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Effects of matrix viscoelasticity on viscous and viscoelastic drop deformation in a shear flow

Published online by Cambridge University Press:  25 April 2008

NISHITH AGGARWAL  and
KAUSIK SARKAR
NISHITH AGGARWAL
Affiliation:
Department of Mechanical Engineering, University of Delaware, Newark, DE-19716, USAsarkar@udel.edu
KAUSIK SARKAR
Affiliation:
Department of Mechanical Engineering, University of Delaware, Newark, DE-19716, USAsarkar@udel.edu
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Abstract

The deformation of a Newtonian/viscoelastic drop suspended in a viscoelastic fluid is investigated using a three-dimensional front-tracking finite-difference method. The viscoelasticity is modelled using the Oldroyd-B constitutive equation. Matrix viscoelasticity affects the drop deformation and the inclination angle with the flow direction. Numerical predictions of these quantities are compared with previous experimental measurements using Boger fluids. The elastic and viscous stresses at the interface, polymer orientation, and the elastic and viscous forces in the domain are carefully investigated as they affect the drop response. Significant change in the drop inclination with increasing viscoelasticity is observed; this is explained in terms of the first normal stress difference. A non-monotonic change – a decrease followed by an increase – in the steady-state drop deformation is observed with increasing Deborah number (De) and explained in terms of the competition between increased localized polymer stretching at the drop tips and the decreased viscous stretching due to change in drop orientation angle. The transient drop orientation angle is found to evolve on the polymer relaxation time scale for high De. The breakup of a viscous drop in a viscoelastic matrix is inhibited for small De, and promoted at higher De. Polymeric to total viscosity ratio β was seen to affect the result through the combined parameter βDe indicating a dominant role of the first normal stress difference. A viscoelastic drop in a viscoelastic matrix with matched relaxation time experiences less deformation compared to the case when one of the phases is viscous; but the inclination angle assumes an intermediate value between two extreme cases. Increased drop phase viscoelasticity compared to matrix phase leads to decreased deformation.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 601 , 25 April 2008 , pp. 63 - 84
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Aggarwal, N. & Sarkar, K. 2007 a Deformation and breakup of a viscoelastic drop in a Newtonian matrix under steady shear. J. Fluid Mech. 584, 1–21.CrossRefGoogle Scholar
Aggarwal, N. & Sarkar, K. 2007 b Rheology of an emulsion of viscoelastic drops in steady shear. J. Non-Newtonian Fluid Mech. 150, 19–31.CrossRefGoogle Scholar
Chaffey, C. E. & Brenner, H. 1967. A second order theory for shear deformation of drops. J. Colloid Interface Sci. 24, 258–269.CrossRefGoogle Scholar
Chinyoka, T., Renardy, Y. Y., Renardy, A. & Khismatullin, D. B. 2005 Two-dimensional study of drop deformation under simple shear for Oldroyd-B liquids. J. Non-Newtonian Fluid Mech. 130, 45–56.CrossRefGoogle Scholar
Cox, R. G. 1969 The deformation of a drop in a general time-dependent fluid flow. J. Fluid Mech. 37, 601–623.CrossRefGoogle Scholar
Elmendorp, J. J. & Maalcke, R. J. 1985 A study on polymer blending microrheology. 1. Polymer Engng Sci. 25, 1041–1047.CrossRefGoogle Scholar
Flumerfelt, R. W. 1972 Drop breakup in simple shear fields of viscoelastic fluids. Indust. Engng Chem. Fund. 11, 312–318.CrossRefGoogle Scholar
Greco, F. 2002 Drop deformation for non-Newtonian fluids in slow flows. J. Non-Newtonian Fluid Mech. 107, 111–131.CrossRefGoogle Scholar
Guido, S., Simeone, M. & Greco, F. 2003 Deformation of a Newtonian drop in a viscoelastic matrix under steady shear flow – Experimental validation of slow flow theory. J. Non-Newtonian Fluid Mech. 114, 65–82.CrossRefGoogle Scholar
Hobbie, E. K. & Migler, K. B. 1999 Vorticity elongation in polymeric emulsions. Phys. Rev. Lett. 82, 5393–5396.CrossRefGoogle Scholar
Hooper, R. W., de Almeida, V. F., Macosko, C. W. & Derby, J. J. 2001 Transient polymeric drop extension and retraction in uniaxial extensional flows. J. Non-Newtonian Fluid Mech. 98, 141–168.CrossRefGoogle Scholar
Khismatullin, D. B., Renardy, Y. & Renardy, M. 2006 Development and implementation of VOF-PROST for 3D viscoelastic liquid–liquid simulations. J. Non-Newtonian Fluid Mech. 140, 120–131.CrossRefGoogle Scholar
Levitt, L., Macosko, C. W. & Pearson, S. D. 1996 Influence of normal stress difference on polymer drop deformation. Polymer Engng Sci. 36, 1647–1655.CrossRefGoogle Scholar
Li, X. Y. & Sarkar, K. 2005 a Effects of inertia on the rheology of a dilute emulsion of drops in shear. J. Rheol. 49, 1377–1394.CrossRefGoogle Scholar
