Skip to main content Accessibility help
×
Home
Hostname: page-component-78dcdb465f-hcvhd Total loading time: 9.499 Render date: 2021-04-17T03:58:45.228Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Deformation and breakup of a viscoelastic drop in a Newtonian matrix under steady shear

Published online by Cambridge University Press:  25 July 2007

NISHITH AGGARWAL
Affiliation:
University of Delaware, Newark, USA
KAUSIK SARKAR
Affiliation:
University of Delaware, Newark, USA

Abstract

The deformation of a viscoelastic drop suspended in a Newtonian fluid subjected to a steady shear is investigated using a front-tracking finite-difference method. The viscoelasticity is modelled using the Oldroyd-B constitutive equation. The drop response with increasing relaxation time λ and varying polymeric to the total drop viscosity ratio β is studied and explained by examining the elastic and viscous stresses at the interface. Steady-state drop deformation was seen to decrease from its Newtonian value with increasing viscoelasticity. A slight non-monotonicity in steady-state deformation with increasing Deborah number is observed at high Capillary numbers. Transient drop deformation displays an overshoot before settling down to a lower value of deformation. The overshoot increases with increasing β. The drop shows slightly decreased alignment with the flow with increasing viscoelasticity. A simple ordinary differential equation model is developed to explain the various behaviours and the scalings observed numerically. The critical Capillary number for drop breakup is observed to increase with Deborah number owing to the inhibitive effects of viscoelasticity, the increase being linear for small Deborah number.

Type
Papers
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below.

