1. (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.

2. (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

3. (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

4. (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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

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

13. 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

14. 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

15. 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

16. 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