11 Flat Flow - iypt solutions
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11 Flat Flow by Kamila SouΔková Task Fill a thin gap between two large transparent horizontal parallel plates with a liquid and make a little hole in the center of one of the plates. Investigate the flow in such a cell, if a different liquid is injected through the hole. 2 3 11 12 Interfacial Pressure π1 π2 π β π ≈ Δππ = π1 − π2 = 2 π interfacial pressure due to curved surface must be overcome π ≈ 10−1 , π ≈ 10−4 ⇒ Δππ ≈ 103 pressures we work with are ~104 to 105 ⇒ negligible 13 Understood Phenomena • flow through a porous medium – Darcy’s Law: π£ = π − π π»π Figure from Hornberger et al. (1998) • flow in a small gap – Hele-Shaw equation: π£ = k : permeability of medium h : gap between plates β2 − π»π 12π π : viscosity of liquid ∇p : pressure gradient 14 Position, Velocity • measure how fast the interface moves – at various places – take the average position / cm 5 4 3 weight on syringe removed 2 1 0 0 0.5 1 1.5 2 time / s 2.5 3 15 3.5 Formation of Instabilities 16 Formation of Instabilities small instability pressure differences instability grows “fingers” 17 Slow-Motion Video of Patterns 300 fps 18 stabilizing destabilizing What Affects Instabilities? promotes disturbances • high viscosity of liquid in the gap → pressure differences • big pressure gradient tries to dampen out disturbances • surface tension 19 Low Viscosity → No Fingers low viscosity symmetrical situation easier to spread out evenly no “fingers” 20 Low Viscosity → No Fingers Ink (less viscous) Glycerol (more viscous) 21 PATTERN COMPLEXITY MEASUREMENTS Equipment • Liquid : – more viscous: glycerol, motor oil – less viscous: water (colored), ink, ethanol • Plates – plexiglass (Hele–Shaw cell): – 25 x 25 cm – gap: distance set by weights – hole: size customized to the syringe 23 Equipment 24 Equipment 25 What to Measure quantify the instabilities • count perturbances – tells how “interesting” the pattern is – to make more objective: 26 VISCOSITY 27 Viscosity of Medium water → motor oil 10W water → motor oil 5W Viscosity < π = 0.14 Pa.s π = 0.17 Pa.s 28 Viscosity of Medium water → glycerol water → motor oil 15W Viscosity < π = 0.41 Pa.s π = 1.48 Pa.s 29 # of Fingers vs Viscosity of Medium Number of perturbances 30 25 20 15 10 5 0 0 0.5 1 Viscosity [Pa.s] 1.5 2 30 Viscosity of Injected Liquid old ink (more viscous) → glycerol new ink (less viscous) → glycerol Viscosity > 31 DISTANCE OF THE PLATES 32 Gap Size 1mm (Water → glycerol) ≈0.8 mm 33 ≈0.4mm ≈0.2mm ≈0.1mm 34 # of Fingers vs Gap Size 45 Number of perturbances 40 35 30 25 20 15 10 5 0 0 0.2 0.4 0.6 Gap [mm] 0.8 1 1.2 35 PRESSURE IN SYRINGE 36 Pressure in Syringe changed by putting weights on syringe p = 15 kPa (300g) p = 22 kPa (450g) p = 49 kPa (1 kg) 37 # of Fingers vs Pressure 30 Number of perturbances 25 20 15 10 5 0 0 10 20 30 40 50 60 pressure [kPa] 38 INTERFACIAL TENSION 39 Interfacial Tension Ethanol: Water: π = 0.009 Pa.s πΎ = 0.0225 N/m π = 0.001 Pa.s πΎ = 0.072 N/m → glycerol → glycerol 40 Air → water ( = 0.072 N/m) Higher surface tension → more rounded Air → water with detergent (δ = 0.025 N/m) Lower surface tension → greater instability 41 FURTHER INVESTIGATION 42 An analogous experiment • Jose A. Miranda, Michael Widom: Radial Fingering in a Hele-Shaw Cell: a weakly nonlinear analysis, Physica D 120(1998) 315-328 π= 2 ππ (π2 − π1 ) π β3 πΎ Q = flow [m3/s] R = initial radius [m] π1 = viscosity of injected liquid [Pa.s] π2 = viscosity of medium [Pa.s] h = gap width [m] πΎ = interfacial surface tension [N.m] 43 Viscosity π2 Number of perturbances 30 25 20 15 10 π= 5 2 ππ (π2 − π1 ) π β3 πΎ 0 0 0.5 1 Viscosity [Pa.s] 1.5 2 44 Gap Size h 45 Number of perturbances 40 π= 35 30 2 ππ (π2 − π1 ) π β3 πΎ 25 20 15 10 5 0 0 0.2 0.4 0.6 Gap [mm] 0.8 1 1.2 45 Pressure p Number of perturbances 30 25 20 15 2 ππ (π2 − π1 ) π β3 πΎ π= 10 5 π ∝ π − π0 0 0 10 20 30 40 50 60 pressure [kPa] 46 Interfacial Tension πΎ Ethanol → glycerol πΎ = 0,0225 N/m Water → glycerol πΎ = 0,072 N/m π ∝ 1/ πΎ 47 Conclusion • observed the phenomenon • showed when it does not work • explained the mechanism of emerging patterns 48 Conclusion • Proved assumed influences on the phenomenon: οpressure οviscosity < οgap between the plates > < οsurface tension < 49 Thank you for your attention! • Proved assumed influences on the phenomenon: οpressure οviscosity < οgap between the plates > < οsurface tension < 50 Thank you for your attention ! 11. Flat Flow Kamila SouΔková APPENDIX Changing temperature of glycerol 80°C → the same patterns Data for experiments Motor oil → glycerol (β ν = 1119,5 . 10-6 m2/s)
