Yield Stress Fluids, Meeting #8

Yield Stress Fluids, Meeting #8 Goldin, Pfeffer and Shinnar. (1972). Break-up of a Capillary Jet of a Non-Newtonian Fluid having a Yield Stress. Thomas J. Ober August 24, 2009 Part of the summer 2009 Reading Group: Yielding, Yield Stresses, Viscoelastoplasticity Non-Newtonian Fluids (NNF) Laboratory, led by Prof. Gareth McKinley

Lord Kelvin - Revolutionary Scientist William Thomson Professor, University of Glasgow Contributions to thermodynamics: Kelvin scale, influence on 1st & 2nd law Contributions to electrical engineering: Transatlantic cable, mirror galvanometer, siphon recorder Kelvin Gallery University of Glasgow University of Glasgow Tower 2

Lord Kelvin - Revolutionary Scientist William Thomson Artificial glacier 1887 Professor, University of Glasgow Contributions to thermodynamics: Kelvin devised this model to show Kelvin scale, influence on 1st & 2nd law the behavior of aether (the substance Contributions to electrical engineering: Transatlantic cable, mirror galvanometer, siphon recorder in which light was once thought to travel). The aether needed to be very rigid for fast motions while at the same time not impede slow motions. If you hit the cobbler s wax hard it will shatter, yet sit it at the top of a stair and it will slowly slide down over time, like a fluid. Kelvin Gallery University of Glasgow Hunterian Museum & Kelvin Gallery. University of Glasgow. 2

Lord Kelvin - Revolutionary Scientist William Thomson Artificial glacier 1887 Professor, University of Glasgow Contributions to thermodynamics: Kelvin devised this model to show Kelvin scale, influence on 1st & 2nd law the behavior of aether (the substance Contributions to electrical engineering: Transatlantic cable, mirror galvanometer, siphon recorder in which light was once thought to travel). The aether needed to be very rigid for fast motions while at the same time not impede slow motions. If you hit the cobbler s wax hard it will shatter, yet sit it at the top of a stair and it will slowly slide down over time, like a fluid. παντα ρει Everything Flows! Kelvin Gallery University of Glasgow Hunterian Museum & Kelvin Gallery. University of Glasgow. 2

Abstract - What was Accomplished? Accomplished: Newtonian and inelastic non-newtonian jets show similar behavior. Normal forces in elastic fluids inhibit breakup. Instabilities in inelastic NNF s due to structural relaxation time. Motivation: Control atomization of NNF s and suspensions. Prevent settling of suspended material high η at low Allow for easy atomization low η at high!!!! Flame thrower Promote atomization Fiber spinning Inhibit atomization Spray paint Promote atomization 3

1. Introduction - Putting Paper in Context In past work : Linear stability analysis: for same ηo, NNF jet break-ups faster than Newtonian jet. Problems: Experimentally, elasticity produces stable jet. Analysis assumes relaxed fluid at nozzle exit. Rationalize contradiction with structural relaxation time. Other timescales: Maxwellian, ψ1/2η, time to regain η (thixotropy) Here: Choose NNF s with minimal elasticity and large thixotropy. Isolate influence of shear-thinning. Gouldin, M., Yerushalmi, J., Pfeffer, R. and R. Shinnar. (1969). Stability of viscoelastic capillary jets. Journal of Fluid Mechanics, 38 689. 4

2. Theoretical Relations Growth rate of symmetrical disturbances with no air resistance: 2 2 2 Ka F Ka Ka 2 3" o ( ) 1 2! $ ( ) & F! + = ' % ( ) ( ) where Ka=2πa/λ 1 For a<<λ, F1 F2 1. Observed growth rate is maximum growth rate. Given a & λ K α α/ (Ka)=0 α* Assume: L CV =! * Weber found: L 2a # a 2 find! 12 = C + " # We 3 We Re solve for $ % & 2# a 3 find We = 2 2! av " Re = 2aV! " o Analytical solution accounting for air drag exists, but must be solved numerically. find F1, F2 = ratio of Bessel functions ρ = density η = viscosity σ = surface tension ao = capillary radius a = jet radius K = wave number λ = disturbance wavelength α = disturbance growth rate α* = observed growth rate L = break-up length V = jet velocity Weber, C. (1931). Z. Angew. Math. Mech., 11 136. 5

