(Photo descriptive du sujet) Problem 11 – Flat Flow IYPT 2012 Bad Salgau French Team (Photo descriptive du sujet) Fill a thin gap between two large transparent horizontal parallel plates with a liquid and make a little hole in the centre of one of the plates. Investigate the flow in such a cell, if a different liquid is injected through the hole.

(Photo descriptive du sujet) Key questions How to produce such a pattern ? What are the relevant parameters ? How to explain this phenomenon ? (Photo descriptive du sujet) IYPT 2012 Bad Saulgau French Team

(Re)producing Pattern : Parameters Injected fluid (i) + + Injection pressure and speed Host fluid (h), between the plates + Surface tension Modifiers e Sucre / eau : densité en mPa.s 40/60 : 6 50/50 : 15 60/40 : 50 Eau : 1 + Surface tension modifiers → such as added surfactant - Thickness « e » - Material of the plates IYPT 2012 Bad Saulgau Better results with one hydrophobic fluid and a water-based solution (h or i irrelevant) French Team

(Re)producing Pattern Hand injection Easy Allow control over relevant parameters (Semi)-Repoducible experiments → Good enough for qualitative quantitative experiments Room for improvement : medical IV-push (mechanical push-seringe) IYPT 2012 Bad Saulgau French Team

Rating pattern How to « rate » pattern ? VS Direct observation : Easy, a bit biased/unscientific → « Does this look unstable ? » Individual features analysis : Easy, Objective results, Low sensibility → « How many fingers ? Mean width ? Computerized methods : Complex+++, Great accuracy, No bias → Area/perimeter ratio, image feature detection VS « Manual » methods chosen : yield sufficient result, considering our time-constraint French Team

Impact of viscosities Low R ratio High R ratio → Fingering → Instability High R ratio → Spreads uniformly → Stable flow → Perturbation are dampened (but still existing) → Unstable → Initial conditions are influential. → Chaotic system Warning : given the parameters, the flow is always strictly laminar IYPT 2012 Bad Saulgau French Team

Impact of viscosities Method : The host fluid is constant (sunflower oil) while viscosity is adjusted in the injected fluid (water with variable amount of sugar dissolved) IYPT 2012 Bad Saulgau French Team

Plate spacing Low spacing → Larger finger →Vertical homogeneous → Initial Cond. = important High spacing → Smaller finger →(Hint of) 3D behaviour → Initial Cond. = less determinant IYPT 2012 Bad Saulgau French Team

Plate spacing IYPT 2012 Bad Saulgau French Team Methods : Put layers of paper between the plate (1 layer = 1 a.u). While determining paper width is possible, it does not readily translate to plate spacing. IYPT 2012 Bad Saulgau French Team

Plate spacing More spacing allow movements along the third axis & Plate spacing Why is spacing relevant ? e More spacing allow movements along the third axis Not a strictly 2D behaviour Injection « front » not vertical Instabilities / inhomogeneity along the third axis → Not desirable for a simple theoretical model → Not studied here :-) IYPT 2012 Bad Saulgau French Team

Surface tension Higher surface tension → Intial perturbation determines future. → Less finger → Less/No « subfinger » Lower surface tension → Initial perurbation not determinant → High instabilty / Turbulence-like features → More finger → Subfinger frequent IYPT 2012 Bad Saulgau Method : Soap added to injected solution French Team

Injection Pressure and Speed High injection pressure : → More finger/subfinger → Tubulent-like. Low/Normal injection pressure IYPT 2012 Bad Salgau French Team

Initial Perturbations Defects in the glass in the fluid repartition in the injection method Prone to infinitesimal change Similar to nucleation site (boiling) → Chaotic phenomenon → Inherently non-reproducible → Limited power of prediction of theoretical model :-) We will still provide a gist-of-thing explaination IYPT 2012 Bad Saulgau French Team

A 1D model of the instability From Darcy's Law ... U : Velocity field n : Viscosity of both fluids K : Constant p : pressure field V : Instant velocity F : Frictional force A : Area of the plates → for a capillary Xf And the following approximation : > 0 We have : C = Non-dependant coefficient Physically, the interface Xf is going away faster with time : the density in the capilar decreases Faster front speed  → Increased inertia of injected fluid → Infinitesimal perturbation amplified to the macroscopic → Finger formation IYPT 2012 Bad Saulgau French Team

A 1D model of the instability How do parameters effect the pattern in theory ? Such that an increase in is a increase in Xf speed, thus in instabities. Instabilities are also linked to the difference What are the effect of surface tension ? Reduced surface tension → energically favorabler fingering (though not included in theoreticam model) IYPT 2012 Bad Saulgau French Team

Thanks for your attention Conclusion Main parameter : Viscosity of both liquid Main mechanism : Micro-perturbation amplification What to explore then : Other theoretical models : Saffman-Taylor instabilities, percolation/ lubrication How to produce such pattern with computer/simulation ? Thanks for your attention IYPT 2012 Bad Saulgau French Team