nach oben

Rheologica Acta

Erschienen in:

01.08.2010 | Review

Models for the deformation of a single ellipsoidal drop: a review

verfasst von: Mario Minale

Erschienen in: Rheologica Acta | Ausgabe 8/2010

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

Dilute polymer blends and immiscible liquid emulsions are characterized by a globular morphology. The dynamics of a single drop subjected to an imposed flow field has been considered to be a valuable model system to get information on dilute blends. This problem has been studied either theoretically by developing exact theories for small drop deformations or by developing simplified models often based on phenomenological assumptions. In this paper, a critical overview of the available models for the dynamics of a single drop is presented, discussing four different systems, namely the Newtonian system, where a single Newtonian drop is immersed in an infinite Newtonian matrix; the non-Newtonian system, where at least one of the components, the drop fluid or the matrix one, is non-Newtonian; the confined Newtonian system, where the matrix is confined and wall effects alter the drop dynamics; and the confined non-Newtonian system.

Nächster Artikel Large amplitude oscillatory extension of soft polymeric networks

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Almusallam AS, Larson RG, Solomon MJ (2000) Constitutive model for prediction of ellipsoidal droplet shapes and stresses in immiscible blends. J Rheol 44:1055–1083CrossRefADS

Almusallam AS, Larson RG, Solomon MJ (2004) Comprehensive constitutive model for immiscible blends of Newtonian polymers. J Rheol 48:319–348CrossRefADS

Astarita G, Marrucci G (1974) Principles of non-Newtonian fluid mechanics. McGraw-Hill, New York

Batchelor GK (1970) The stress system in a suspension of force-free particles. J Fluid Mech 41:545–570MATHCrossRefMathSciNetADS

Bentley BJ, Leal LG (1986) An experimental investigation of drop deformation and breakup in steady two-dimensional linear flows. J Fluid Mech 167:241–283MATHCrossRefADS

Cardinaels R, Verhulst K, Renardy Y, Moldenaers P (2008) Transient droplet behavior and droplet breakup during bulk and confined shear flow in blends with one viscoelastic component: experiments, modelling and simulations. In: Proceedings of the XVth international congress on rheology, August 2008, Monterey, USA, pp 1405–1407

Carotenuto C, Grosso M, Maffettone PL (2008) Fourier transform rheology of dilute immiscible polymer blends: a novel procedure to probe blend morphology. Macromolecules 41:4492–4500CrossRefADS

Chaffey CE, Brenner H (1967) A second-order theory for shear deformation of drops. J Colloid Interface Sci 24:258–269CrossRef

Cristini V, Tan Y (2004) Theory and numerical simulation of droplet dynamics in complex flows—a review. Lab Chip 4:257–264CrossRefPubMed

Cristini V, Hooper RW, Macosko CW, Simeone M, Guido S (2002) A numerical and experimental investigation of lamellar blend morphologies. Ind Eng Chem Res 41:6305–6311CrossRef

de Bruijn RA (1989) Deformation and breakup of drops in simple shear flow. PhD thesis, Eindhoven University of Technology

Delaby I, Ernst B, Germain Y, Muller R (1995) Droplet deformation during elongational flow in blends of viscoelastic fluids. Small deformation theory and comparison with experimental results. Rheol Acta 34:525–533CrossRef

Doi M, Ohta T (1991) Dynamics and rheology of complex interfaces. J Chem Phys 95:1242–1248CrossRefADS

Dressler M, Edwards BJ (2004a) The influence of matrix viscoelasticity on the rheology of polymer blends. Rheol Acta 43:257–282CrossRef

Dressler M, Edwards BJ (2004b) Rheology of polymer blends with matrix viscoelasticity and narrow droplet size distribution. J Non-Newton Fluid Mech 120:189–205MATHCrossRef

Dressler M, Edwards BJ, Windhab EJ (2008) An examination of droplet deformation and break-up between concentrically rotating cylinders. J Non-Newton Fluid Mech 152:86–100MATHCrossRef

