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[091-104]-M. Norouzi.Fm Korea-Australia Rheology Journal, Vol.26, No.1, pp.91-104 (February 2014) www.springer.com/13367 DOI: 10.1007/s13367-014-0010-8 Analytical solution for creeping motion of a viscoelastic drop falling through a Newtonian fluid B.Z. Vamerzani1, M. Norouzi1,* and B. Firoozabadi2 1Mechanical Engineering Department, Shahrood University of Technology, Shahrood, Iran 2Mechanical 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 Newtonian model. The obtained results have a suitable agreement with experimental observations. In this sce- Motion and shape of the axisymmetric drop falling nario, the dimple shape occurs while the viscoelastic under gravity in an immiscible fluid has become a stresses dominating on the surface tension (Taylor, 1934; benchmark problem in fluid dynamics and has a wide Stone, 1994). Their experimental results indicated that range of applications in petroleum (liquid-liquid extrac- increasing the drop volume led to a toroidal shape of tion) and medicine processing (Penicillin manufacture), falling drop. Gurkan (1989) considered a falling power- metals extraction (copper production), painting and law drop in a Newtonian fluid. Kishore et al. (2008) used wastewater treatment. The problem of a falling viscous a finite difference technique to obtain the drag coeffi- drop has been solved in absence of inertia by Hadamard cient of power-law drops at moderate Reynolds numbers. (1911) and Rybczynski (1911). They analytically derived More recently, Smagin et al. (2011) implemented vari- terminal velocity and drag force led to obtaining a spher- ation of the integral equation method to simulate the sed- ical shape of viscous drop. Later, Taylor and Acrivos imentation of a viscoelastic drop in a Newtonian liquid. (1964) conducted a theoretical investigation by means of Furthermore, Mukherjee and Sarkar (2011) performed a a singular-perturbation solution of the axisymmetric numerical study on the viscoelastic drop deformation equation of motion. They showed that at low Reynolds and revealed that the drop shape changes from spherical (Re 1) and finite capillary numbers, the drop shape to oblate and sedimentation velocity decreases contrarily remains exactly spherical while for higher values of Rey- of viscous drop. German and Bertola (2010) experi- nolds numbers, the drop takes an oblate shape. Also, mentally demonstrated that the formation of viscoelastic Sostarecz and Belmonte (2003) conducted experimental drops under gravity by capillary breakup is different and analytical analysis for polymer falling drop in a vis- from the Newtonian and power law drops. Aggarwal and cous fluid. Their analytical solution was based on the Sarkar (2007) numerically studied the deformation of a third order constitutive equation and presented that the viscoelastic drop suspended in a Newtonian fluid using falling drop takes the oblate shape (the shape with a dim- front-tracking finite-difference method. Their results ple at the rear end). The third order constitutive equation shown that, a slight non-monotonicity in steady state is a retarded-motion expansion model which is defined deformation is caused by increasing the Deborah number based on the third order deviation from Newtonian at high capillary numbers. Aminzadeh et al. (2012) behavior. The model is defined based on the Taylor investigated experimentally the motion of the Newtonian series expansion of shear rate tensor and its convected and non-Newtonian drops at Reynolds numbers of derivatives up to third order possible terms around the 50<Re<500. The falling Newtonian drops in viscoelastic matrix have been considered by Levrenteva et al. (2009) exper- *Corresponding author: [email protected] imentally, Potapov et al. (2006) and Singh and Denn © 2014 The Korean Society of Rheology and Springer 91 B.Z. Vamerzani, M. Norouzi and B. Firoozabadi (2008) numerically. You et al. (2008) numerically sim- Oldroyd-B constitutive equation is a quasi-linear model ulated the motion of a drop in a viscoelastic two-phase that predicts a constant viscosity for any polymeric solu- system by implementing the FENE-R model. They com- tion. Therefore, the model could be useful for creeping bined artificial compressibility and the flux difference viscoelastic flows in which the viscosity of solution is splitting scheme for overcoming the numerical instability equal to the viscosity at zero shear rate (η˜ p = η˜ 0,p and caused by high elasticity of the viscoelastic phase. Wag- η˜ = η˜ 0 ). The viscosity ratio is βη= ˜ 0,p ⁄η˜ 0 , here, η˜ 0 is the ner and Sattery (1971) investigated the slow flow past a zero shear-rate viscosity of the droplet andη˜ 0,p is the zero drop. They employed third order constitutive equation shear-rate viscosity of the polymer. The minimum value for viscoelastic drop and using double perturbation and ofββ refers to a pure Newtonian fluid (= 0,η˜ 0,p and matched asymptotic expansion method simultaneously. η˜ 0 = η˜ s ). The maximum value of β refers to pure poly- They used the inertia term for interior flow (viscoelastic meric fluid. In this case, the viscosity ratio is equal to one drop) and Wiessenberg number (defined based on the (β = 1 ). Therefore, the range of viscosity ratio for any vis- retardation time) for the exterior flow as the perturbation coelastic solution is 0≤≤β 1 . parameter. In this study η refers yo the viscosity of Newtonian exterior According to the literature, the previous studies for fall- ⎛⎞vol 13⁄ fluid. Taking the drop equivalent radius R = ⎜⎟----------3 , “vol” ing/rising viscoelastic drop in liquid phases are limited to ⎝⎠4π experimental and numerical cases. This subject is related is briefed volume of falling drop, and the terminal velocity to the complicated form of analytical solutions. Previous U∞ as the dimensionless scales for the length and velocity, analytical study employed third order model for simulate we can define the Deborah number for interior flow as fol- viscoelastic drop. Third order model is member of lows: retarded-motion expansion family. These models cannot describe recoil, material memory, stress relaxation, oscil- (1) latory response, time-dependent elongational flows and so on. where λ is relaxation time of polymeric solution. The Rey- In this paper, the creeping motion of a viscoelastic fall- nolds and capillary numbers for the exterior flow are also ing drop through a viscous fluid is studied analytically. defined as follows: The perturbation method is used to solve both exterior ρRU ηU ∞ Ca ∞ (viscous fluid) and interior (viscoelastic drop) flows. The Re = -------------η - , = ----------Γ (2) Oldroyd-B model is used as the constitutive equation and it is shown that the results of present analytical solution where ρ is the density of exterior fluid and Γ is the surface in estimating the terminal velocity and drop shape have tension. U0 is the typical interior flow speed which is a more agreement with experimental results than the U∞ η˜ 0 defined as U0 = -------------- . Here, k = ----- (this choice will be other previous analytical investigations. Here, we inves- ()k+1 η tigate rigorously the effects of viscosity ratio (β), vis- explained in section 3). In this investigation, Deborah and cosity ratio between the drop and the external fluid (k), capillary numbers are considered as the perturbation Deborah and capillary dimensionless numbers on the parameters. Therefore, both of dimensionless numbers drop motion and deformation. Also, we investigate should be smaller than one (O(De)<1 and O(Ca)<1). The effects of the Deborah and capillary numbers on the analytical results of present study indicates that if the Deb- velocity fields (streamlines) for internal and external orah and capillary numbers be greater than 0.8 and 0.7, flows. respectively, the deviation of analytical results from exper- imental observations are considerable. In this paper, we 2. Problem Setup obtain the shape and motion of a polymer drop falling for different values of β, k, De and Ca. Steady shape of polymeric drop falling through a qui- escent viscous fluid is obtained by balance of the net grav- 3. Mathematical Description itational force, surface tension, pressure, internal viscoelastic stresses and external viscous stresses. Here, it For mathematical description, we used perturbation is supposed that the viscoelastic drop is made by dis- analysis to solve the exterior and interior flows. Here, it is solving a polymeric additive (by viscosity equal to η˜p ) in supposed that the falling drop motion is creeping. There- a Newtonian solvent (by viscosity equal to η˜s ). In this fore, we can neglect the inertia effects and describe the study, we used the Oldroyd-B model as the constitutive interior and exterior flows using the Stokes equation. In equation for the viscoelastic drop. Based on this consti- order to determine the drop shape, we assume that the tutive equation, the viscosity of a viscoelastic drop is exterior and interior flows are axisymmetric, the drop defined as η˜ = η˜ s +η˜ p . It is important to remember that shape is steady (time-independent) and the inertia effect is 92 Korea-Australia Rheology J., Vol.
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