Convective Instability in quasi 2D foams Eric Janiaud, Stefan Hutzler, Denis Weaire. Trinity College, Dublin

Convection in (3d) foams  Add liquid on top of a foam :  if Q> Threshold: plastic flow of bubbles.  Add liquid on top of a foam :  if Q> Threshold: plastic flow of bubbles. L. Alonso

Analogy with Thermal convection Driving force: Damping:     + d  Gravity thought density variation Viscosity: energy dissipated in T1 Capillary diffusion: reduces density variation Plastic yield stress: capillary stress balances non uniform hydrostatic pressure    + d  Gravity thought density variation

Difference with Thermal convection Heat diffusion Gaussian continuous media Newtonian liquid Heat diffusion Gaussian continuous media Newtonian liquid Liquid diffusion non linear (chemistry) Discrete network Yield stress+dilatency Liquid diffusion non linear (chemistry) Discrete network Yield stress+dilatency Ra = F Archimede / F viscosity Nu = Q convection /Q no convection Pr = D mvt / D heat Pe = T diffusion / T convection ?

Issues Mix Rheology and drainage Onset: dilatency, surfactants, bubble size,  gradient... After the onset : Shear bands, efficiency on liquid transport, size sorting... Mix Rheology and drainage Onset: dilatency, surfactants, bubble size,  gradient... After the onset : Shear bands, efficiency on liquid transport, size sorting...

From 3D to 2D...  Decoration theorem:  Quasi 2D in a Hele-Shaw : Film Vertex “decorated” Pseudo Plateau Border No liquid flow between vertex Drainage is possible !

Quasi 2D foams in Hele-Shaw cell

What is the Volume fraction?  Wetting Plateau border V~R 2 L  Film V~L 2 e  “Perpendicular” Plateau border V~R 2 d  Vertices V~R 3  Wetting Plateau border V~R 2 L  Film V~L 2 e  “Perpendicular” Plateau border V~R 2 d  Vertices V~R 3 Needs experiments and Evolver Simulations Cox

Hele-Shaw Cell setup  Sensitive to the wetting film  Bidisperse foam Area ~90 and 130 pixel  Sensitive to the wetting film  Bidisperse foam Area ~90 and 130 pixel Projector Q

Steady state Drainage  Injection point  Uniform: free convection  Local: forced convection  Injection point  Uniform: free convection  Local: forced convection Q Q What is the solution to the drainage equation? What is the caracteristic length for roll?

Local injection Q

Vertical Velocity Velocity (pi/s) Horizontal coordinate (pi) 3 pictures Smooth profile = No shear Band

Non uniform liquid Fraction How to measure the volume fraction?

Image Analysis : Vertices Velocity: tracking bubbles T1: neighbourhood relations Local Volume fraction: ration of Pixel Stress/Strain field: Vertex position Capillary pressure curvature measurement

Plugin for ImageJ  Public/free Java software developed by the NIH  Object oriented easy implementation  Already rich  Multiplatform  Public/free Java software developed by the NIH  Object oriented easy implementation  Already rich  Multiplatform

Perspectives  Forced 2D convection impose and measure  measure the velocity field  Free 2D convection homogenous liquid input !!! boundary conditions !!! destabilisation length?  Back to 3D convection  Forced 2D convection impose and measure  measure the velocity field  Free 2D convection homogenous liquid input !!! boundary conditions !!! destabilisation length?  Back to 3D convection