Li, X. Y. & Sarkar, K. 2005 b Numerical investigation of the rheology of a dilute emulsion of drops in an oscillating extensional flow. J. Non-Newtonian Fluid Mech. 128, 71–82.CrossRefGoogle Scholar
Maffettone, P. L. & Greco, F. 2004 Ellipsoidal drop model for single drop dynamics with non-Newtonian fluids. J. Rheol. 48, 83–100.CrossRefGoogle Scholar
Maffettone, P. L. & Minale, M. 1998. Equation of change for ellipsoidal drops in viscous flow. J. Non-Newtonian Fluid Mech. 78, 227–241.CrossRefGoogle Scholar
Maffettone, P. L., Greco, F., Simeone, M. & Guido, S. 2005 Analysis of start-up dynamics of a single drop through an ellipsoidal drop model for non-Newtonian fluids. J. Non-Newtonian Fluid Mech. 126, 145–151.CrossRefGoogle Scholar
Mighri, F., Ajji, A. & Carreau, P. J. 1997 Influence of elastic properties on drop deformation in elongational flow. J. Rheol. 41, 1183–1201.CrossRefGoogle Scholar
Mighri, F., Carreau, P. J. & Ajji, A. 1998 Influence of elastic properties on drop deformation and breakup in shear flow. J. Rheol. 42, 1477–1490.CrossRefGoogle Scholar
Minale, M. 2004 Deformation of a non-Newtonian ellipsoidal drop in a non-Newtonian matrix: extension of Maffettone–Minale model. J. Non-Newtonian Fluid Mech. 123, 151–160.CrossRefGoogle Scholar
Peskin, C. 1977 Numerical analysis of blood flow in the heart. J. Comput. Phys. 25, 220–252.CrossRefGoogle Scholar
Pillapakkam, S. B. & Singh, P. 2001 A level-set method for computing solutions to viscoelastic two-phase flow. J. Comput. Phys. 174, 552–578.CrossRefGoogle Scholar
Rallison, J. M. 1984 The deformation of small viscous drops and bubbles in shear flows. Annu. Rev. Fluid Mech. 16, 45–66.CrossRefGoogle Scholar
Ramaswamy, S. & Leal, L. G. 1999 a The deformation of a Newtonian drop in the uniaxial extensional flow of a viscoelastic liquid. J. Non-Newtonian Fluid Mech. 88, 149–172.CrossRefGoogle Scholar
Ramaswamy, S. & Leal, L. G. 1999 b The deformation of a viscoelastic drop subjected to steady uniaxial extensional flow of a Newtonian fluid. J. Non-Newtonian Fluid Mech. 85, 127–163.CrossRefGoogle Scholar
Sarkar, K. & Schowalter, W. R. 2000 Deformation of a two-dimensional viscoelastic drop at non-zero Reynolds number in time-periodic extensional flows. J. Non-Newtonian Fluid Mech. 95, 315–142.CrossRefGoogle Scholar
Sarkar, K. & Schowalter, W. R. 2001 Deformation of a two-dimensional drop at non-zero Reynolds number in time-periodic extensional flows: numerical simulation. J. Fluid Mech. 436, 177–206.CrossRefGoogle Scholar
Sibillo, V., Simeone, M. & Guido, S. 2004 Break-up of a Newtonian drop in a viscoelastic matrix under simple shear flow. Rheol. Acta 43, 449–456.CrossRefGoogle Scholar
Stone, H. A. 1994 Dynamics of drop deformation and breakup in viscous fluids. Annu. Rev. Fluid Mech. 26, 65–102.CrossRefGoogle Scholar
Tagvac, T. 1972 Drop deformation and breakup in simple shear fields. PhD thesis. Chemical Engineering Department, University of Houston, Texas.Google Scholar
Taylor, G. I. 1932 The viscosity of a fluid containing small drops of another fluid. Proc. R. Soc. Lond. 138, 41–48.Google Scholar
Toose, E. M., Geurts, B. J. & Kuerten, J. G. M. 1995 A boundary integral method for 2-dimensional (non)-Newtonian drops in slow viscous-flow. J. Non-Newtonian Fluid Mech. 60, 129–154.CrossRefGoogle Scholar
Torza, S., Mason, S. G. & Cox, R. G. 1972 Particle motions in sheared suspensions. 27. Transient and steady deformation and burst of liquid drops. J. Colloid Interface Sci. 38, 395–411.CrossRefGoogle Scholar
Tryggvason, G., Bunner, B., Ebrat, O. & Taubar, W. 1998 Computation of multiphase flows by a finite difference front tracking method. I. multi-fluid flows. 29th Computational Fluid Dynamics Lecture Series 1998–03. Von Karman Institute of Fluid Dynamics, Sint-Genesius-Rode, Belgium.Google Scholar
Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Taubar, W., Han, J., Nas, S. & Jan, Y. J. 2001 A front-tracking method for the computations of multiphase flow. J. Comput. Phys. 169, 708–759.CrossRefGoogle Scholar
Tretheway, D. C. & Leal, L. G. 2001 Deformation and relaxation of Newtonian drops in planar extensional flows of a Boger fluid. J. Non-Newtonian Fluid Mech. 99, 81–108.CrossRefGoogle Scholar
Tucker, C. L. & Moldenaers, P. 2002 Microstructural evolution in polymer blends. Annu. Rev. Fluid Mech. 34, 177–210.CrossRefGoogle Scholar
Unverdi, S. O. & Tryggvason, G. 1992 A front-tracking method for viscous, incompressible multi-fluid flows. J. Comput. Phys. 100, 25–37.CrossRefGoogle Scholar
Vanoene, H. 1972 Modes of dispersion of viscoelastic fluids in flow. J. Colloid Interface Sci. 40, 448–467.CrossRefGoogle Scholar
Yue, P. T., Feng, J. J., Liu, C. & Shen, J. 2005 a Transient drop deformation upon start up shear in viscoelastic fluids. Phys. Fluids 17, 123101.CrossRefGoogle Scholar
Yue, P. T., Feng, J. J., Liu, C. & Shen, J. 2005 b Viscoelastic effects on drop deformation in steady shear. J. Fluid Mech. 540, 427–437.CrossRefGoogle Scholar