3. Experimental Results - Test Fluids Carbopol 934: slightly crosslinked, small N1, no recoil/drag reduction MPA 60: thixotropic, gellation time (hrs) >> jetting time Silica particles: used to gellify material & control flow properties Measured: Surface tension at elevated temperatures (gelled structure temporarily destroyed), extrapolate to room temperature Yield stress & shear stress with viscometer at low rates, with capillary tube at high rates 6

3. Experimental Results - Test Fluids Carbopol 934: slightly crosslinked, small N1, no recoil/drag reduction MPA 60: thixotropic, gellation time (hrs) >> jetting time Silica particles: used to gellify material & control flow properties Measured: Carbopol MPA 60 Silica Gel Surface tension at elevated temperatures (gelled structure temporarily destroyed), extrapolate to room temperature Yield stress & shear stress with viscometer at low rates, with capillary tube at high rates 6

3. Experimental Results - Procedures Pressure transducer Instron/ Pressurized N2 Piston Filter 50-125 µm Capillary Jet Apparatus mounted on concrete blocks to dampen vibrations. Hypodermic needle nozzles, checked for eccentricities. Nozzle l/d > 100 typically 7

3. Experimental Results - Qualitative Results 0.2% Carbopol and 24% MPA 60 break up like Newtonian fluid Figure 3 0.2% Carbopol 2ao=0.0414 cm V=438 cm/s downstream Figure 4 MPA 60 2ao=0.0414 cm V=258 cm/s downstream 0.6% Carbopol breaks up randomly due to frictional air drag downstream Figure 5 0.6% Carbopol 2ao=0.0414 cm V=4710 cm/s 8

3. Experimental Results - Qualitative Results 12.4% Silica breaks up due to random large amplitude disturbances No break-up within 1.5 m Break-up due to large amplitude random disturbances Break-up due to onset of turbulence downstream Figure 6 2ao=0.180 cm V=1080 cm/s Figure 7 2ao=0.180 cm V=1950 cm/s Figure 8 2ao=0.180 cm V=2850 cm/s All inelastic NNF s could be atomized. 9

3. Experimental Results - Quantitative Results Use Newtonian wall shear rate for characteristic shear rate. ( )= ( ) #!"! Solve for disturbance growth with air resistance (numerically): F! 1! = 4V a o 2 + K 4Va o Use empirical corrections for F3 and C based on Ohnesorge#. ' 1 Re 2 3" ( Ka) F Ka Ka o 2! $ 1 = ( ) & ' % ( ) 2 3 # a 2# a n ' 2 2 o n ' ao V = ( ) 2! 2 n '! 1 K ' 8 ( ) # V 2 ( Ka ) 3 F % A 2 2# a n ' 3 " L We 12 CV o = Z o =! * Re o Weber, C. (1931). Z. Angew. Math. Mech., 11 136. Grant, R.P. and S. Middleman. (1966). A.I.Ch.E J., 12 669. 10

3. Experimental Results - Quantitative Results 0.2% Carbopol Use of ηo Apparently (at 0.1 faster s -1 ) break-up gives than large errors. that of Newtonian fluid of Evidently, the comparable fluids viscosity. are not able to relax back their unyielded state, so characteristic η is not ηo. Evidence of structural relaxation time? MPA 60 11

3. Experimental Results - Quantitative Results 0.2% Carbopol Agreement between experiment and theory not particularly stellar. Use of ηo Apparently (at 0.1 faster s -1 ) break-up gives than large errors. that of Newtonian fluid of Evidently, the comparable fluids viscosity. are not able to relax back their unyielded state, so characteristic η is not ηo. Evidence of structural relaxation time? MPA 60 11

3. Experimental Results - Quantitative Results 0.2% Carbopol Agreement between experiment and theory not particularly stellar. Use of ηo (at 0.1 s -1 ) gives large errors. Evidently, the fluids are not able to relax back their unyielded state, so characteristic η is not ηo. Evidence of structural relaxation time? 11