References

Barthès-Biesel, D. & Acrivos, A. 1973 Deformation and burst of liquid droplet freely suspended in a linear shear field. J. Fluid Mech. 61, 121.CrossRefGoogle 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, 241283.CrossRefGoogle Scholar
de Bruijn, R. A. 1989 Deformation and breakup of droplets in simple shear flows. PhD thesis, Eindhoven University of Technology, Eindhoven, The Netherlands.Google Scholar
Chaffey, C. E. & Brenner, H. 1967 A second-order theory for shear deformation of drops. J. Colloid Interface Sci. 24, 258269.CrossRefGoogle Scholar
Gauthier, F., Goldsmith, H. L. & Mason, S. G. 1971 Particle motions in non-Newtonian media. II: Poiseuille flow. Trans. Soc. Rheol. 15, 297330.CrossRefGoogle Scholar
Grace, H. P. 1982 Dispersion phenomena in high viscosity immiscible fluid systems and application of static mixers as dispersion devices in such systems. Chem. Engng Commun. 14, 225277.CrossRefGoogle Scholar
Greco, F. 2002 Drop deformation for non-Newtonian fluids in slow flows. J. Non-Newtonian Fluid Mech. 107, 111131.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, 6582.CrossRefGoogle Scholar
Hooper, R. W., De Almedia, 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, 141168.CrossRefGoogle Scholar
Khismatullin, D., Renardy, Y. & Renardy, M. 2006 Development and implementation of VOF-PROST for 3D viscoelastic liquid–liquid simulations. J. Non-Newtonian Fluid Mech. 140, 120131.CrossRefGoogle Scholar
Lerdwijitjarud, W., Larson, R. G., Sirivat, A. & Solomon, M. J. 2003 Influence of weak elasticity of dispersed phase on droplet behavior in sheared polybutadiene/poly(dimethyl siloxane) blends. J. Rheol. 47 (1), 3758.CrossRefGoogle Scholar
Levitt, L. & Macosko, C. W. 1996 Influence of normal stress difference on polymer drop deformation. Polymer Engng Sci. 36, 16471655.CrossRefGoogle Scholar
Li, X. & Sarkar, K. 2005 a Drop dynamics in an oscillating extensional flow at finite inertia. Phys. Fluids 17, 027103.CrossRefGoogle Scholar
Li, X. & Sarkar, K. 2005 b Rheology of a dilute emulsion of drops in finite Reynolds number oscillator extensional flows. J. Non-Newtonian Fluid Mech. 128, 7182.CrossRefGoogle Scholar
Li, X. & Sarkar, K. 2005 c Effects of inertia on the rheology of a dilute emulsion of drops in shear. J. Rheol. 49, 13771394.CrossRefGoogle Scholar
Li, X. & Sarkar, K. 2005 d Negative normal stress elasticity of emulsion of viscous drops at finite inertia. Phys. Rev. Lett. 95, 256001.CrossRefGoogle ScholarPubMed
Li, X. & Sarkar, K. 2006 Drop deformation and breakup in a vortex for finite inertia. J. Fluid Mech. 564, 123.CrossRefGoogle Scholar
Maffetone, P. L. & Greco, F. 2004 Ellipsoidal drop model for single drop dynamics with non-Newtonian fluids. J. Rheol. 48, 83100.CrossRefGoogle Scholar
Maffetone, P. L. & Minale, M. 1998 Equation of change for ellipsoidal drops in viscous flow. J. Non-Newtonian Fluid Mech. 78, 227241.CrossRefGoogle Scholar
Mighri, F. A., Carreau, P. J. & Ajji, A. 1998 Influence of elastic properties on drop deformation and breakup in shear flow. J. Rheol. 42, 14771490.CrossRefGoogle Scholar
Mighri, F. A., Ajji, A. & Carreau, P. J. 1997 Influence of elastic properties on drop deformation in elongational flow. J. Rheol. 41, 11831201.CrossRefGoogle Scholar
Perera, M. G. N. & Walters, K. 1977 Long-range memory effects in flows involving abrupt changes in geometry part I: flows associated with L-shaped and T-shaped geometries. J. Non-Newtonian Fluid Mech. 2, 4981.CrossRefGoogle Scholar
Peskin, C. 1977 Numerical analysis of blood flow in the heart. J. Comput Phys. 25, 220252.CrossRefGoogle Scholar
Pilapakkam, S. B. & Singh, P. 2004 A level-set method for computing solutions to viscoelastic two-phase flow. J. Comput. Phys. 174, 552578.CrossRefGoogle Scholar
Rajagopalan, D., Armstrong, R. C. & Brown, R. A. 1990 Finite element methods for calculations of steady, viscoelastic flow using constitutive equations with a Newtonian viscosity. J. Non-Newtonian Fluid Mech. 36, 159192.CrossRefGoogle Scholar
Rallison, J. M. 1984 The deformation of small viscous drops and bubbles in shear flows. Annu. Rev. Fluid Mech. 16, 4566.CrossRefGoogle Scholar
Rallison, J. M. & Hinch, E. J. 1988 Do we understand the physics in the constitutive equation? J. Fluid Mech. 29, 3755.Google Scholar
Ramaswamy, S. & Leal, L. G. 1999 a The deformation of a viscoelastic drop subjected to steady uniaxial extensional flow of a Newtonian fluid. J. Non-Newtonian Fluid Mech. 85, 127163.CrossRefGoogle Scholar
Ramaswamy, S. & Leal, L. G. 1999 b The deformation of a Newtonian drop in the uniaxial extensional flow of a viscoelastic liquid. J. Non-Newtonian Fluid Mech. 88, 149172.CrossRefGoogle Scholar
Renardy, Y. & Cristini, V. 2001 Effects of inertia on drop breakup under shear. Phys. Fluids 13, 713.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, 315342.CrossRefGoogle Scholar
Sarkar, K. & Schowalter, W. R. 2001 a Deformation of a two-dimensional drop at non-zero Reynolds number in time-periodic extensional flows: numerical simulation. J. Fluid Mech. 436, 177206.Google Scholar
Sarkar, K. & Schowalter, W. R. 2001 b Deformation of a two-dimensional viscous drop time-periodic extensional flows: analytical treatment. J. Fluid Mech. 436, 207230.Google 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, 449456.CrossRefGoogle Scholar
Stone, H. A. 1994 Dynamics of drop deformation and breakup in viscous fluids. Annu. Rev. Fluid Mech. 26, 65102.CrossRefGoogle Scholar
Sun, J. & Tanner, R. I. 1994 Computation of steady flow past a sphere in a tube using a PTT integral model. J. Non-Newtonian Fluid Mech. 54, 379403.CrossRefGoogle Scholar
Sun, J., Phan-Thien, N. & Tanner, R. I. 1996 An adaptive viscoelastic stress splitting scheme and its applications: AVSS/SI and AVSS/SUPG. J. Non-Newtonian Fluid Mech. 65, 7591.CrossRefGoogle Scholar
Taylor, G. I. 1932 The viscosity of a fluid containing small drops of another fluid. Proc. R. Soc. Lond. A 138, 4148.CrossRefGoogle Scholar
Taylor, G. I. 1934 The formation of emulsions in definable fields of flow. Proc. R. Soc. Lond. A 146, 501523.CrossRefGoogle Scholar
Toose, E. M., Guerts, B. J. & Kuerten, J. G. M. 1995 A boundary integral method for two-dimensional (non)-Newtonian drops in slow viscous flow. J. Non-Newtonian Fluid Mech. 60, 129154.CrossRefGoogle Scholar
Torza, S., Coz, R. G. & Mason, S. G. 1972 Particle motions in sheared suspensions XXVII. Transient and steady deformation and burst of liquid drops. J. Colloid Interface Sci. 38, 395411.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 1998–2003. Von Kármán 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 computations of multiphase flow. J. Comput Phys. 169, 708759.CrossRefGoogle Scholar
Tucker, C. L. iii & Moldenaers, P. 2002 Microstructural evolution in polymer blends. Annu. Rev. Fluid Mech. 34, 177210.CrossRefGoogle Scholar
Unverdi, S. O. & Tryggvason, G 1992 A front-tracking method for viscous, incompressible multi-fluid flows. J. Comput. Phys. 100, 2537.CrossRefGoogle Scholar
Varanasi, P. P., Ryan, M. E. & Stroeve, P. 1994 Experimental study on the breakup of model viscoelastic drops in uniform shear flow. Ind. Engng Chem. Res. 33, 18581866.CrossRefGoogle Scholar
Yu, W., Bousmina, M., Zhou, C. & Tucker, C. L. iii 2004 Theory for drop deformation in viscoelastic systems. J. Rheol. 48, 417438.CrossRefGoogle Scholar
Yue, P., Feng, J. J., Liu, C. & Shen, J. 2005 a Viscoelastic effects on drop deformation in steady shear. J. Fluid Mech. 540, 427437.CrossRefGoogle Scholar
Yue, P., Feng, J. J., Liu, C. & Shen, J. 2005 b Transient drop deformation upon start up shear in viscoelastic fluids. Phys. Fluids 17, 123101.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 264 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 17th April 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Deformation and breakup of a viscoelastic drop in a Newtonian matrix under steady shear
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Deformation and breakup of a viscoelastic drop in a Newtonian matrix under steady shear
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Deformation and breakup of a viscoelastic drop in a Newtonian matrix under steady shear
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *