In the present study, an analytical solution based on a double perturbation method is obtained to simulate the creeping motion of a Boger drop descending in an immiscible viscoelastic quiescent medium. The perturbation parameters are considered to be the Deborah number defined based on the relaxation times of interior and exterior media and the Capillary number. The Oldroyd-B model is implemented as a constitutive equation for both interior and exterior phases. The analytical solutions are obtained up to second order of perturbation expansion for both phases. The motion of the droplet in the surrounding medium is considered to be creeping. Effects of different parameters including, Deborah numbers, viscosity ratios, elasto-capillary number, and viscosity ratio between the drop and external flows are investigated on the shape, motion, and stream function in detail. In contrast to the generally spherical shape of creeping drops in Newtonian regimes, it is shown that the effect of elasticity in a non-Newtonian scenario can feature prolate or oblate shapes. It is shown that an increment in elastic properties of interior fluid is followed by decrement in the terminal velocity of the droplet, while an increment in elasticity of exterior fluid has a contrary effect and increases the terminal velocity. The results have shown suitable agreements with previously reported experimental results.
Viscoelastic drop Viscoelastic media Creeping motion Oldroyd-B model Deborah number Elasto-capillary number Perturbation method
This is a preview of subscription content, log in to check access.
The authors would like to express their gratitude to Professor Morton Denn of the Levich Institute, City College of New York, USA, for his valuable discussions and guidance during the present research.
Hadamard J (1911) Mouvement permanent lent d’une sphere liquide et visqueuse dans un liquide. C R Acad Sci Paris 152:1735–1738zbMATHGoogle Scholar
Rybczynski W (1911) Bull. Uber die fortschreitende Bewegung einer flüssigen Kugel in einem zähen Medium. Bull Acad Sci Crac 1:40–46Google Scholar
Bentley BJ, Leal LG (1986) An experimental investigation of drop deformation and breakup in steady, two-dimensional linear flows. J Fluid Mech 167:241–283CrossRefzbMATHGoogle Scholar
Mukherjee S, Sarkar K (2011) Viscoelastic drop falling through a viscous medium. Phys Fluids 23(1):013101CrossRefGoogle Scholar
Kishore N, Chhabra RP, Eswaran V (2008) Effect of dispersed phase rheology on the drag of single and of ensembles of fluid spheres at moderate Reynolds numbers. Chem Eng J 141:387–392CrossRefGoogle Scholar
You R, Borhan A, Haj-Hariri H (2008) A finite volume formulation for simulating drop motion in a viscoelastic two-phase system in a viscoelastic two-phase system. J Non Newton Fluid Mech 153:109–129CrossRefzbMATHGoogle Scholar
Acharya A, Mashelkar RA (1978) Motion of liquid drops in rheologically complex fluids. Can J Chem Eng 56(1):19–25CrossRefGoogle Scholar
Acharya A, Mashelkar RA, Ulbrecht J (1978) Mechanics of bubble motion and deformation in non-newtonian media. Chem Eng Sci 32(8):863–872CrossRefGoogle Scholar
Mohan V, Nagarajan R, Venkateswarlu D (1972) Fall of drops in non-Newtonian media. Can J Chem Eng 50:37CrossRefGoogle Scholar
Wagner MG, Slattery JC (1971) Slow flow of a non-Newtonian fluids past a droplet. AIChE J 17:1198–1207CrossRefGoogle Scholar
Dairenieh LS, McHugh AJ (1985) Viscoelastic fluid flow past a submerged spheroidal body. J Non Newton Fluid Mech 19:81–111CrossRefzbMATHGoogle Scholar
Batchelor GK (1967) An introduction to fluid dynamics. Cambridge University, CambridgezbMATHGoogle Scholar
Vamerzani BZ, Norouzi M, Firoozabadi B (2014) Analytical solution for creeping motion of a viscoelastic drop falling through a Newtonian fluid. Korea Aust Rheol J 26(1):91–104CrossRefGoogle Scholar
Vamerzani BZ, Norouzi M, Firoozabadi B (2016) Theoretical and experimental study on the motion and shape of viscoelastic falling drops through Newtonian media. Rheol Acta 55(11–12):935–955CrossRefGoogle Scholar
Bird RB, Armstrong RC, Hassager O (1987) Dynamics of polymeric liquids, fluid dynamics, vol 1, 2nd edn. Wiley, New YorkGoogle Scholar
White FM (2006) Viscose fluid flow, 3rd edn. McGraw-Hill, New YorkGoogle Scholar