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

© 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. 26, No. 1 (2014) An analytical solution for creeping motion of a viscoelastic drop falling through a Newtonian fluid negligible for both flows. η˜ 0 = η˜ 0,p +η˜ s (9a) The exterior fluid is Newtonian and incompressible, thus its flow is described by the Stokes equation: λ1η˜ s λ2 = ------.(9b) (3a) η˜ 0 Finally, the Oldroyd-B constitutive equation is obtained (3b) in the following form: T where = 2ηDˆ and Dˆ = --1-()∇uˆ +∇uˆ . Here, is stress tensor, (10) 2 Dˆ is deformation tensor, uˆ is velocity field, pˆ is pressure where, and g is the acceleration of gravity. The Oldroyd-B constitutive equation has been used pre- The governing equations for interior flow are as follows: viously for modeling the solid sphere falling and the bub- ble rising in a viscoelastic fluid (Arigo and McKinley, (4a) 1998; Pillapakkam et al., 2007; Singh et al., 2000; Singh and Joseph, 2000). It seems that the Oldroyd-B model (4b) could be suitable for this problem because unlike the The interior fluid is modeled using Oldroyd-B consti- retarded-motion expansions constitutive equations (such tutive equation which is given by: as the second order or third order models), it presents the relaxation time constant of the viscoelastic fluid which is (5) necessary for proper definition of the Deborah number. Here, η˜ 0 is the zero shear-rate viscosity, λ1 is the relax- This model is a quasi-linear one and predicts constant val- ation time, λ2 is the retardation time, Dˆ is the rate of ues for viscometric functions (Bird et al., 1987). Here, the deformation tensor, and dˆ is the upper convected deriv- interior flow of the falling drop is creeping so the material ative defined by: modules tends to their value in zero-shear rate (constant values) and it is in accordance with specifications of Old- (6) royd-B model. The boundary conditions at the interface of the droplet when λ2 = 0 , Eq.(5) is reduced to the upper convected and exterior fluid are as follows: Maxwell constitutive equation (UCM) and if λ1 ==λ2 0 , - the normal velocity components of two fluids are zero: the Oldroyd-B model is simplified to Newtonian model (11a) with viscosity η˜ 0 . The Oldroyd-B constitutive equation for a viscoelastic - tangential velocity components are continuous: solution could be obtained by considering the UCM model (11b) for the polymeric additives and Newtonian behavior for the solvent. This form of Oldroyd-B constitutive equation - tangential stress components are as follows: is considered in current study to model to the viscoelastic (11c) contribution of polymeric additives and to Newtonian solvent contribution . - and normal stress components could be expressed as follows (Batchelor, 1967): (7) where (11d)

(8a) where n is the unit vector normal to the droplet surface; and R1 and R2 are the principle radii of curvature of the sur- (8b) face. We assumed that the drop is falling in an infinite bath and origin of coordinate system is fixed at center of mass Here,η˜ 0,p is the viscosity of the polymeric contribution of the drop. in zero shear-rate andη˜ s is the viscosity of the Newtonian In the present work, the perturbation method is used to solvent. In current study, both exterior and interior flows solve the governing equations for both exterior and inte- are assumed to be creeping (γ· →0 ), therefore viscosity rior flows. The perturbation parameter is Deborah number (η˜ ) in Oldroyd-B model for creeping motion is consid- and the second order expansion is used similar to the work ered equal to zero shear rate viscosity (η˜ 0 ) .Now, the vis- of Taylor and Acrivos (1964). In addition, different ref- cosity of solution and relation between time constants erence velocity and reference stress are used to obtain the could be expressed as follows: dimensionless form of governing equations for the exte-

Korea-Australia Rheology J., Vol. 26, No. 1 (2014) 93 B.Z. Vamerzani, M. Norouzi and B. Firoozabadi rior and interior flows: (21b) Therefore, the dimensionless form of constitutive equa- tion and normal stress boundary condition are: Both exterior and interior flows are incompressible and axisymmetric; thus, we have the standard stream function (12a) in spherical coordinates (Happel and Brenner, 1965). Therefore, the velocity fields are calculated for each order as (White, 2006): (12b) (22a) (13) and (22b)

(14) 4. Solution The velocity field, pressure distribution and deformation tensor of the drop (interior) could be expressed as follows: In this study, we employed the perturbation method to solve the stream function for both interior and exterior (15) flows. It is also possible to derive the applied forces in the drop surface based on the obtained stream function and (16) pressure distribution of the interior and exterior flows. By substituting the Eqs. (15) and (16) into the momentum (17) equation, and collecting the same order terms, we found the following equations for both interior and exterior Note operator whereas dependences on the velocity flows (It is important to remember that the expansion of field: velocity and pressure field for interior and exterior flows are similar): (18) (23a) Then, is expressed as follows: . (23b) The stress tensor expansion is By replacing the velocity terms (u˜ 0 and u0 ) by stream (19) functions with same order of magnitude and eliminating By substituting the velocity vector, deformation tensor, the pressure (vorticity method), we obtain the Stokes operator and stress tensor in Eq. (13), we obtain equation for both interior and exterior flows: (20a) (24a)

