Abstract
In this paper, an analytical solution for steady creeping motion of viscoelastic drop falling through a viscous Newtonian fluid is presented. The Oldroyd-B model is used as the constitutive equation. The analytical solutions for both interior and exterior flows are obtained using the perturbation method. Deborah number and capillary numbers are considered as the perturbation parameters. The effect of viscoelastic properties on drop shape and motion are studied in detail. The previous empirical studies indicated that unlike the Newtonian creeping drop in which the drop shape is exactly spherical, a dimpled shape appears in viscoelastic drops. It is shown that the results of the present analytical solution in estimating the terminal velocity and drop shape have a more agreement with experimental results than the other previous analytical investigations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aggarwal, N. and K. Sarkar, 2007, Deformation and breakup of a viscoelastic drop in a Newtonian matrix under steady shear, J. Fluid Mech. 584, 1–21.
Aminzadeh, M., A. Maleki, B. Firoozabadi and H. Afshin, 2012, On the motion of Newtonian and non-Newtonian liquid drops, Sci. Iran. B 19(5), 1265–1278.
Arigo, M.T. and G.H. McKinley, 1998, An experimental investigation of negative wakes behind sphere settling in a shearthinning viscoelastic fluid, Rheol. Acta 37, 307–327.
Batchelor, G.K., 1967, An introduction to fluid dynamics, Cambridge University, Cambridge.
Bird, R.B., R.C. Armstrong and O. Hassager, 1987, Dynamics of Polymeric Liquids, fluid dynamics, Vol. 1, second ed., Wiley, New York.
Chakravarthy, S.R., 1984, Relaxation methods for unfactored implicit upwind schemes, AIAA J. 84–0165.
German, G. and V. Bertola, 2010, Formation of viscoplastic drops by capillary breakup, Phys. Fluids 22, 33101.
Gurkan, T., 1989, Motion of a circulating power-law drop translating through Newtonian fluid at intermediate Reynolds numbers, Chem. Eng. Commun. 80, 53–67.
Hadamard, J., 1911, Mouvement permanent lent d’une sphere liquide et visqueuse dans un liquide, C. R. Acad. Sci. Paris 152, 1735–1738.
Happel, J. and H. Brenner, 1965, Low Reynolds number hydrodynamics, Prentice-Hall, New Jersey.
Joseph, D.D., G.S. Beavers and R.L. Fosdick, 1972, The free surface on a liquid between cylinders rotating at different speeds speeds, Part 1, Arch. Rational Mech. Anal. 49, 321–380.
Joseph, D.D., G.S. Beavers and R.L. Fosdick, 1973, The free surface on a liquid between cylinders rotating at different speeds speeds, Part 2, Arch. Rational Mech. Anal. 49, 381–401.
Kishore, N., R.P. Chhabra and V. Eswaran, 2008, Effect of dispersed phase rheology on the drag of single and of ensembles of fluid spheres at moderate Reynolds numbers, Chem. Eng. J. 141, 387–392.
Landau, L. and I. Lifshitz, 1959, Fluid mechanics, Pergamon, Oxford.
Levrenteva, O.M., Y. Holenberg and A. Nir, 2009, Motion of viscous drops in tubes filled with yield stress fluid, Chem. Eng. Sci. 64(22), 4772–4786.
Mckinley, G.H., 2005, Dimensionless groups for understanding free surface flows of complex fluids, Sco. Rheol. Bulletin 74(2), 6–9.
Mukherjee, S. and K. Sarkar, 2011, Viscoelastic drop falling through a viscous medium, Phys. Fluids 23, 013101.
Payne, L.E. and W.H. Pell, 1960, The Stokes flow problem for a class of axially symmetric bodies, J. Fluid Mech. 7, 529–342.
Pillapakkam, S.B., P. Singh, D. Blackmore and N. Aubry, 2007, Transient and steady state of a rising bubble in a viscoelastic fluid, J. Fluid Mech. 589, 215–252.
Potapov, A., R. Spivak, O.M. Laverenteva and A. Nir, 2006, Simulation of motion and deformation of Newtonian drops in Bingham fluid, Ind. Eng. Chem. Res. 45(21), 6985–6995.
Rybczynski, W., 1911, Uber die fortschreitende Bewegung einer flüssigen Kugel in einem zähen Medium, Bull. Acad. Sci. Cracovie 1, 40–46.
Singh, J.P. and M.M. Denn, 2008, Interacting two-dimensional bubbles and droplets in a yield stress fluids, Phys. Fluids 20, 040901.
Singh, P. and D.D. Joseph, 2000, Sedimentation of a sphere near a vertical wall in an Oldroyd-B fluid, J. Non-Newtonian Fluid Mech. 94, 179–203.
Singh, P., D.D. Joseph, T.I. Hesla, R. Glowinski and T.W. Pan, 2000, A distributed Lagrange multiplier/fictitious domain method for viscoelastic particulate flows, J. Non-Newtonian Fluid Mech. 91, 165–188.
Smagin, I., M. Pathak, O.M. Lavrenteva and A. Nir, 2011, Motion and shape of an axisymmetric viscoplastic drop slowly falling through a viscous fluid, Rheol. Acta 50, 361–374.
Smolka, L., 2002, On the motion of Newtonian and non-Newtonian liquid filaments: stretching, beading, blistering, pinching, Ph.D. Thesis, The Pennsylvania State University, USA.
Sostarez, M.C. and A. Belmonte, 2003, Motion and shape of a viscoelastic drop falling through a viscose fluid, J. Fluid Mech. 497, 235–252.
Stone, H.A., 1944, Dynamics of drop deformation and breakup in viscous fluids, Annu. Rev. Fluid Mech. 26, 65–102.
Taylor, G.I., 1934, The formation of emulsions in definable fields of flow, Proc. R. Soc. London. A 146, 501–523.
Taylor, T.D. and A. Acrivos, 1984, On the deformation and drag of a falling viscous drop at low Reynolds number, J. Fluid Mech. 18, 466–476.
Wagner, M.G. and J.C. Slattery, 1971, Slow flow of a non-Newtonian fluids past a droplet, AIChE J. 17, 1198–1207.
White, F.M., 2006, Viscose fluid flow, third ed., McGraw-Hill, New York.
You, R., A. Borhan and H. Haj-Hariri, 2008, A finite volume formulation for simulating drop motion in a viscoelastic twophase system in a viscoelastic two-phase system, J. Non-Newtonian Fluid Mech. 153, 109–129.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Vamerzani, B.Z., Norouzi, M. & Firoozabadi, B. Analytical solution for creeping motion of a viscoelastic drop falling through a Newtonian fluid. Korea-Aust. Rheol. J. 26, 91–104 (2014). https://doi.org/10.1007/s13367-014-0010-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13367-014-0010-8