PROBLEM N°13 EGG WHITE PEARLS
Team Ecole polytechnique
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 1 Egg white Pearls
Egg whites are separated from the yolk and put into a syringe. From the syringe, the egg white is ejected into heated oil while the tip is in motion (video on IPT website). How does the size of the egg white pearls produced depend on the various parameters such as the temperature of the oil, ejection and motion speed, nozzle diameter or the non-Newtonian properties of egg whites?
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 2 Experimental setup
Colored egg white
Oil
High-speed camera 1000fps
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 3 Variability between experiments
Egg white: approx. 90% water, 10% proteins (ovalbumin)
Viscoelastic, shear-thinning fluid
Inhomogeneous repartition of proteins
Variability : • between successive experiments • between different eggs
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 4 First observations
Big pearls: free shape, several cm
Medium pearls: rounded shape, 1-2cm
Small pearls: elongated, <1cm
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 5 Formation of big pearls Inhomogeneities
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 6 Formation of big pearls Inhomogeneities
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 7 Formation of medium pearls
Instability similar to Kelvin-Helmholtz
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 8 Formation of medium pearls
Egg white
Elastic force
+ 푅 푅 Shear - 푅++
푅− Shear + 푅−−
Maizena in oil
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 9 Transition medium to big pearls
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 10 Formation of small pearls
Rayleigh-Plateau instability
Elastic behavior ↔ surface tension
Small pearls Formation
Z. Liu - Instability of a viscoelastic liquid jet with axisymmetric and asymmetric disturbances, ijmulflow 2008
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 11
Translation and injection speed (cm)
Injection speed (cm/s)
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 12
Translation and injection speed (cm)
Translation speed (cm/s)
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 13 Speed does not influence pearl size
Kelvin-Helmholtz dispersion relation for speed: 휔 ∝ 푘푈 Time spend falling: 푡 = ℎ/푈 U
Growth of instability: 휔푡 ∝ 푘ℎ no speed dependence ! h
True for injection speed ≫ translation speed
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 14 Influence of oil temperature
Dilatation is negligible
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 15
Influence of oil temperature (cm)
Temperature (C)
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 16 Influence of oil temperature
Temperature increases ⇒ viscosity decreases Viscous K-Helmholtz dispersion relation: 휔 ≈ 푘2μ − 푓 푘
−푓′ Fastest growing wavenumber: 푘 = 2휇
Wavelength 2휋Τ푘 decreases with temperature
Sahasrabudhe - Density, viscosity, and surface tension of five vegetable oils at elevated temperatures: Measurement and modeling, IJFP 2017 Funada - Viscous potential flow analysis of Kelvin-Helmholtz instability in a channel, JFM 2001 ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 17 Influence of oil temperature
T=27C η=1.5mPa.s Silicon oil
T=38C T=107C η=40mPa.s η=7mPa.s
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 18
Influence of nozzle diameter (cm)
Nozzle diameter (mm)
Diameter increase ⇒ Shear reduction ⇒ slower instability ⇒ less small pearls
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 19
Height of oil (cm)
Height of oil (cm)
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 20 Height of oil
Low heights: instability creates droplets decrease of big pearls
Higher heights (>3cm): instability has stabilized more dispersion of pearls less merging
Hypothesis: • Fixed pearl size before merging • Uniform diffusion of pearls in a cone • Fixed height of pearls 푛푢푚푏푒푟 표푓 푝푒푎푟푙푠 푏푒푓표푟푒 푚푒푟푔𝑖푛푔 σ 푎푟푒푎 표푓 표푛푒 푝푒푎푟푙 ≈ 푛푢푚푏푒푟 표푓 푝푒푎푟푙푠 푎푓푡푒푟 푚푒푟푔𝑖푛푔 푏표푡푡표푚 푎푟푒푎 h 1 퐿 ∝ ℎ
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 21 Conclusion
Three kinds of pearls:
Big pearls → accumulation and inhomogeneities
Medium pearls → Kelvin-Helmholtz instability driven by shear-thinning behavior
Small pearls → Rayleigh-Plateau instability driven by viscoelastic behavior
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 22 Conclusion
How does the size of the egg white pearls produced depend on the various parameters such as the temperature of the oil, ejection and motion speed, nozzle diameter or the non-Newtonian properties of egg whites? Influence on the size of pearls : Decreases with temperature of the oil → Viscosity decreases Constant with speed (translation/injection) → 휔푡 = 푘ℎ
1 Decreases with oil height → merging of pearls 퐿 ∝ Decreases with diameter → Slower instability ℎ
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 23 PROBLEM N°13 EGG WHITE PEARLS ADDITIONAL SLIDES
Team Ecole polytechnique
ECOLE POLYTECHNIQUE – International Physicists’ ’ ’ ’ Tournament 2018 24