(20b) (24b) where the operator E4 ≡E2E2 and E2 is defined as: (20c) Here, the perturbation method is also used for Newto- (25) nian exterior flow. Similar to interior flow, we should expand the velocity, pressure and the deformation tensor Furthermore, Happel and Brenner (1965) represented as a function of Deborah number for the exterior flow the solution of Stokes equation (Eqs. (24a) and (24b)) in which is related to the interaction of interior and exterior spherical coordimates as: flows. The stress tensor of Newtonian exterior flow is expressed as . The stream func- (26) tions of the interior and exterior flows are expanded as fol- lows: where µθ= cos and the Gegenbauer polynomial ()Qn()µ are closely related to the Legendre polynomial Pn()µ as: (21a)

94 Korea-Australia Rheology J., Vol. 26, No. 1 (2014) An analytical solution for creeping motion of a viscoelastic drop falling through a Newtonian fluid

(27) drag force for O()1 as follows:

In order to obtain the finite solution, we should consider (31) An ==Bn 0 , n≥1 for the interior flow and Dn = 0 , n≥1 , and Cn = 0 , n≥2 for the exterior flow. By applying the In steady state situation, the force exerted on the drop is boundary conditions of Eq. (26) at r()θ = 1 , the solution zero ()F = 0 . Therefore, we could obtain U∞ which has of zero order terms of the stream functions for creeping been reported by Hadamard (1911) and Rybczynski the interior and exterior flows are obtained as follows: (1911) UHR as the first time:

⎛⎞ 2 2k+2 ()ρ˜ –ρ gR (28a) U∞ ≡UHR = ⎜⎟------.(32) ⎝⎠9k+6 η (28b) In this investigation, zero order of Deborah number indi- cates falling of Newtonian drops in Newtonian liquids. where Therefore, the terminal velocity of Hadamard (1911) and Rybczynski (1911) is similar to us at zero order of Deb- (28c) orah term. In this situation, we have: According to Eqs. (12a) and (12b), the dimensionless ⎛⎞ δp 1 ⎜⎟1 1 interior and exterior velocities are defined based on U0 = ------+ ----- (33) Ca⎝⎠R1 R2 and U∞ (typical interior and exterior velocities). These definitions were presented by Sostarecz and Belmonte where, δp is the ambient pressure jump and will be cal- (2003) to have a better normalization for velocity of these culated in higher order. For higher order terms, the exterior 4 phases. The difference between the definitions of refer- flow is an undriven Stokes flow (E = 0 for each i). The ence velocities is related to the difference of viscosity of governing equation for the interior flow must be calculated interior and exterior fluids. The velocity terms of interior for each order. For the interior flow in ODe() , we obtain and exterior flows (u˜ 0 and u0 ) can be obtained from the equation of motion as: definition of stream function (Eqs. (22a) and (22b)). The (34) zero order pressure distribution is also determined based on the momentum equation (Eqs. (23a) and (23b)). The By eliminating pressure, we have: velocity, pressure and stress terms for any orders are pre- (35) sented in Appendix. The applied forces on the surface of the drop are buoyancy and drag: By applying the boundary conditions and using of Eq. (26), we find: FF= D +FB (29) where (36a)

(30a) (36b) Here, we used the Pyne and Pell (1960) theorem to cal- culate the drag force. They derived drag force as follow: where

(30b) (36c)

If the fluid is not rest at infinity, the above formula is not For calculating drag force, it is necessary to consider the applicable, so if ψ∞ denotes the stream function corre- stream function as in Eq. (30b), but the drag sponding to the fluid motion at infinity, then the stream force remains unchanged so the terminal velocity remains function ψ∞ –ψ gives a state of rest at infinity. Therefore, in its previous value. We considered that the viscoelastic the above formula becomes follow: drop is deformed from a sphere to a shape described by r = 1+ζµ(), where µθ= cos . For the small deformation (30c) from the spherical shape (maxζ 1), the surface tension term may be linearized as follows (Landau and Lifshitz, 2 2 r sin θ 1959): where ψ∞ = ------which is the stream function for free 2 streaming flow (Payne and Pell, 1960). We calculated the (37)

Korea-Australia Rheology J., Vol. 26, No. 1 (2014) 95 B.Z. Vamerzani, M. Norouzi and B. Firoozabadi

The normal stress boundary condition (Eq. (14)) order. For convenience in solving the boundary conditions becomes on the surface of the drop at r()θ = 1+ζ()µ , we used the Taylor expansion in a neighborhood of sphere shape (38) (r()θ = 1 ). The interfacial boundary conditions (which will be evaluated at r=1) are The function ζ should satisfy the two conditions: (39) These equations are linearizations of the conditions in which the volume of the drop must remain constant and the center of mass of the drop must remain at the origin of coordinate system, respectively. At the terminal velocity of the drop (Eq. (32)), we have

(40) where

(41)

