- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Fluids

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Numerical study on rheology of emulsions and bubbly suspensions with elastoviscoplastic matrix fluids in simple shear

Kazi Tassawar Iqbal*

Daulet Izbassarov

Luca Brandt

Outi Tammisola†

  • Department of Environment, Land and Infrastructure Engineering (DIATI), Politecnico di Torino, Turin, Italy

  • *Contact author: ktassawar.iqbal@gmail.com
  • †Contact author: outi@mech.kth.se
PDF

Phys. Rev. Fluids 11, 053302 – Published 15 May, 2026

DOI: https://doi.org/10.1103/s7sj-y8zx

Abstract

We investigate the rheology of biphasic fluid systems comprising an elastoviscoplastic (EVP) matrix fluid containing initially spherical, monodisperse, viscous droplets through interface-resolved numerical simulations. The interface is captured using the level-set method, and the EVP phase is modeled with the Saramito constitutive equation. We explore the effects of the volume fraction, capillary, Weissenberg, and Bingham numbers, considering both dilute and semidilute regimes (dispersed phase volume fractions from 0.16% to 20%) at two density and viscosity ratios. The constitutive curve shows a negative curvature, consistent with previous studies on emulsion rheology with coalescing drops. Increasing the capillary and Bingham numbers show qualitatively similar trends: increasing drop deformation with both results in a decrease of the system's relative viscosity. This is further supported by stress budget analysis, which shows that yield stress increases the effective viscosity of the matrix fluid, thereby increasing the effective capillary number and reducing the effective viscosity ratios. The bulk rheology shows a complex relationship with the Weissenberg number (Wi). In dilute systems, droplet deformation increases with Wi, while bubbles exhibit a nonmonotonic trend. Relative viscosity rises with Wi for emulsions up to Wi=1.75 due to localized EVP stresses, then slightly decreases. For bubbly suspensions, the viscosity shows little variation at low Wi but decreases at larger values. At semidilute concentrations, similar trends emerge with EVP stresses dictating the system's bulk rheology. At high Wi, increased EVP stresses, reduced coalescence, larger interfacial stress, and near-wall droplet migration collectively increase emulsion relative viscosity.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (87)

