Hydrodynamics in Manchester

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"Effect of a local perturbation on a bubble propagating in a constricted Hele-Shaw channel"

This is an experimental research.

Salary : no

Key words : Bubble, Hele-Shaw channel, Saffman-Taylor instability, nonlinear dynamical systems, transition to disorder

Requirements : Basic knowledge in coding.
We use a Labview code to run the experiment and control the camera, the syringe pumps, and the motor linear stage on which the camera sits, as well as a Python code for image analysis. The intern does not need to be a Labview/Python/Matlab expert as we will teach him/her how to run the existing codes and modify them as needed.
The experimental setup is already fully operational.

See this Gallery of fluid motion video for an simple illustration of the work already done : https://www.youtube.com/watch?v=lTOG0KTBM6k&list=PLgxD9DiwxLGrbN4o-AIH2DzD3DjeDRXEO&index=35

Description :
The system is a Hele-Shaw channel (40mm * 1mm rectangular channel of length 170cm) partially occluded by a "rail" (band of adhesive tape) of thickness 20um placed at the centre of the channel. The channel is filled with oil and we investigate the propagation of an air bubble when injecting oil from one side of the channel. Without occlusion, the bubble always ends up propagating steadily at the centre of the channel as this is the only stable steady state of the system. Adding the occlusion adds a new stable mode of propagation where the bubble sits off the rail. This is now a bistable system. This simple system turns out to exhibit a wide range of phenomena that are found in many other nonlinear dynamical systems. Here, capturing the dynamics is very easy since we only need to detect the air-oil interface using a simple camera. Therefore, this system can be thought of as a playground to study nonlinear dynamics in general, with an access to fine details that can not be achieved in some other systems such as shear flows (Poiseuille, Taylor-Couette), due to the complex 3D structure of the flow.

During this internship, the intern will focus on the following question : "How does the bubble choose between these two possible modes of propagation (on-rail and off-rail)". Numerical and preliminary experimental results suggests that the boundary between the two states consists of an unstable periodic orbit where the bubble "wobbles" on the rail. This is called an "edge state" as it sets the edge between the two available stable states (on-rail and off-rail). The experimental work will consist of, starting from a symmetric on-rail propagation, destabilising the bubble by a localised perturbation (for example a small cylinder filling the 1mm thickness of the channel) and observe the transient evolution and final outcome of the bubble. Depending on the amplitude of the perturbation (parametrised by the size of the cylinder and its distance to the channel centreline) the bubble is expected to transiently oscillate (signature of the underlying unstable periodic orbit) and either return back to symmetric propagation for a small perturbation (damped oscillations) or exhibiting growing oscillations leading to either the bubble splitting in two or "falling" off the rail and therefore reaching the off-rail propagation mode for a large perturbation.
This "edge state" tracking is relevant since it is a key concept in other flows such as the transition to turbulence in pipe flows. Indeed, for intermediate Reynolds numbers where both laminar and fully turbulent flows can be observed, a local perturbbation of the laminar flow might lead to either decaying back to laminar flow (small perturbation) of growing to turbulence (large perturbation), where the edge between the two scenarios is described by "edge states" which are weakly unstable solution of the Navier-Stokes equations.

Another question to answer if time is left is the effect of the same local perturbation on the bubble at high flow rate where the propagation of the bubble is about to become disordered. Indeed, like in the transition to turbulence in pipe flows, there is a transition to a disordered state (disordered air-oil front propagation with tip-splittings) beyond some value of the Capillary number (flow rate) which depends on the amount of perturbation in the system (wall roughness), although the trivial state (laminar flow in pipes / steady symmetric finger in Hele-Shaw channels) in linearly stable at all Reynolds / Capillary numbers (subcritical transition). In pipe flows, below the threshold, we know that a local perturbation can lead to a fully turbulent state. Is is also the case for Saffman-Taylor fingers (or bubbles), or will the finger always return to a symmetric steady propagation mode below the threshold ?

Litterature :

Bubble propagation in Hele-Shaw channels with centred constrictions
https://iopscience.iop.org/article/10.1088/1873-7005/aaa5cf/meta

The influence of invariant solutions on the transient behaviour of an air bubble in a Hele-Shaw channel
https://arxiv.org/abs/1907.12932

Contact Anne Juel

<Anne.Juel@manchester.ac.uk>

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Prochain Séminaire de la FIP :
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Retrouvez toutes les informations pour vos stages :
Stages L3
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Actualités : Séminaire de Recherche ICFP
du 25 au 29 novembre 2019 :

Retrouvez le programme complet

Emploi du temps :
Emploi du temps L3
Emploi du temps M1 ICFP
Emploi du temps M2 ICFP

Contact - Secrétariat de l’enseignement :
Tél : 01 44 32 35 60
enseignement@phys.ens.fr