Rapidly yawing spheroids in viscous shear flow: Emergent loss of symmetry

Abstract

We investigate the emergent three-dimensional (3D) dynamics of a rapidly yawing spheroidal swimmer interacting with a viscous shear flow. We show that the rapid yawing generates non-axisymmetric emergent effects, with the active swimmer behaving as an effective passive particle with two orthogonal planes of symmetry. We also demonstrate that this effective asymmetry generated by the rapid yawing can cause chaotic behaviour in the emergent dynamics, in stark contrast to the emergent dynamics generated by rapidly rotating spheroids, which are equivalent to those of effective passive spheroids. In general, we find that the shape of the equivalent effective particle under rapid yawing is different to the average shape of the active particle. Moreover, despite having two planes of symmetry, the equivalent passive particle is not an ellipsoid in general, except for specific scenarios in which the effective shape is a spheroid. In these scenarios, we calculate analytically the equivalent aspect ratio of the effective spheroid. We use a multiple scales analysis for systems to derive the emergent swimmer behaviour, which requires solving a nonautonomous nonlinear 3D dynamical system, and we validate our analysis via comparison to numerical simulations.

Figures (7)

• Figure 1: Schematic of (a) physical setup and (b) Euler angle definitions, with laboratory ($\e{i}$) & swimmer-fixed ($\ehat{i}$) frames denoted by black and green arrows, respectively. The Euler angle rotations in (b) occur in the order $\phi$, $\theta$, $\psi$. The swimmer self-generates a rapid yawing via the time-dependent angular velocity $\angvel(\tstandard) = \Om \Amp \cos \left(\Om \tstandard\right) \ehat{2}$ (curved purple arrows), a time-dependent translational velocity $\Vel(\tstandard)$ (straight purple arrow in (a)), and interacts with a far-field shear flow $\flowvel = y \e{3}$ (blue arrows).
• Figure 2: Numerical solutions of the full rotational dynamics \ref{['eq: full gov eq']}--\ref{['eq: f functions']} (solid blue lines), compared to: (1) ignoring the slow evolution, by setting $\fgen = 0$ in \ref{['eq: full gov eq']} (solid red lines) and (2) the asymptotic solutions, consisting of the leading-order solutions we derive in \ref{['eq: mu def mod']} and the emergent slow evolution equations we derive in \ref{['eq: slow evolution']}, where the latter are solved numerically (dotted black lines). We use parameter values $\Breth = 0.9$, $\Amp = 2$, and $\Om = 3$ with initial conditions $(\theta,\psi, \phi) = (\pi/6, \pi/12, \pi/12)$. We see that the emergent (asymptotic) dynamics we derive in the limit of large $\Om$ agree well with the full dynamics, even for moderate values of $\Om$.
• Figure 3: Numerical solutions of the full translational dynamics \ref{['eq: translational dynamics']}, which also depend on the solution to the full rotational dynamics \ref{['eq: full gov eq']}--\ref{['eq: f functions']} (solid blue lines), compared to: (1) ignoring the slow evolution, by setting $\fgen = 0$ in \ref{['eq: full gov eq']} (solid red lines) and (2) the asymptotic solutions, from the emergent slow evolution equations we derive in \ref{['eq: slow ev trans']}, solved numerically (dotted black lines). We use the same parameter values as in Figure \ref{['fig: orientation dynamics']}, and additionally $\Vel(\tstandard)$ is defined in \ref{['eq: Vel def']}, with $(\velav_1, \velav_2, \velav_3) = (-0.2, 0.5, 0.2)$, $(\velosc_1, \velosc_2, \velosc_3) = (0.2, 0.6, 0.5)$, $(\velps_1, \velps_2, \velps_3) = (\pi/2, \pi/4, -\pi/4)$, and initial conditions $\bs{X}(0) = \bs{0}$.
• Figure 4: The effective coefficients \ref{['eq: Effective coefficients']}, obtained by comparing \ref{['eq: slow evolution']} to \ref{['eq: slow evolution outline']}. The marked stars along the $x$-axis are the values of $\Amp$ at which \ref{['eq: B effective ellipsoid relationship']} is satisfied (i.e. $J_0(2 \Amp) = 0$) and therefore when the effective shape is an ellipsoid. In fact, in these cases the effective shape is constrained further to a spheroid, whose aspect ratio is given in \ref{['eq: effective aspect ratio']}.
• Figure 5: Poincaré section for the classic Jeffery's equations, which are equivalent to setting $\Amp = 0$ in our emergent equations \ref{['eq: slow evolution']}--\ref{['eq: Effective coefficients']}. Using the Poincaré map outlined in the main text, we use $\Amp = 0$, $\Breth = 0.99$ and iterate up to $n = 500$. The full 2D phase space is obtained by exploiting reflectional symmetry across $\newtheta = \pi/2$ and translational symmetry in $\newpsi \mapsto \newpsi + \pi$. No chaos is possible for the classic Jeffery's equations, as observed here.