Flow analysis of the low Reynolds number swimmerC. elegans

Abstract

Swimming cells and microorganisms are a critical component of many biological processes. In order to better interpret experimental studies of low Reynolds number swimming, we combine experimental and numerical methods to perform an analysis of the flow field around the swimming nematode Caenorhabditis elegans. We first use image processing and particle tracking velocimetry to extract the body shape, kinematics, and flow fields around the nematode. We then construct a three-dimensional model using the experimental swimming kinematics and employ a boundary element method to simulate flow fields, obtaining very good quantitative agreement with experiment. We use this numerical model to show that calculation of flow shear rates using purely planar data results in significant underestimates of the true three-dimensional value. Applying symmetry arguments, validated against numerics, we demonstrate that the out-of-plane contribution can be accounted for via incompressibility and therefore simply calculated from particle tracking velocimetry. Our results show how fundamental fluid mechanics considerations may be used to improve the accuracy of measurements in biofluiddynamics.

Thrifty Swimming With Shear-Thinning: A Note on Out-of-Plane Effects for Undulatory Locomotion Through Shear-Thinning Fluids

“…Furthermore, by incompressibility, w z = −u x − v y . Using these conditions, which are valid for a range of non-Newtonian flows, the 3D formula for shear rate in the midplane can be written in terms of the available 2D data [18],…”

“…Figure 1(c,d) shows the streamlines at a particular phase of the nematode beating cycle generated experimentally from particle tracking velocimetry in a Newtonian and a representative shear-thinning fluid, respectively. We note that the body shapes are approximate (for more details on the techniques and data, see [8,10,18]).…”

“…Experimental data acquisition of flow fields driven by microswimmers is typically limited to a 2D slice at a swimmer's midplane; the flow shear rate depends upon velocity derivatives, and as such a highly-resolved differentiable flow field is required to probe the effects of shear-thinning rheology. However, while 2D data are sufficient to accurately measure the flow field around a planar swimmer, the shear rate and therefore the flow dynamics are dependent upon out-of-plane flow derivatives, which must be properly incorporated into the analysis [18]. Ignoring these effects results in relative errors in the shear rate of 25% to 40% for C. elegans in a Newtonian fluid [18].…”

“…However, while 2D data are sufficient to accurately measure the flow field around a planar swimmer, the shear rate and therefore the flow dynamics are dependent upon out-of-plane flow derivatives, which must be properly incorporated into the analysis [18]. Ignoring these effects results in relative errors in the shear rate of 25% to 40% for C. elegans in a Newtonian fluid [18].…”

“…This relative error in shear rate indicates that non-Newtonian effects on both locomotion and the resulting flow field in complex fluids may be larger than anticipated. Examples in which underestimating the local shear rate may yield significant errors include the impact of elastic stretching on locomotion, measured by the Weissenberg number Wi = λ Eγ where λ E is the longest relaxation time of the fluid, and the effects of swimming through a generalised Newtonian fluid, whose shearthinning or shear-thickening behaviour is indicated by the Carreau number Cr = λ Crγ where λ Cr is a timescale that represents the onset of shear-thinning effects [18].…”