  1. N. J. Balmforth, I. A. Frigaard, and G. Ovarlez, Yielding to stress: Recent developments in viscoplastic fluid mechanics, Annu. Rev. Fluid Mech. 46, 121 (2014).
  2. R. Gref, Y. Minamitake, M. T. Peracchia, V. Trubetskoy, V. Torchilin, and R. Langer, Biodegradable long-circulating polymeric nanospheres, Science 263, 1600 (1994).
  3. I. Cohen, H. Li, J. L Hougland, M. Mrksich, and S. R Nagel, Using selective withdrawal to coat microparticles, Science 292, 265 (2001).
  4. N. J. Zuidam and V. Nedovic (eds.), Encapsulation Technologies for Active Food Ingredients and Food Processing (Springer, New York, NY, 2010), Vol. 410.
  5. A. Einstein, Eine neue Bestimmung der Moleküldimensionen, Ann. Phys. 324, 289 (1906).
  6. A. Einstein, Berichtigung zu meiner Arbeit: Eine neue Bestimmung der Moleküldimensionen, Ann. Phys. 339, 591 (1911).
  7. A. Fröhlich and R. Sack, Theory of the rheological properties of dispersions, Proc. A 185, 415 (1946).
  8. M. Mooney, The viscosity of a concentrated suspension of spherical particles, J. Colloid Sci. 6, 162 (1951).
  9. G. K. Batchelor, The stress system in a suspension of force-free particles, J. Fluid Mech. 41, 545 (1970).
  10. G. K. Batchelor, The stress generated in a non-dilute suspension of elongated particles by pure straining motion, J. Fluid Mech. 46, 813 (1971).
  11. H. Brenner, Suspension rheology in the presence of rotary Brownian motion and external couples: Elongational flow of dilute suspensions, Chem. Eng. Sci. 27, 1069 (1972).
  12. G. K. Batchelor and J. T. Green, The determination of the bulk stress in a suspension of spherical particles to order c2, J. Fluid Mech. 56, 401 (1972).
  13. G. K. Batchelor and J. T. Green, The hydrodynamic interaction of two small freely-moving spheres in a linear flow field, J. Fluid Mech. 56, 375 (1972).
  14. H. Eilers, Die Viskosität von Emulsionen hochviskoser Stoffe als Funktion der Konzentration, Kolloid-Z. 97, 313 (1941).
  15. F. Ferrini, D. Ercolani, B. De Cindio, L. Nicodemo, L. Nicolais, and S. Ranaudo, Shear viscosity of settling suspensions, Rheol. Acta 18, 289 (1979).
  16. I. M. Krieger and T. J. Dougherty, A mechanism for non-Newtonian flow in suspensions of rigid spheres, Trans. Soc. Rheol. 3, 137 (1959).
  17. G. I. Taylor, The viscosity of a fluid containing small drops of another fluid, Proc. R. Soc. Lond. A 138, 41 (1932).
  18. A. C. Rust and M. Manga, Effects of bubble deformation on the viscosity of dilute suspensions, J. Non-Newton. Fluid Mech. 104, 53 (2002).
  19. J. K. Mackenzie, The elastic constants of a solid containing spherical holes, Proc. Phys. Soc. Sec. B 63, 2 (1950).
  20. E. W. Llewellin, H. M. Mader, and S. D. R. Wilson, The rheology of a bubbly liquid, Proc. R. Soc. Lond. A 458, 987 (2002).
  21. R. Pal, Rheological constitutive equation for bubbly suspensions, Ind. Eng. Chem. Res. 43, 5372 (2004).
  22. K. Ohie, Y. Tasaka, and Y. Murai, Rheology of dilute bubble suspensions in unsteady shear flows, J. Fluid Mech. 983, A39 (2024).
  23. J. G. Oldroyd, The elastic and viscous properties of emulsions and suspensions, Proc. A 218, 122 (1953).
  24. N. A. Frankel and A. Acrivos, The constitutive equation for a dilute emulsion, J. Fluid Mech. 44, 65 (1970).
  25. J. M. Rallison, A numerical study of the deformation and burst of a viscous drop in general shear flows, J. Fluid Mech. 109, 465 (1981).
  26. M. R. Kennedy, C. Pozrikidis, and R. Skalak, Motion and deformation of liquid drops, and the rheology of dilute emulsions in simple shear flow, Comput. Fluids 23, 251 (1994).
  27. H. Xi and C. Duncan, Lattice Boltzmann simulations of three-dimensional single droplet deformation and breakup under simple shear flow, Phys. Rev. E 59, 3022 (1999).
  28. J. Li, Y. Y Renardy, and M. Renardy, Numerical simulation of breakup of a viscous drop in simple shear flow through a volume-of-fluid method, Phys. Fluids 12, 269 (2000).
  29. H. Zhou and C. Pozrikidis, The flow of ordered and random suspensions of two-dimensional drops in a channel, J. Fluid Mech. 255, 103 (1993).
  30. M. Loewenberg and E. J. Hinch, Numerical simulation of a concentrated emulsion in shear flow, J. Fluid Mech. 321, 395 (1996).
  31. N. Phan-Thien and D. C Pham, Differential multiphase models for polydispersed suspensions and particulate solids, J. Non-Newton. Fluid Mech. 72, 305 (1997).
  32. R. Pal, Viscosity–concentration equation for emulsions of nearly spherical droplets, J. Colloid Interface Sci. 231, 168 (2000).
  33. S. Caserta and S. Guido, Vorticity banding in biphasic polymer blends, Langmuir 28, 16254 (2012).
