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Non-Newtonian slender drops in a nonlinear extensional flow

Published online by Cambridge University Press:  27 August 2019

Moshe Favelukis Open the ORCID record for Moshe Favelukis [Opens in a new window]
Moshe Favelukis*
Affiliation:
Department of Chemical Engineering, Shenkar – College of Engineering and Design, Ramat-Gan, 5252626, Israel
*
†Email address for correspondence: [email protected]
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Abstract

In this theoretical report we explore the deformation and stability of a power-law non-Newtonian slender drop embedded in a Newtonian liquid undergoing a nonlinear extensional creeping flow. The dimensionless parameters describing this problem are: the capillary number $(Ca\gg 1)$, the viscosity ratio $(\unicode[STIX]{x1D706}\ll 1)$, the power-law index $(n)$ and the nonlinear intensity of the flow $(|E|\ll 1)$. Asymptotic analytical solutions were obtained near the centre and close to the end of the drop suggesting that only Newtonian and shear thinning drops $(n\leqslant 1)$ with pointed ends are possible. We described the shape of the drop as a series expansion about the centre of the drop, and performed a stability analysis in order to distinguish between stable and unstable stationary states and to establish the breakup point. Our findings suggest: (i) shear thinning drops are less elongated than Newtonian drops, (ii) as non-Newtonian effects increase or as $n$ decreases, breakup becomes more difficult, and (iii) as the flow becomes more nonlinear, breakup is facilitated.

JFM classification

Low-Reynolds-number flows: Slender-body theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 877 , 25 October 2019 , pp. 561 - 581
Copyright
© 2019 Cambridge University Press 

