© 2014 The Korean Society of Rheology and Springer 91 Korea-Australia Rheology Journal, Vol.26, No.1, pp.91-104 (February 2014) DOI: 10.1007/s13367-014-0010-8 www.springer.com/13367 Analytical solution for creeping motion of a viscoelastic drop falling through a Newtonian fluid B.Z. Vamerzani 1 , M. Norouzi 1, * and B. Firoozabadi 2 1 Mechanical Engineering Department, Shahrood University of Technology, Shahrood, Iran 2 Mechanical Engineering Department, Sharif University of Technology, Tehran, Iran (Received August 2, 2013; final revision received December 17, 2013; accepted January 22, 2014) 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 vis- coelastic drops. It is shown that the results of the present analytical solution in estimating the terminal veloc- ity and drop shape have a more agreement with experimental results than the other previous analytical investigations. Keywords: viscoelastic drop, creeping fluid, perturbation solution, Oldroyd-B model 1. Introduction Motion and shape of the axisymmetric drop falling under gravity in an immiscible fluid has become a benchmark problem in fluid dynamics and has a wide range of applications in petroleum (liquid-liquid extrac- tion) and medicine processing (Penicillin manufacture), metals extraction (copper production), painting and wastewater treatment. The problem of a falling viscous drop has been solved in absence of inertia by Hadamard (1911) and Rybczynski (1911). They analytically derived terminal velocity and drag force led to obtaining a spher- ical shape of viscous drop. Later, Taylor and Acrivos (1964) conducted a theoretical investigation by means of a singular-perturbation solution of the axisymmetric equation of motion. They showed that at low Reynolds (Re 1) and finite capillary numbers, the drop shape remains exactly spherical while for higher values of Rey- nolds numbers, the drop takes an oblate shape. Also, Sostarecz and Belmonte (2003) conducted experimental and analytical analysis for polymer falling drop in a vis- cous fluid. Their analytical solution was based on the third order constitutive equation and presented that the falling drop takes the oblate shape (the shape with a dim- ple at the rear end). The third order constitutive equation is a retarded-motion expansion model which is defined based on the third order deviation from Newtonian behavior. The model is defined based on the Taylor series expansion of shear rate tensor and its convected derivatives up to third order possible terms around the Newtonian model. The obtained results have a suitable agreement with experimental observations. In this sce- nario, the dimple shape occurs while the viscoelastic stresses dominating on the surface tension (Taylor, 1934; Stone, 1994). Their experimental results indicated that increasing the drop volume led to a toroidal shape of falling drop. Gurkan (1989) considered a falling power- law drop in a Newtonian fluid. Kishore et al. (2008) used a finite difference technique to obtain the drag coeffi- cient of power-law drops at moderate Reynolds numbers. More recently, Smagin et al. (2011) implemented vari- ation of the integral equation method to simulate the sed- imentation of a viscoelastic drop in a Newtonian liquid. Furthermore, Mukherjee and Sarkar (2011) performed a numerical study on the viscoelastic drop deformation and revealed that the drop shape changes from spherical to oblate and sedimentation velocity decreases contrarily of viscous drop. German and Bertola (2010) experi- mentally demonstrated that the formation of viscoelastic drops under gravity by capillary breakup is different from the Newtonian and power law drops. Aggarwal and Sarkar (2007) numerically studied the deformation of a viscoelastic drop suspended in a Newtonian fluid using front-tracking finite-difference method. Their results shown that, a slight non-monotonicity in steady state deformation is caused by increasing the Deborah number at high capillary numbers. Aminzadeh et al. (2012) investigated experimentally the motion of the Newtonian and non-Newtonian drops at Reynolds numbers of 50<Re<500. The falling Newtonian drops in viscoelastic matrix have been considered by Levrenteva et al. (2009) exper- imentally, Potapov et al. (2006) and Singh and Denn *Corresponding author: mnorouzi@shahroodut.ac.ir