  34. M. E. Rosti, F. De Vita, and L. Brandt, Numerical simulations of emulsions in shear flows, Acta Mech. 230, 667 (2019).
  35. F. De Vita, M. E. Rosti, S. Caserta, and L. Brandt, On the effect of coalescence on the rheology of emulsions, J. Fluid Mech. 880, 969 (2019).
  36. S. D. Hudson, Wall migration and shear-induced diffusion of fluid droplets in emulsions, Phys. Fluids 15, 1106 (2003).
  37. F. De Vita, M. E. Rosti, S. Caserta, and L. Brandt, Numerical simulations of vorticity banding of emulsions in shear flows, Soft Matter 16, 2854 (2020).
  38. A. R. Malipeddi and K. Sarkar, Shear-induced collective diffusivity down a concentration gradient in a viscous emulsion of drops, J. Fluid Mech. 868, 5 (2019).
  39. I. Girotto, A. Scagliarini, R. Benzi, and F. Toschi, Lagrangian statistics of concentrated emulsions, J. Fluid Mech. 986, A33 (2024).
  40. P. Srivastava, A. R. Malipeddi, and K. Sarkar, Steady shear rheology of a viscous emulsion in the presence of finite inertia at moderate volume fractions: Sign reversal of normal stress differences, J. Fluid Mech. 805, 494 (2016).
  41. R. B. Reboucas, N. N. Nikolova, and V. Sharma, Modeling drop deformations, rheology of dilute to dense emulsions, Curr. Opin. Colloid Interface Sci. 77, 101904 (2025).
  42. H Vanoene, Modes of dispersion of viscoelastic fluids in flow, J. Colloid Interface Sci. 40, 448 (1972).
  43. J. J. Elmendorp, A study on polymer blending microrheology, Polym. Eng. Sci. 26, 418 (1986).
  44. F. Mighri, P. J. Carreau, and A. Ajji, Influence of elastic properties on drop deformation and breakup in shear flow, J. Rheol. 42, 1477 (1998).
  45. S. Guido, M. Simeone, and F. Greco, 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 (2003).
  46. P. Yue, J. J. Feng, C. Liu, and J. Shen, Viscoelastic effects on drop deformation in steady shear, J. Fluid Mech. 540, 427 (2005).
  47. N. Aggarwal and K. Sarkar, Deformation and breakup of a viscoelastic drop in a Newtonian matrix under steady shear, J. Fluid Mech. 584, 1 (2007).
  48. N. Aggarwal and K. Sarkar, Effects of matrix viscoelasticity on viscous and viscoelastic drop deformation in a shear flow, J. Fluid Mech. 601, 63 (2008).
  49. S. Mukherjee and K. Sarkar, Effects of viscosity ratio on deformation of a viscoelastic drop in a Newtonian matrix under steady shear, J. Non-Newton. Fluid Mech. 160, 104 (2009).
  50. D. Izbassarov and O. Tammisola, Dynamics of an elastoviscoplastic droplet in a Newtonian medium under shear flow, Phys. Rev. Fluids 5, 113301 (2020).
  51. P. Saramito, A new constitutive equation for elastoviscoplastic fluid flows, J. Non-Newton. Fluid Mech. 145, 1 (2007).
  52. H. Zhang, W. Gong, W. Yuan, and B. Meng, Numerical investigation on the deformation and breakup of an elastoviscoplastic droplet in simple shear flow, Phys. Fluids 36, 053111 (2024).
  53. M. E. Rosti and L. Brandt, Suspensions of deformable particles in a Couette flow, J. Non-Newton. Fluid Mech. 262, 3 (2018).
  54. S. Mukherjee and K. Sarkar, Effects of matrix viscoelasticity on the lateral migration of a deformable drop in a wall-bounded shear, J. Fluid Mech. 727, 318 (2013).
  55. B. Feneuil, K. T. Iqbal, A. Jensen, L. Brandt, O. Tammisola, and A. Carlson, Experimental and numerical investigation of bubble migration in shear flow: Deformability-driven chaining and repulsion, Phys. Rev. Fluids 8, 063602 (2023).
  56. J. Michele, R. Pätzold, and R. Donis, Alignment and aggregation effects in suspensions of spheres in non-Newtonian media, Rheol. Acta 16, 317 (1977).
  57. M. Sussman, P. Smereka, and S. Osher, A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys. 114, 146 (1994).
  58. Z. Ge, J.-C. Loiseau, O. Tammisola, and L. Brandt, An efficient mass-preserving interface-correction level set/ghost fluid method for droplet suspensions under depletion forces, J. Comput. Phys. 353, 435 (2018).
  59. J. U Brackbill, D. B Kothe, and C. Zemach, A continuum method for modeling surface tension, J. Comput. Phys. 100, 335 (1992).
  60. F. Picano, W.-P. Breugem, D. Mitra, and L. Brandt, Shear thickening in non-Brownian suspensions: An excluded volume effect, Phys. Rev. Lett. 111, 098302 (2013).
  61. M. E. Rosti and S. Takagi, Shear-thinning and shear-thickening emulsions in shear flows, Phys. Fluids 33, 083319 (2021).
  62. K. Verhulst, R. Cardinaels, P. Moldenaers, Y. Renardy, and S. Afkhami, Influence of viscoelasticity on drop deformation and orientation in shear flow: Part 1. Stationary states, J. Non-Newton. Fluid Mech. 156, 29 (2009).
