Session P01: Minisymposia: Symmetry in Fluid Dynamics

Symmetry-breaking instabilities in the bouncing droplet system

We consider the dynamics of bouncing millimetric droplets, levitating on the surface of a vibrating liquid bath. We review recent work that characterizes and rationalizes theoretically a number of symmetry-breaking instabilities that arise as the driving parameter, the vibrational acceleration, is increased progressively. Instabilities of individual droplets, droplet pairs and lattices of various geometries are considered in turn. The relation between this bouncing-droplet system and a number of microscopic systems is briefly discussed.   

Symmetry breaking due to surface tension and geometry

Multiphase flows occur in an enormous variety of fluid mechanics situations. The influence of surface tension at the fluid-fluid interface can be the driver for symmetry breaking, as has been demonstrated in many cases in the literature. In this talk I give recent examples of our studies of free surface flows where the concepts of symmetry, symmetry breaking, and self-similarity occur and possibly intersect. In particular, I will discuss (1) the manner in which the geometry of an ordinary pressure-driven channel flow can influence, and control,  viscous fingering instabilities, (2) the case of a ``lifted” Hele-Shaw cell, where time-dependent strategies  can either suppress the viscous fingering instability or maintain a series of non-splitting viscous fingers during the displacement process, and (3) a three-dimensional film drainage flow  where an underlying structure of the partial differential equation for the film shape allows a transformation from three independent variables to a single variable, and the solution of the corresponding ordinary differential equation is in excellent agreement with experimental measurements. 

 

    

Time flies and memory lingers: altering the stability of synchronized states in hydrodynamically coupled oscillators.

Minimal models of coupled oscillators have been used to shed light on the emergence of synchronized beating of nearby flagella or metachronal waves of arrays of cilia.  Here we focus on oscillators whose motion is driven by two alternating traps, one active and one not.  The activation state of the trap is governed by a geometric switch that depends upon the oscillator's distance to it.  In Stokes flow, it has been shown both analytically and experimentally that two in-tandem spherical colloids, each with their its own pair of traps (all co linear), will tend towards anti-phase oscillations, with in-phase oscillations unstable.    Here we show that adding some memory to the system can stabilize the in-phase oscillation. This memory may come from fluid inertia (unsteady Stokes or Navier-Stokes), weak elastic coupling of oscillators, or an altered switch that checks for a stall-velocity in addition to geometric information.   This stabilization of in-phase synchrony is robust – we have considered rigid spherical oscillators  in 3D Stokes and 3D unsteady Stokes, elastic filament oscillators in 3D Stokes, and flexible droplets in 2D Stokes and 2D Navier Stokes – and all demonstrate this phenomenon   

Symmetric Fluid Flows and Similarity Variables

The role of similarity variables in symmetric flows is investigated in fluid dynamics. The most important type of symmetry is the dilation symmetry which is

considered in detail. In some cases certain constants are not uniquely determined

and this is called incomplete similarity. In this case extra boundary conditions or

conservation laws are necessary to complete the solution. Examples of similarity

solutions in fluid dynamics are given.   

Symmetry, spatio-temporal self-similarity and scaling in wall turbulence

Self-similarity of the statistics of wall turbulence has been well-studied. The properties of the mean velocity profile and a description of the inertial (logarithmic) region are well-documented, while work in the last two decades has unraveled some of the complexities of the scaling of the Reynolds stresses and spectral representations of the fluctuations. An incomplete list of recent models for the scaling behavior for these flows include the Lie group symmetry analysis of Oberlack (1999), the work of She et al. (2017), and the mean momentum balance (MMB) approach of Klewicki et al. (2007). The attached eddy hypothesis (AEH), e.g. Perry & Chong (1982), and recent extensions (Marusic & Monty, 2019) provide a static structural analog replicating much of the observed behavior. Recently explored equation- and data-driven analysis techniques such as resolvent analysis (McKeon & Sharma, 2010; Hwang & Cossu, 2010; McKeon, 2017; Jovanovic, 2021) and spectral proper orthogonal decomposition (Towne et al, 2018) permit the investigation of instantaneous coherent structure.

I will discuss symmetry in the context of spatio-temporal coherent structure, as admitted by the Navier-Stokes equations and the ``anatomy” of the turbulent boundary layer, including some associated analytical developments with regards to the linear and nonlinear terms in the Navier-Stokes equations for flows over a range of Mach number and various boundary conditions.   

Chiral symmetry avoidance through a cascade of wrinkles

We describe how a line initially dividing the plane spontaneously breaks its chiral symmetry by actively sustaining a cascade of successive wrinkles of ever smaller sizes. The line is an oil thread deposited at the surface of a still water bath, interacting with its environment by Marangoni stresses.

The thread surface tension is lower than that of the underlying bath by an amount $\Delta\sigma$. This driving force directed from the thread core to the outer environment first thickens the thread by spreading the oil concentration gradient over a length scale $\ell$. 

This chiral left/right (-/+) symmetry soon breaks into a meandering motion, any small displacement of the thread centerline being amplified by the difference of the surface tension gradients on each side of the thread $\Delta\sigma/\ell\vert_- \neq \Delta\sigma/\ell\vert_+$, a difference further increasing as the thread deforms. 

We provide a detailed description of this newly reported instability, of its most unstable wavelength and associated growth rate. We also explain why and how this destabilization process repeats iteratively, leading to an accelerated cascade of fractal wrinkles, which ends-up in a finite time when molecular diffusion finally erases the driving force concentration gradients.