Motorhead or Masochist:

Spinning and Whipping Dynamics

of an Axially Loaded Micro-Filament

Feng Ling, Hanliang Guo, Eva Kanso

April 14, 2018

Motivation

Biological filaments across difference length scales

 

Simplicity, Flexibility, and Maneuverability

 

Generalization of planar follower force motion

     Bayly and Dutcher (2016), De Canio, Lauga, and Goldstein (2017)

Feng Ling                                                                                        SoCalFluids XII (2018)                                                                                    04/14/2018

Andrej Vilfan

Noah (2014)

BioVisions

Flagella Beating

D. Meng, M. Cao,T.  Oda, and J. Pan, Journal of Cell Science (2014)

Feng Ling                                                                                        SoCalFluids XII (2018)                                                                                    04/14/2018

Soft Slender Robots

Feng Ling                                                                                        SoCalFluids XII (2018)                                                                                    04/14/2018

R. Faubel, C. Westendorf, E. Bodenschatz, and G. Eichele, Science (2016)

Elastic Rod Model

Balance of Linear Momentum

 

 

Balance of Angular Momentum

 

 

Constitutive Relations

Assumptions:

No Inertia

No Shear

No Twist (Yes Torsion)

Resistive Force

Feng Ling                                                                                        SoCalFluids XII (2018)                                                                                    04/13/2018

Body Frame

Stiffness

Re \(=0\)

Feng Ling                                                                                        SoCalFluids XII (2018)                                                                                    04/14/2018

Elastic Rod Model

Balance of Linear Momentum

 

 

Balance of Angular Momentum

 

 

Constitutive Relations

Body Frame

Stiffness

There exists Multiple Forms of Periodic Solutions despite \(\mathbf{f}^a\neq\mathbf{f}^a(t)\)

Spinning and Whipping

Numerical solutions with \(\delta s<L/80\)

under Power Law Force Profiles

 

 

 

 

 

 

(a) Effect of Total Force

(b) Effect of Tip Dominance

Feng Ling                                                                                        SoCalFluids XII (2018)                                                                                    04/14/2018

Large active axial force closer to the

tip cause spinning to whipping

transition

 

GOALs:

  • Obtain a simplified model
  • Understand the cause of transition

Feng Ling                                                                                        SoCalFluids XII (2018)                                                                                    04/14/2018

Spinning and Whipping

Bead-Spring Model

Forces at the Beads

  • Tip Activation:

 

  • Drag Forces:

 

  • Torsional Springs:

Feng Ling                                                                                        SoCalFluids XII (2018)                                                                                    04/14/2018

Linearized system has planar symmetry

\(l\)

Bead-Spring Model

Two Links can reproduce both spinning and whipping

but spinning seems stable in general

Feng Ling                                                                                        SoCalFluids XII (2018)                                                                                    04/13/2018

Bead-Spring Model

Three Links reaches an almost planar solution in the limit of large forces

however the required tip force overwhelms the spring/drag force

Feng Ling                                                                                        SoCalFluids XII (2018)                                                                                    04/13/2018

Path To Transition

How do we obtain 3D to 2D transition?

Feng Ling                                                                                        SoCalFluids XII (2018)                                                                                    04/13/2018

Let's crank up the number of beads...

1 link: inextensibility \(\Rightarrow\) stable

2 links: spinning is stable at large \(\mathbf{F}_{tip}\)

3 links: vertical 'windmill' at large \(\mathbf{F}_{tip}\)

4 links: horizontally precessing 'windmill'

5 links: whipping-like 'windmill'

6 links: 3D to 2D transition!

Path To Transition

Why 6?

Feng Ling                                                                                        SoCalFluids XII (2018)                                                                                    04/14/2018

Need 3 links to account for effects of torsion (not twist)

Need 3 links near the base to properly 'cancel out' the accumulation of centerline torsion?

Further Analysis

Stability about the periodic solution (Floquet Multipliers)

 

Require numerical study due to nonlinear system and 'essentially' nonlinear phenomenon

Feng Ling                                                                                        SoCalFluids XII (2018)                                                                                    04/14/2018

Linearization decouples angles and can only give planar motion

Thanks!

Feng Ling                                                                                        SoCalFluids XII (2018)                                                                                    04/14/2018

 

  • F. Ling, H. Guo, E. Kanso, “Instability-triggered Oscillations of Active Microfilament” In preparation
  • P. V. Bayly, S. K. Dutcher SK, “Steady dynein forces induce flutter instability and propagating waves in mathematical models of flagella” J. R. Soc. Interface (2016) 13: 20160523
  • G. De Canio, E. Lauga, R. E. Goldstein, “Spontaneous Oscillations of Elastic Filaments Induced by Molecular Motors” J. R. Soc. Interface (2017) 14: 20170491
  • M. Gazzola, Levi H. Dudte, Andrew G. McCormick, L. Mahadevan, “Dynamics of soft filaments that can stretch, shear, bend and twist.” ArXiv Preprint (2016)
  • M. Bergou, M. Wardetzky, S. Robinson, B. Audoly, E. Grinspun, “Discrete Elastic Rods.” ACM Trans. Graph. 27, 3, Article 63 (2008)

SoCal Fluids XII

By Feng Ling

SoCal Fluids XII

Using bead-spring model to (attempt to) understand how/why the spinning trajectory transition into whipping trajectory for a fixed-free micro-filament under tip follower load.

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