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Transcript
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. ି 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. ଵ ଵଵ ଵଶ ଶ 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.