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References

Acrivos, A. & Lo, T. S. 1978 Deformation and breakup of a single slender drop in an extensional flow. J. Fluid. Mech. 86, 641–672.10.1017/S0022112078001329Google Scholar
Antanovskii, L. K. 1996 Formation of a pointed drop in Taylor’s four-roller mill. J. Fluid. Mech. 327, 325–341.10.1017/S0022112096008567Google Scholar
Bentley, B. J. & Leal, L. G. 1986 An experimental investigation of drop deformation and breakup in steady, two-dimensional linear flows. J. Fluid Mech. 167, 241–283.10.1017/S0022112086002811Google Scholar
Booty, M. R. & Siegel, M. 2005 Steady deformation and tip-streaming of a slender bubble with surfactant in an extensional flow. J. Fluid Mech. 544, 243–275.10.1017/S0022112005006622Google Scholar
Brady, J. F. & Acrivos, A. 1982 The deformation and breakup of a slender drop in an extensional flow: inertial effects. J. Fluid Mech. 115, 443–451.10.1017/S0022112082000846Google Scholar
Briscoe, B. J., Lawrence, C. J. & Mietus, W. G. P. 1999 A review of immiscible fluid mixing. Adv. Colloid. Interface Sci. 81, 1–17.10.1016/S0001-8686(99)00002-0Google Scholar
Buckmaster, J. D. 1972 Pointed bubbles in slow viscous flow. J. Fluid. Mech. 55, 385–400.10.1017/S0022112072001910Google Scholar
Buckmaster, J. D. 1973 The bursting of pointed drops in slow viscous flow. Trans ASME J. Appl. Mech. E 40, 18–24.10.1115/1.3422923Google Scholar
Chhabra, R. P. 2006 Bubbles, Drops, and Particles in Non-Newtonian Fluids, 2nd edn. CRC Press.10.1201/9781420015386Google Scholar
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops and Particles. Academic Press.Google Scholar
Delaby, I., Ernst, B., Germain, Y. & Muller, R. 1994 Droplet deformation in polymer blends during uniaxial elongational flow: influence of viscosity ratio for large Capillary numbers. J. Rheol. 38, 1705–1720.10.1122/1.550568Google Scholar
Favelukis, M. 2016 A slender drop in a nonlinear extensional flow. J. Fluid Mech. 808, 337–361.10.1017/jfm.2016.646Google Scholar
Favelukis, M. & Nir, A. 2001 Deformation of a slender bubble in a non-Newtonian liquid in an extensional flow. Chem. Engng Sci. 56, 4643–4648.10.1016/S0009-2509(01)00125-7Google Scholar
Favelukis, M., Lavrenteva, O. M. & Nir, A. 2005 Deformation and breakup of a non-Newtonian slender drop in an extensional flow. J. Non-Newtonian Fluid Mech. 125, 49–59.10.1016/j.jnnfm.2004.09.006Google Scholar
Favelukis, M., Lavrenteva, O. M. & Nir, A. 2006 Deformation and breakup of a non-Newtonian slender drop in an extensional flow: inertial effects and stability. J. Fluid Mech. 563, 133–158.10.1017/S0022112006001042Google Scholar
Favelukis, M., Lavrenteva, O. M. & Nir, A. 2012 On the evolution and breakup of slender drops in an extensional flow. Phys. Fluids 24, 043101.10.1063/1.3701373Google Scholar
Han, C. D. & Funatsu, K. 1978 An experimental study of droplet deformation and breakup in pressure-driven flows through converging and uniform channels. J. Rheol. 22, 113–133.10.1122/1.549475Google Scholar
Hinch, E. J. 1980 The evolution of slender inviscid drops in an axisymmetric straining flow. J. Fluid Mech. 101, 545–553.10.1017/S0022112080001784Google Scholar
Hinch, E. J. & Acrivos, A. 1979 Steady long slender droplets in two-dimensional straining motion. J. Fluid Mech. 91, 401–414.10.1017/S0022112079000227Google Scholar
Howell, P. D. & Siegel, M. 2004 The evolution of a slender non-axisymmetric drop in an extensional flow. J. Fluid Mech. 521, 155–180.10.1017/S002211200400148XGoogle Scholar
Levitskiy, S. P. & Shulman, Z. P. 1995 Bubbles in Polymeric Liquids. Technomic Publishing Company.Google Scholar
Milliken, W. J. & Leal, L. G. 1991 Deformation and breakup of viscoelastic drops in planar extensional flows. J. Non-Newtonian Fluid Mech. 40, 355–379.10.1016/0377-0257(91)87018-SGoogle Scholar
Rallison, J. M. 1984 The deformation of small viscous drops and bubbles in shear flows. Annu. Rev. Fluid. Mech. 16, 45–66.10.1146/annurev.fl.16.010184.000401Google Scholar
Sadhal, S. S., Ayyaswamy, P. S. & Chung, J. N. 1997 Transport Phenomena with Drops and Bubbles. Springer.10.1007/978-1-4612-4022-8Google Scholar
Sherwood, J. D. 1984 Tip streaming from slender drops in a nonlinear extensional flow. J. Fluid Mech. 144, 281–295.10.1017/S0022112084001609Google Scholar
Stone, H. A. 1994 Dynamics of drop deformation and breakup in viscous fluids. Annu. Rev. Fluid. Mech. 26, 65–102.10.1146/annurev.fl.26.010194.000433Google Scholar
Sumanasekara, U. R. & Bhattacharya, S. 2017 Detailed finer features in spectra of interfacial waves for characterization of a bubble-laden drop. J. Fluid Mech. 831, 698–718.10.1017/jfm.2017.607Google Scholar
Sumanasekara, U. R., Azese, M. N. & Bhattacharya, S. 2017 Transient penetration of a viscoelastic fluid in a narrow capillary channel. J. Fluid Mech. 830, 528–552.10.1017/jfm.2017.576Google Scholar
Taylor, G. I. 1934 The formation of emulsions in definable fields of flow. Proc. R. Soc. Lond. A A146, 501–523.10.1098/rspa.1934.0169Google Scholar
Taylor, G. I. 1964 Conical free surfaces and fluid interfaces. In Proceedings of the 11th International Congress on Applied Mechanics, Munich, pp. 790–796. Springer.Google Scholar
Tretheway, D. C. & Leal, L. G. 1999 Surfactant and viscoelastic effects on drop deformation in 2-D extensional flow. AIChE J. 45, 929–937.10.1002/aic.690450503Google Scholar
Zapryanov, Z. & Tabakova, S. 1999 Dynamics of Bubbles, Drops and Rigid Particles. Kluwer Academic Publishers.10.1007/978-94-015-9255-0Google Scholar