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Unsteady Swimming of Small Organisms
Shiyan wang and Arezoo Ardekani
Aerospace and Mechanical Engineering
University of Notre Dame
CBET-1066545
Introduction
Small planktonic organisms ubiquitously display unsteady or impulsive motion to attack a
prey or escape a predator in natural environments. Despite this, the role of unsteady
forces such as history and added mass forces on the low Reynolds number propulsion of
small organisms, e.g. Paramecium, is poorly understood. In this work, we derive the
fundamental equation of motion for an organism swimming by the means of surface
distortion in a non-uniform background flow field at a low Reynolds number regime. We
show that the history and added mass forces are important as the product of Reynolds
number and Strouhal number increases above unity.
The fundamental equation of motion for an organism swimming by the means of surface
deformation:
Force due to pressure gradient of undisturbed flow
Buoyancy force
Stokes drag
Time-dependent swimming of an algal
cell (Jerry Gollub, Haverford College)
Escape of Paramecium from a localized heating
50 mW laser, (Hamel et al. PNAS, 2011)
Added mass force
Basset force due translation
Stokes drag due surface deformation
Basset force due surface deformation
Equation of motion for unsteady Squirmer:
Blake, J Fluid Mech., 1970
Stokes drag
Basset force
Added mass force
Blake’s solution
Background flow field
Organism velocity
Surface velocity
Fluids viscosity
Swimmer’s radius
Laplace variable
Legendre polynomial of the first kind of degree n
(a) The ratio of velocity amplitude predicted by the present analysis normalized by the one
predicted by the Blake solution is plotted for two different squirmers with an identical
mean velocity. (b) Amplitude of oscillation of Basset, added mass, and Stokes forces for the
same organisms.
In the limit of
:
In the limit of
:
In order to model the jumping behavior, we calculate the time dependent swimming
velocity with a surface motion described as,
The fundamental equation motion for a jumping organisms can be used to calculate its
velocity decay
The velocity decay of a jumping
copepod can be well predicted
after considering all the unsteady
hydrodynamic forces.
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Unsteady hydrodynamic forces affect the swimming velocity of small organisms. (a) The
dimensionless swimming velocity of the organism is compared with the Blake solution
where unsteady hydrodynamic forces are neglected. (b) The Basset and added mass forces
acting on the organism are larger than the hydrodynamic forces predicted by quasi-steady
Stokes equations.
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ଵଵ
ଵଶ
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Conclusions
• We have derived the fundamental equation of motion for an unsteady swimmer in
a non-uniform flow.
• Transient behavior of the microorganism cannot be captured by using quasi-steady
Stokes equations
• Long term swimming velocity is strongly affected by Basset and added mass forces.
• Our results show that unsteady inertial effects can lead to propulsion of an
unsteady squirmer even for the cases with zero streaming parameters.