  63. D. Izbassarov, M. E. Rosti, M. N. Ardekani, M. Sarabian, S. Hormozi, L. Brandt, and O. Tammisola, Computational modeling of multiphase viscoelastic and elastoviscoplastic flows, Int. J. Numer. Methods Fluids 88, 521 (2018).
  64. C.-W. Shu, High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Rev. 51, 82 (2009).
  65. R. R. Nourgaliev and T. G. Theofanous, High-fidelity interface tracking in compressible flows: Unlimited anchored adaptive level set, J. Comput. Phys. 224, 836 (2007).
  66. C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys. 77, 439 (1988).
  67. A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comput. 22, 745 (1968).
  68. M. S. Dodd and A. Ferrante, A fast pressure-correction method for incompressible two-fluid flows, J. Comput. Phys. 273, 416 (2014).
  69. Ul. Schumann and R. A. Sweet, Fast Fourier transforms for direct solution of Poisson's equation with staggered boundary conditions, J. Comput. Phys. 75, 123 (1988).
  70. P. Costa, A FFT-based finite-difference solver for massively-parallel direct numerical simulations of turbulent flows, Comput. Math. Appl. 76, 1853 (2018).
  71. Y. Holenberg, O. M. Lavrenteva, A. Liberzon, U. Shavit, and A. Nir, PTV and PIV study of the motion of viscous drops in yield stress material, J. Non-Newton. Fluid Mech. 193, 129 (2013).
  72. K. Hyun, S. H. Kim, K. H. Ahn, and S. J. Lee, Large amplitude oscillatory shear as a way to classify the complex fluids, J. Non-Newton. Fluid Mech. 107, 51 (2002).
  73. C. J. Dimitriou, R. H. Ewoldt, and G. H. McKinley, Describing and prescribing the constitutive response of yield stress fluids using large amplitude oscillatory shear stress (LAOStress), J. Rheol. 57, 27 (2013).
  74. D. Fraggedakis, Y. Dimakopoulos, and J. Tsamopoulos, Yielding the yield-stress analysis: A study focused on the effects of elasticity on the settling of a single spherical particle in simple yield-stress fluids, Soft Matter 12, 5378 (2016).
  75. P. Saramito, A new elastoviscoplastic model based on the Herschel–Bulkley viscoplastic model, J. Non-Newton. Fluid Mech. 158, 154 (2009).
  76. R. Cardinaels, S. Afkhami, Y. Renardy, and P. Moldenaers, An experimental and numerical investigation of the dynamics of microconfined droplets in systems with one viscoelastic phase, J. Non-Newton. Fluid Mech. 166, 52 (2011).
  77. G. Astarita and G. Apuzzo, Motion of gas bubbles in non-Newtonian liquids, AIChE J. 11, 815 (1965).
  78. N. Dubash and I. A. Frigaard, Propagation and stopping of air bubbles in Carbopol solutions, J. Non-Newton. Fluid Mech. 142, 123 (2007).
  79. E. Chaparian, M. N. Ardekani, L. Brandt, and O. Tammisola, Particle migration in channel flow of an elastoviscoplastic fluid, J. Non-Newton. Fluid Mech. 284, 104376 (2020).
  80. D. Izbassarov, M. E Rosti, L. Brandt, and O. Tammisola, Effect of finite Weissenberg number on turbulent channel flows of an elastoviscoplastic fluid, J. Fluid Mech. 927, A45 (2021).
  81. S. Habibi, K. T. Iqbal, M. N. Ardekani, E. Chaparian, L. Brandt, and O. Tammisola, Numerical study of particle suspensions in duct flow of elastoviscoplastic fluids, J. Fluid Mech. 1007, A36 (2025).
  82. S. Osher, R. Fedkiw, and K. Piechor, Level set methods and dynamic implicit surfaces, Appl. Mech. Rev. 57, B15 (2004).
  83. M. Yang, S. Krishnan, and E. S. G. Shaqfeh, Numerical simulations of the rheology of suspensions of rigid spheres at low volume fraction in a viscoelastic fluid under shear, J. Non-Newton. Fluid Mech. 233, 181 (2016).
  84. F. Mangani, G. Soligo, A. Roccon, and A. Soldati, Influence of density and viscosity on deformation, breakage, and coalescence of bubbles in turbulence, Phys. Rev. Fluids 7, 053601 (2022).
  85. M. S. Abdelgawad, S. J. Haward, A. Q. Shen, and M. E. Rosti, From yield stress to elastic instabilities: Tuning the extensional behavior of elastoviscoplastic fluids, PNAS Nexus 3, pgae227 (2024).
  86. G. Soligo, A. Roccon, and A. Soldati, Coalescence of surfactant-laden drops by phase field method, J. Comput. Phys. 376, 1292 (2019).
  87. K. T. Iqbal, Data repository: Numerical study on rheology of emulsions and bubbly suspensions with elastoviscoplastic matrix fluids in simple shear, Github (2026) https://github.com/KTH-Complex-fluids-group/2026_Simple-shear-EVP-matrix-fluid-rheology.git.

Outline

Information

Phys. Rev. Fluids 11, 053302– Published 15 May, 2026
  • Received 30 April 2025
  • Accepted 25 February 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by Bibsam.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation