Pursuit and Synchronization in Hydrodynamic Dipoles

Abstract

We study theoretically the behavior of a class of hydrodynamic dipoles. This study is motivated by recent experiments on synthetic and biological swimmers in microfluidic Hele-Shaw type geometries. Under such confinement, a swimmer's hydrodynamic signature is that of a potential source dipole, and the long-range interactions among swimmers are obtained from the superposition of dipole singularities. Here, we recall the equations governing the positions and orientations of interacting asymmetric swimmers in doubly-periodic domains, and focus on the dynamics of swimmer pairs. We obtain two families of 'relative equilibria'-type solutions that correspond to pursuit and synchronization of the two swimmers, respectively. Interestingly, the pursuit mode is stable for large tail swimmers whereas the synchronization mode is stable for large head swimmers. These results have profound implications on the collective behavior reported in several recent studies on populations of confined microswimmers.

Emergence of in-line swimming patterns in zebrafish pairs

“…is the unit vector orthogonal tov f (t) forming a local right-handed coordinate system and ∇ is the gradient operator in the global coordinate system. With respect to the paper by Filella et al (2018), we note the difference in sign between ( 8) and ( 5) therein, which arises from our choice to work with transverse versus aligned dipoles, that is, T-dipoles instead of A-dipoles, as defined by Kanso & Tsang, 2014. One can verify that the sign is indeed negative by considering a fish swimming in the x-direction ( f = 0,v f =î, andv ⊥ f =ĵ) in a shear flow (U f = Yî with > 0); in this case, the turn rate in ( 8) is − such that the fish will turn clockwise, as one would expect also based on the work of Tchieu et al (2012).…”

School cohesion, speed and efficiency are modulated by the swimmers flapping motion

“…Existing mathematical models of flow interactions in fish schools vary in the degree of fidelity to the fluid dynamics and sensory-feedback control at the swimmer level. Ideal flow models – based on a dipolar far-field approximation (Tchieu, Kanso & Newton 2012) – with no feedback control have been used to assess the effect of passive flow interactions on the stability of pairwise (Kanso & Tsang 2014, 2015) and diamond lattice formations (Tsang & Kanso 2013) and the advantages of flapping out of phase (Kanso & Newton 2009). This far-field flow model coupled to visual feedback control, either in the form of behavioural rules (Filella et al.…”

Mathematical modelling of bacterial quorum sensing and biofilm extracellular polymeric substances

“…Reference [213] proposed a model with a set of equations for cell cell interaction through hydrodynamics which agrees with the experimental observation of vertex arrays. Hydrodynamic interactions of dipoles in confined space were analytically and numerically studied [215,216,217].…”

“…In reference [217], a dipole is comprised of a head and a tail connected by a rod of invariant length. Analysis shows that when the tail of the dipole is larger than the head, pursuit mode (one dipole follows another) is stable and when the head is larger than the tail, synchronization mode (dipoles swim in parallel) is stable.…”

“…> > > > > :ż n = Ue ia n + µw(z n ) a n = Re[n 1 dw dz ie 2ia n + n 2w ie ia n ] w(z n ) = Â j6 =n s e ia j (z n z j ) 2 (7.5) z n is the location of n th dipole, represented by a complex number; U is the speed of a dipole; a n is orientation of n th dipole; µ is the translation coefficient; w is the fluid velocity represented by a complex number; n 1 = (l h µ h +l t µ t l h +l t , n 2 = (µ h µ t )/l; subscript h means head and t means tail; l h and l t are parameters that enforce the distance between head and tail of the dipole is constant l. Reference [217] used this model and concluded that for swimmers like bacteria with larger head and smaller tail, alignment is the stable mode of hydrodynamic interactions, thus proving that hydrodynamic interaction leads to cell cell alignment. Similar with method 1, this model assumes constant swimming speed of the dipole, which should be adjusted to address how alignment increases average speed.…”