Edwards BJ, Dressler M (2003) A rheological model with constant approximate volume for immiscible blends of ellipsoidal droplets. Rheol Acta 42:326–337CrossRef

El Afif A, El Omari M (2009) Low and mass transport in blends of immiscible viscoelastic polymers. Rheol Acta 48:285–299CrossRef

Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc Lond A 241:376–396MATHCrossRefMathSciNetADS

Frankel NA, Acrivos A (1970) The constitutive equation for a dilute emulsion. J Fluid Mech 44:65–78MATHCrossRefADS

Grace HP (1982) Dispersion phenomena in high viscosity immiscible fluid systems and applications of static mixers as dispersion devices. Chem Eng Commun 14:225–227CrossRef

Greco F (2002) Drop deformation for non-Newtonian fluids in slow flows. J Non-Newton Fluid Mech 107:111–131MATHCrossRef

Grizzuti N, Bifulco O (1997) Effects of coalescence and break-up on the steady-state morphology of an immiscible polymer blend in shear flow. Rheol Acta 36:406–415

Grmela M, Ait-Kadi A (1998) Rheology of inhomogeneous immiscible blends. J Non-Newton Fluid Mech 77:191–199MATHCrossRef

Grmela M, Ottinger HC (1997) Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Phys Rev E 56:6620–6626CrossRefMathSciNetADS

Grmela M, Ait-Kadi A, Utracki LA (1998) Blends of two immiscible and rheologically different fluids. J Non-Newton Fluid Mech 77:253–259MATHCrossRef

Grmela M, Bousmina M, Palierne J (2001) On the rheology of immiscible blends. Rheol Acta 40:560–569CrossRef

Gu JF, Grmela M (2008) Flow properties of immiscible blends: Doi–Ohta model with active advection. Phys Rev E 78:056302/1–056302/17MathSciNetADS

Guido S, Greco F (2004) Dynamics of a liquid drop in a flowing immiscible liquid. Rheol Rev 2:99–142

Guido S, Villone M (1998) Three dimensional shape of a drop under simple shear flow. J Rheol 42:395–415CrossRefADS

Guido S, Minale M, Maffettone PL (2000) Drop shape dynamics under shear-flow reversal. J Rheol 44:1385–1399CrossRefADS

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-Newton Fluid Mech 114:65–82CrossRef

Han CD (1981) Multiphase flow in polymer processing. Academic, London

Happel J, Brenner H (1983) Low Reynolds number hydrodynamics. Martinus Nijhoff Publishers, Dordrecht

Jackson NE, Tucker CL III (2003) A model for large deformation of an ellipsoidal droplet with interfacial tension. J Rheol 47:659–682CrossRefADS

Janssen PJA, Anderson PD (2007) Boundary-integral method for drop deformation between parallel plates. Phys Fluids 19:043602/1–043602/11CrossRefADS

Jansseune J, Mewis J, Moldenaers P, Minale M, Maffettone PL (2000) Rheology and rheological morphology determination in immiscible polymer blends. J Non-Newton Fluid Mech 93:153–165MATHCrossRef

Jansseune J, Vinckier I, Moldenaers P, Mewis J (2001) Transient stresses in immiscible model polymer blends during start-up flows. J Non-Newton Fluid Mech 99:167–181MATHCrossRef

Kennedy MR, Pozrikidis C, Skalak R (1994) Motion and deformation of liquid drops, and the rheology of dilute emulsions in simple shear flow. Comput Fluids 23:251–278MATHCrossRef

Khakhar DV, Ottino JM (1986) Deformation and breakup of slender drops in linear flows. J Fluid Mech 166:265–285MATHCrossRefADS

Lacroix C, Grmela M, Carreau PJ (1998) Relationships between rheology and morphology for immiscible molten blends of polypropylene and ethylene copolymers under shear flow. J Rheol 42:41–62CrossRefADS

