, corresponding to the setup in Figure 10. Dashed lines in the graph correspond to the swimmer with centerline initialized ${W}_{d}=0.5$ above the wall and the corresponding solid line is for a swimmer in free-space (no wall). (

**b**) Angle differences (degrees) in trajectories of swimmers with different head radii initialized ${W}_{d}=0.5$ above a wall in comparison to the corresponding solo swimmer in free space. All simulations are utilizing the baseline parameters in Table 1 with amplitude $\beta =0.1$ and resistance parameter $\alpha $ is varied.

**Figure 12.**Filament (${r}_{h}=0$) initialized ${W}_{d}=0.5$ above the wall with tail traces up to $t=15$; (

**a**) $\alpha =1$ and (

**b**) $\alpha =8$. All simulations are utilizing the baseline parameters in Table 1 with amplitude $\beta =0.1$.

**Figure 13.**Two swimmers initialized ${I}_{d}=0.5$ apart in the vertical direction, shown in free-space in the left hand column and initialized ${W}_{d}=0.5$ above an elastic wall in the right column. End of the tail traces are shown for $t=0-12$ and the tail beat form over four beat periods ($t=12-14$) are overlayed and shown in gray. All simulations are utilizing the baseline parameters in Table 1 with amplitude $\beta =0.1$ and resistance parameter is varied: (

**a**,

**b**) are $\alpha =0$, (

**c**,

**d**) are $\alpha =4$ and (

**e**,

**f**) are $\alpha =8$.

**Figure 14.**Corresponding to the setup in Figure 13: (

**a**) y-distance to the wall for top swimmer S1 and bottom swimmer S2 at $t=12$. (

**b**) Angle differences (degrees) in swimming trajectories of top swimmer S1 with respect to time, in comparison to corresponding solo swimmer in free-space. All simulations are utilizing ${r}_{h}=0.1$ and the baseline parameters in Table 1 with amplitude $\beta =0.1$.

$\mathcal{T}$, characteristic time scale, | 0.05 s (beat frequency 20 Hz) |

$\mathcal{L}$, characteristic length scale | 50 $\mathsf{\mu}$m |

$\mathcal{V}$, characteristic velocity scale | $\mathcal{L}/\mathcal{T}$ (wavespeed) |

$\mu $, viscosity | 0.001 kg m${}^{-1}$ s${}^{-1}$ |

$\mathcal{F}$, characteristic force scale | $\mu \mathcal{V}\mathcal{L}$ |

L, tail length | 1 |

${r}_{h}$, head radius | 0.1 |

$\kappa $, wavenumber | $2\pi $ |

$\beta $, amplitude | varied [0.1,0.3] |

$\sigma $, frequency | $4\pi $ |

${S}_{C}$, curvature stiffness of tail | 8 |

${S}_{T}$, tensile stiffness of tail | 10,000 |

${S}_{N}$, tensile stiffness of neck | 500 |

${S}_{H}$, tensile stiffness of head | 10,000 |

${S}_{H,C}$, curvature stiffness of head | 0.8 |

${S}_{N,A}$, stiffness for tail-head connecting angle | 10,000 |

**Table 2.**Angle differences at $t=15$, calculated using Equation (10). $S1$ and $S2$ correspond to the top and bottom swimmers, respectively, when centerlines are initialized a distance ${I}_{d}=0.5$ apart. F corresponds to filament (no head).

Parameters | $\mathit{\alpha}=0$ | $\mathit{\alpha}=1$ | $\mathit{\alpha}=4$ | $\mathit{\alpha}=8$ |
---|---|---|---|---|

S1, $\beta =0.1$, ${r}_{h}=0.1$ | 4.1427 | 4.0591 | 3.3784 | 2.6849 |

S2, $\beta =0.1$, ${r}_{h}=0.1$ | 2.6702 | 2.5901 | 2.1684 | 1.4504 |

S1, $\beta =0.2$, ${r}_{h}=0.1$ | 29.8417 | 10.0230 | 12.0022 | 21.1165 |

S2, $\beta =0.2$, ${r}_{h}=0.1$ | 2.7297 | 3.1615 | 6.3438 | 15.9360 |

S1, $\beta =0.1$, ${r}_{h}=0$ (F) | 8.5972 | 8.3903 | 6.1374 | 3.4372 |

S2, $\beta =0.1$, ${r}_{h}=0$ (F) | 0.7264 | 0.7264 | 0.4022 | 1.0445 |

S1, $\beta =0.2$, ${r}_{h}=0$ (F) | 5.3680 | 5.3680 | 6.1029 | 3.6030 |

S2, $\beta =0.2$, ${r}_{h}=0$ (F) | 1.5309 | 1.2868 | 0.7523 | 0.3240 |

**Table 3.**Angle Differences at $t=12$ for swimmers with ${r}_{h}=0.1$ and $\beta =0.1$ that are initialized with centerlines ${I}_{d}=0.5$ apart. S1 is the top swimmer, above S2. The cases with a W also have an elastic wall ${W}_{d}=0.5$ below the centerline of S2.

Parameters | $\mathit{\alpha}=0$ | $\mathit{\alpha}=1$ | $\mathit{\alpha}=4$ | $\mathit{\alpha}=8$ |
---|---|---|---|---|

W, S1 | 9.9491 | 11.4553 | 10.5367 | 7.9992 |

W, S2 | 3.2190 | 4.1000 | 1.9124 | 1.9891 |

S1 | 4.4893 | 4.3786 | 3.5096 | 2.5974 |

S2 | 2.3740 | 2.3133 | 2.0395 | 1.5294 |

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**MDPI and ACS Style**

Jeznach, C.; Olson, S.D. Dynamics of Swimmers in Fluids with Resistance. *Fluids* **2020**, *5*, 14.
https://doi.org/10.3390/fluids5010014

**AMA Style**

Jeznach C, Olson SD. Dynamics of Swimmers in Fluids with Resistance. *Fluids*. 2020; 5(1):14.
https://doi.org/10.3390/fluids5010014

**Chicago/Turabian Style**

Jeznach, Cole, and Sarah D. Olson. 2020. "Dynamics of Swimmers in Fluids with Resistance" *Fluids* 5, no. 1: 14.
https://doi.org/10.3390/fluids5010014