3. Experimental Results - Quantitative Results NNF jets match Newtonian jets if Newtonian wall shear rate is used as characteristic shear rate. Use of ηo gives order of magnitude disagreement. Similarity to Newtonian jets further evidence of structural relaxation time? Characteristic wall shear rate! = 4V a o We = 2 2! av " Re o n ' ao V = ( ) 2! 2 n '! 1 K ' 8 n ' " 12

3. Experimental Results - Quantitative Results Weber found relation for disturbance wavelength for Newtonian jet 12 with no air resistance: & ) Authors measured disturbance wavelength to find apparent viscosity.! 2a = " 2( 1+ ' 3# o + 2a$% * Differences in viscosities indicate structural formation within the liquid during the lifetime of the jet. 13

4. Discussion of Results - Observations & Commentary Observations: Pseudoplastics exhibit great variability in jetting behavior. Pseudoplastics having large N1 are harder to atomize. Yield stress does seem to correlate with jetting behavior. Commentary: LVE theory predicts increased instability; experimentally N1 inhibits break-up. Predict stabilizing yield stress, but experimentally σy irrelevant. Some non-lve relaxation process may dictate jetting behavior. Heavily sheared MPA exhibits time-dependent η. Dilute solutions break up sooner than comparable Newtonian fluid. Possibly only due to definition of characteristic shear rate. 14

4. Discussion of Results - Concluding Points 15

4. Discussion of Results - Concluding Points Viscous/jellified solutions can be atomized if they are inelastic & highly sheared. 15

4. Discussion of Results - Concluding Points Viscous/jellified solutions can be atomized if they are inelastic & highly sheared. Inelastic, shear-thinning fluids have appreciable structural? relaxation times. than flow time? Elastic solutions regain viscosity much faster even if they have large relaxation times. Maxwellian? Shear-thinning, inelastic properties may promote wave growth and break-up in unexplained way. /? 15

Possible Analytical Approach - Partly considered... Consider critical dimensionless groups. Length Scales Time Scales Pressure/ Stress 2ao λm ρv 2 L ψ1/2η τy Suggested 3rd axis if a structural time is important! = " V 1 struct 2 a o OR! 2 = " struct V L λ η/τy ηv/ao α -1 λstruct σ/ao Sv =! y! y Re = 2a o! V Bi = " " V 2 " We 12! We = 2 2! av o Z = = " Re "# 2a o Wi =! V ao 2a o Kao = 2! " V ao 2! = L 2a o We Minimum Re-We plane of true experimental domain Goldin et al. restrict analysis to Re-We plane Re 16

Other Studies (1) Coussot and Gaulard. Gravity flow instability of viscoplastic materials: The ketchup drip. Physical Review E 72.031409. Examine break-up of jets under their own weight. Model flow with Herschel-Bulkley model. Varying trends in break-up length with velocity observed. Fluid τc [Pa] Mayo 15 Gel 51 Bent. 84 Ket. 30 X0 V/πR0 2 Xc 3 1/2 (τc/ρg) λ=x0/xc ν (K/τc) 1/n ρgu/τc 17

Other Studies (2) Ellwood, Georgiou, Papanastasiou, and Wilkes. Laminar jets of Bingham-plastic liquids. J. Rheol 34(6), 1990. Finite-element model of laminar jet break-up of Bingham-like fluids. Constitutive Model* #! y! = 2 % µ + # 1 1" " $% $ % 2 2 exp m ) D ( ) ) D 1 2 & &( D '( '( µ " µ ' µ " µ 0 Carreau Model ' = # +(! ) 2 n 1 ( % $ D & ( " 1) 2 Yield stress postpones break-up time Ty=Rτy/σ should be Cite Gouldin et al. *T. Papanastasiou. J. Rheol. 31, 385 1987. 18

More on Yield Stress Jets - My Own Research Interplay between inertia, viscosity and effective yield stress in high speed jets lead to peculiar, but fascinating consequences. Re =! " DV o Bi = " o! y ( ) 8V D Wi = ( )! 8V D Sv =! " y V 2 19