Larson RG (1999) The structure and rheology of complex fluids. Oxford University Press, New York

Lee HM, Park OO (1994) Rheology and dynamics of immiscible polymer blends. J Rheol 38:1405–1425CrossRefADS

Lerdwijitjarud W, Larson RG, Sirivat A, Solomon MJ (2003) Influence of weak elasticity of dispersed phase on droplet behavior in sheared polybutadiene/poly(dimethyl siloxane) blends. J Rheol 47:37–58CrossRefADS

Maffettone PL, Greco F (2004) Ellipsoidal drop model for single drop dynamics with non-Newtonian fluids. J Rheol 48:83–100CrossRefADS

Maffettone PL, Minale M (1998) Equation of change for ellipsoidal drops in viscous flow. J Non-Newton Fluid Mech 78:227–241 Erratum (1999) J Non-Newton Fluid Mech 84:105–106

Minale M (2004) Deformation of a non-Newtonian ellipsoidal drop in a non-Newtonian matrix: extension of Maffettone–Minale model. J Non-Newton Fluid Mech 123:151–160MATHCrossRef

Minale M (2008) A phenomenological model for wall effects on the deformation of an ellipsoidal drop in viscous flow. Rheol Acta 47:667–675CrossRef

Minale M, Maffettone PL (2003) Morphology estimation from normal stress measurements for dilute immiscible polymer blends. Rheol Acta 42:158–165CrossRef

Minale M, Moldenaers P, Mewis J (1997) Effect of shear history on the morphology of immiscible polymer blends. Macromolecules 30:5470–5475CrossRefADS

Minale M, Mewis J, Moldenaers P (1998) Study of the morphological hysteresis in immiscible polymer blend. AIChE J 44:943–950CrossRef

Minale M, Moldenaers P, Mewis J (1999) Transient flow experiments in a model immiscible polymer blend. J Rheol 43:815–827CrossRefADS

Minale M, Caserta S, Guido S (2010) Microconfined shear deformation of a droplet in an equiviscous non-Newtonian immiscible fluid: experiments and modeling. Langmuir 26:126–132CrossRefPubMed

Okamoto K, Takahashi M, Yamane H, Kashihara H, Watanabe H, Masuda T (1999) Shape recovery of a dispersed droplet phase and stress relaxation after application of step shear strains in a polystyrene polycarbonate blend melt. J Rheol 43:951–965CrossRefADS

Ottinger HC, Grmela M (1997) Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism. Phys Rev E 56:6633–6655CrossRefMathSciNetADS

Ottino JM, De Roussel P, Hansen S, Khakhar DV (2000) Mixing and dispersion of viscous liquids and powdered solids. Adv Chem Eng 25:105–204CrossRef

Palierne JF (1990) Linear rheology of viscoelastic emulsions with interfacial tension. Rheol Acta 29:204–214 Erratum (1991) Rheo Acta 30:497CrossRef

Peters GWM, Hansen S, Meijer HEH (2001) Constitutive modeling of dispersive mixtures. J Rheol 45:659–689CrossRefADS

Rallison JM (1980) Note on the time-dependent deformation of a viscous drop which is almost spherical. J Fluid Mech 98:625–633MATHCrossRefMathSciNetADS

Rallison JM (1984) The deformation of small viscous drops and bubbles in shear flows. Ann Rev Fluid Mech 16:45–66CrossRefADS

Rallison JM, Acrivos A (1978) A numerical study of the deformation and burst of a viscous drop in an extensional flow. J Fluid Mech 89:191–200MATHCrossRefADS

Shapira M, Haber S (1988) Low Reynolds number motion of a droplet between two parallel plates. Int J Multiphase Flow 14:483–506MATHCrossRef

Shapira M, Haber S (1990) Low Reynolds number motion of a droplet in shear flow including wall effects. Int J Multiphase Flow 16:305–321MATHCrossRef

Sibillo V, Simeone M, Guido S, Greco F, Maffettone PL (2006a) Start-up and retraction dynamics of a Newtonian drop in a viscoelastic matrix under simple shear flow. J Non-Newton Fluid Mech 134:27–32MATHCrossRef

Sibillo V, Pasquariello G, Simeone M, Cristini V, Guido S (2006b) Drop deformation in microconfined shear flow. Phys Rev Lett 97:054502CrossRefPubMedADS

Stone HA (1994) Dynamics of drop deformation and breakup in viscous fluids. Ann Rev Fluid Mech 26:65–102CrossRefADS

Stone HA, Leal LG (1989) Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid. J Fluid Mech 198:399–427CrossRefADS

Stone HA, Bentley BJ, Leal LG (1986) An Experimental study of transient effects in the breakup of viscous drops. J Fluid Mech 173:131–158CrossRefADS

Taylor GI (1932) The viscosity of a fluid containing small drop of another fluid. Proc R Soc A 138:41–48MATHCrossRefADS

Taylor GI (1934) The formation of emulsion in definable field of flow. Proc R Soc A 146:501–523CrossRefADS

Toose EM (1997) Simulation of the deformation of non Newtonian drops in viscous flows. PhD thesis, Twente University, The Netherlands

Tretheway DC, Leal LG (1999) Surfactant and viscoelastic effects on drop deformation in 2-D extensional flow. AIChE J 45:929–937CrossRef

Tretheway DC, Leal LG (2001) Deformation and relaxation of Newtonian drops in planar extensional flows of a Boger fluid. J Non-Newton Fluid Mech 99:81–108MATHCrossRef

Truesdell CA, Noll W (1965) The non-linear field theories of mechanics. In: Flugge S (ed) Encyclopedia of physics, vol III/3. Springer, Berlin

Torza S, Cox RG, Mason SG (1972) Particle motions in sheared suspensions. J Colloid Interface Sci 38:395–411CrossRef

Tucker CL, Moldenaers P (2002) Microstructural evolution in polymer blends. Ann Rev Fluid Mech 34:177–210CrossRefMathSciNetADS

Vananroye AP, Van Puyvelde P, Moldenaers P (2006) Effect of confinement on droplet breakup in sheared emulsions. Langmuir 22:3972–3974CrossRefPubMed

Vananroye AP, Van Puyvelde P, Moldenaers P (2007) Effect of confinement on the steady-state behaviour of single droplets during shear flow. J Rheol 51:139–153CrossRefADS

Van Puyvelde P, Moldenaers P (2005) Rheology and morphology development in immiscible polymer blends. Rheology Reviews of the British Society of Rheology, pp 101–145

Van Puyvelde P, Vananroye A, Cardinaels R, Moldenaers P (2008) Review on morphology development of immiscible blends in confined shear flow. Polymer 49:5363–5372CrossRef

Verhulst K, Moldenaers P, Minale M (2007) Drop shape dynamics of a Newtonian drop in a non-Newtonian matrix during transient and steady shear flow. J Rheol 51:261–273CrossRefADS

Verhulst K, Cardinaels R, Renardy Y, Moldenaers P (2008) Steady state droplet deformation and orientation during bulk and confined shear flow in blends with one viscoelastic component: experiments, modeling and simulations. In: Proceedings of the XVth international congress on rheology, August 2008, Monterey, USA, pp 1000–1002

Verhulst K, Cardinaels R, Moldenaers P, Renardy Y, Afkhamib S (2009a) Influence of viscoelasticity on drop deformation and orientation in shear flow. Part 1. Stationary states. J Non-Newton Fluid Mech 156:29–43CrossRef

Verhulst K, Cardinaels R, Moldenaers P, Renardy Y, Afkhamib S (2009b) Influence of viscoelasticity on drop deformation and orientation in shear flow. Part 2: dynamics. J Non-Newton Fluid Mech 156:44–57CrossRef

Vinckier I, Moldenaers P, Mewis J (1996) Relationship between rheology and morphology of model blends in steady shear flow. J Rheol 40:613–631CrossRefADS

Wagner NJ, Ottinger HC, Edwards BJ (1999) Generalized Doi–Ohta model for multiphase flow developed via GENERIC. AIChE J 45:1169–1181CrossRef

Wannaborworn S, Mackley MR, Renardy Y (2002) Experimental observation and matching numerical simulation for the deformation and breakup of immiscible drops in oscillatory shear. J Rheol 46:1279–1293CrossRefADS

Wetzel ED, Tucker CL III (1999) Area tensors for modeling microstructure during laminar liquid–liquid mixing. Int J Multiphase Flow 25:35–61MATHCrossRef

Wetzel ED, Tucker CL III (2001) Droplet deformation in dispersions with unequal viscosities and zero interfacial tension. J Fluid Mech 426:199–228MATHCrossRefADS

Wu Y, Zinchenko AZ, Davis RH (2002) General ellipsoidal model for deformable drops in viscous flows. Ind Eng Chem Res 41:6270–6278CrossRef

Yamane H, Takahashi M, Hayashi H, Kashihara H, Watanabe H, Masuda T (1998) Observation of deformation and recovery of polyisobutylene droplets in a polyisobutylene/polydimethylsiloxane blend after application of step shear strain. J Rheol 42:567–580CrossRefADS

Yu W, Bousmina M (2003) Ellipsoidal model for droplet deformation in emulsions. J Rheol 47:1011–1039CrossRefADS

Yu W, Zhou C (2007) A simple constitutive equation for immiscible blend. J Rheol 51:179–194CrossRefADS

Yu W, Bousmina M, Grmela M, Palierne J, Zhou C (2002a) Quantitative relationship between rheology and morphology in emulsions. J Rheol 46:1381–1399CrossRefADS

Yu W, Bousmina M, Grmela M, Zhou C (2002b) Modeling of oscillatory shear flow of emulsions under small and large deformation fields. J Rheol 46:1401–1418CrossRefADS

Yu W, Bousmina M, Zhou C, Tucker CL III (2004) Theory for drop deformation in viscoelastic systems. J Rheol 48:417–438CrossRefADS

Yu W, Zhou C, Bousmina M (2005) Theory of morphology evolution in mixtures of viscoelastic immiscible components. J Rheol 49:215–236CrossRefADS

Yue P, Feng JJ, Liu C, Shen J (2005a) Transient drop deformation upon startup of shear in viscoelastic fluids. Phys Fluids 17:123101–123107CrossRefADS

Yue P, Feng JJ, Liu C, Shen J (2005b) Diffuse-interface simulations of drop coalescence and retraction in viscoelastic fluids. J Non-Newton Fluid Mech 129:163–176CrossRef

Titel: Models for the deformation of a single ellipsoidal drop: a review
verfasst von: Mario Minale
Publikationsdatum: 01.08.2010
Verlag: Springer-Verlag
Erschienen in: Rheologica Acta / Ausgabe 8/2010
Print ISSN: 0035-4511
Elektronische ISSN: 1435-1528
DOI: https://doi.org/10.1007/s00397-010-0442-0

Premium Partner

Marktübersichten

Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.

Zur Marktübersicht

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Technik"

Springer Professional "Wirtschaft+Technik"

Weitere Artikel der Ausgabe 8/2010

Rheostructural studies of a discotic mesophase pitch at processing flow conditions

Correlation between rheological and electrical properties of polyamides filled with carbon black

Effect of nanofillers on the phase separation and rheological properties of poly(methyl methacrylate)/poly(styrene-co-acrylonitrile) blends

Investigating the transient response of a shear thickening fluid using the split Hopkinson pressure bar technique

Effect of Cloisite and modified Laponite clays on the rheological behavior of TPU–clay nanocomposites

Synthesis and electrorheological properties of polar molecule-dominated TiO 2 particles with high yield stress

Premium Partner

Marktübersichten