Modeling the droplet break-up based on drawing the shape using velocity gradient and normal vector

Document Type : Research Article

Authors

Tarbiat Modares University, Tehran, Iran

Abstract

The present study developed a model to deform a viscous droplet in a viscous matrix by a shear flow based on changing the normal vector. The initial cross-section was assumed to be a regular polygon with 1000 sides instead of a circle or ellipsoid, and also this model was independent of the initial polygon shape. Changing the normal vector and the length of each side of the droplet was a function of the velocity gradient. To calculate the velocity gradient over each side of the shape, the equations of tangential and normal stress, the conservation of mass equation, and the absence of mass transfer equation between two phases were solved simultaneously. By knowing the velocity gradient, normal vectors and the length of each side are calculated; therefore, the new shape can be plotted by drawing sides one after another. The results displayed that the time of the break-up, which this model predicts, coincides with the experimental results. On the other hand, the predicted shape of the droplet at the break-up has logically coincided with the experimental results in the middle range of the Capillary number ratio (1.4 -2.6 critical Capillary number). The drop’s dimensions show less than 30% deviation and its rotation less than 20%. Additionally, the dimension of the end bubble also shows a deviation of less than 40%.

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References

  1. Schowalter, C. Chaffey, H. Brenner, Rheological behavior of a dilute emulsion, Journal of colloid and interface science, 26(2) (1968) 152-160.
  2. Mellema, M. Willemse, Effective viscosity of dispersions approached by a statistical continuum method, Physica A: Statistical Mechanics and its Applications, 122(1-2) (1983) 286-312.
  3. Doi, T. Ohta, Dynamics and rheology of complex interfaces. I, The Journal of chemical physics, 95(2) (1991) 1242-1248.
  4. M. Lee, O.O. Park, Rheology and dynamics of immiscible polymer blends, Journal of Rheology, 38(5) (1994) 1405-1425.
  5. Rallison, The deformation of small viscous drops and bubbles in shear flows, Annual review of fluid mechanics, 16(1) (1984) 45-66.
  6. Vinckier, M. Minale, J. Mewis, P. Moldenaers, Rheology of semi-dilute emulsions: viscoelastic effects caused by the interfacial tension, Colloids and Surfaces A: Physicochemical and Engineering Aspects, 150(1-3) (1999) 217-228.
  7. Pal, Rheology of simple and multiple emulsions, Current opinion in colloid & interface science, 16(1) (2011) 41-60.
  8. Singh, V. Narsimhan, Deformation and burst of a liquid droplet with viscous surface moduli in a linear flow field, Physical Review Fluids, 5(6) (2020) 063601.
  9. Håkansson, L. Brandt, Deformation and initial breakup morphology of viscous emulsion drops in isotropic homogeneous turbulence with relevance for emulsification devices, Chemical Engineering Science, 253 (2022) 117599.
  10. Sahu, A.S. Khair, Dynamics of a viscous drop under an oscillatory uniaxial extensional Stokes flow, International Journal of Multiphase Flow, 146 (2022) 103844.
  11. Li, Y. Renardy, Numerical study of flows of two immiscible liquids at low Reynolds number, SIAM review, 42(3) (2000) 417-439.
  12. Jalili, P. Jalili, Numerical analysis of airflow turbulence intensity effect on liquid jet trajectory and breakup in two-phase cross flow, Alexandria Engineering Journal, 68 (2023) 577-585.
  13. Heydarpoor, N.M. Famili, Polygon model for solution of non-linear velocity gradient of interface and asymmetric break-up of droplet, Physics of Fluids, 36(1) (2024).
  14. Abeyartne, Continuum Mechanics Volume II of Lecture Notes on The Mechanics of Elastic Solids Cambridge, http, web. mit. edu/abeyartne/lecture_notes. html, 11 (2012).
  15. G. Leal, Advanced transport phenomena: fluid mechanics and convective transport processes, Cambridge University Press, 2007.
  16. -C. Léger, Menger curvature and rectifiability, Annals of mathematics, 149(3) (1999) 831-869.
  17. Ivanoff, C. Pinzka, J. Lipman, E1376, The American Mathematical Monthly, 67(3) (1960) 291-292.
  18. R. Marks, Drop breakup and deformation in sudden onset strong flows, University of Maryland, College Park, 1998.
  19. Changzhi, G. Liejin, Experimental study of drop deformation and breakup in simple shear flows, Chinese Journal of Chemical Engineering, 15(1) (2007) 1-5.
AUT Journal of Mechanical Engineering
Volume 7, Issue 3
2023
Pages 261-282

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