The solution of Eq. (40) regarding to the boundary con- ditions defined in Eq. (39) is (45) By applying boundary condition, we have (42) Small values of the capillary number correspond to sit- (46a) uations with dominant surface tension. In particular at zero capillary number, there can be no deformation of the drop from the spherical shape for any Deborah number. Conversely at zero Deborah number, because we are neglecting inertia, there will be nothing to perturb the shape from spherical (Bird et al., 1987). Here, we assume (46b) that De< O()1 and Ca< O()1 . We will now perform a domain perturbation to determine the effect of the deformed drop on the flow fields (Joseph et al., 1972). where Unlike the most of previous studies that used the Webber number (Taylor and Acrivos, 1964) or law order terms in (46c) their analytical studies (Hadamard, 1911; Rybczynski, 1911), the present perturbation solution is obtained up to In order to satisfy the normal stress boundary condition second order terms (second order of Deborah number and defined in Eq. (38), we must determine and and multiplication of Deborah and capillary numbers). We apply them in the bottom equation: perturb the exterior and interior flow as: (43a) (47) (43b) The pressure and stress distribution perturb similar to the velocity field so we have 2 The second order form (O(De )) of governing equations (44a) for the interior flow is:

(44b) (48) in which and are the previously obtained flow field around the sphere. We have assumed By eliminating the pressure from equation of motion, we that the Deborah and capillary numbers are in the same have

96 Korea-Australia Rheology J., Vol. 26, No. 1 (2014) An analytical solution for creeping motion of a viscoelastic drop falling through a Newtonian fluid

The axisymmetric boundary of the drop is obtained by (49) solving Eq. (54) and applying the conditions of î (defined in Eq. (39)):

Using Eq. (26) and applying boundary condition at (56) r(θ)=1, we obtain (50a) 5. Results and Discussion

(50b) 5.1. Validation In this section, the obtained results for motion and where steady shape of the viscoelastic drop falling through a vis- cous fluid are presented. Here, we compare our results with experimental observation and analytical solution of Sostareczand and Belmonte (2003). They used Polydim- ethylsiloxane oil (PDMS) as the exterior viscous fluid –3 with ρ = 0.98g cm and η = 10 P and a polymeric solu- tion of 0.16% xanthan gum by weight in 80:20 glycerol/ water by volume as the drop phase. The properties of –3 polymeric solution are ρ˜ = 1.27g cm , λ≅40 s and sur- face tension between the two fluids and the viscosity ratio –1 are Γ = 25dyn cm and k=50, respectively. A comparison between the terminal velocity of the (51) present analytical solution and the results of previous Considering and using studies is presented in Fig. 1. According to the study of Eq. (29), the total drag force is determined as follows: Smolka (2002), the viscosity of the Newtonian solution of 80:20 glycerol/water by volume is 0.9 poise. The zero shear rate viscosity is increased to the 9poise by dissolving 0.16% xanthan gum additives in the solution. Therefore, the β parameter of Oldroyd-B model for the (52) final polymeric solution can be obtained equal to 0.9 (ηs = 0.9 poise , η˜ 0 = 9 poise and β = 1–η˜ s ⁄η˜ 0 ). Accord- ing to the Fig. 1, increasing the drop volume leads to terminal velocity increment which is arisen from growths The terminal velocity is calculated by balancing the drag of body force. Fig. 1 indicates that the present analytical force with the buoyancy force: solution has a better agreement with experimental data than the previous analytical studies (Hadamard, 1911; (53) Sostarez and Belmonte, 2003). It is important to remem- ber that the Deborah and capillary numbers are used as In the above equation, De and Ca are presented as a the perturbation parameter and increasing the drop vol- function of U∞ . In order to calculate U∞ , we should solve ume increment these two dimensionless parameters. As the above equation in implicit form. At this terminal we assume De<1 and Ca<1, the huge increment in the velocity (steady state condition), we have drop volume leads to inaccuracy of analytical solution. Therefore, the deviation of analytical studies from experimental observation is increased by enhancing the drop volume. Fig. 1 shows that if Deborah number (54) becomes greater than 0.8, the accuracy of the present analytical study is decreased. Later, we show that this where solution is trustable only for Ca<0.7 (refer to section 5.3). According to the Fig. 1, the result of current ana- lytical solution has a more adaption with experimental (55) data in large amount of the drop volume which is indi- cated that the Oldroyd-B constitutive equation is more reliable model for this problem. It is clear that incre-

Korea-Australia Rheology J., Vol. 26, No. 1 (2014) 97 B.Z. Vamerzani, M. Norouzi and B. Firoozabadi

Fig. 3. I: Steady shapes from analytical solution of Sostarez and Belmonte (2003). II: steady shapes obtained from present study using Oldroyd-B model. III: steady shapes from experimental observations of Sostarez and Belmonte (2003). The volume of the drop is 1.4 ml.

Fig. 1. Terminal velocity vs. equivalent radius of drop. In Fig. 2, the obtained steady shape of Oldroyd-B drop is compared with the analytical and experimental results of Sostarecz and Belmonte (2003) for different drop vol- ume. Here, maximum equivalent radius and volume are 0.52 cm and 0.52 ml, (Fig. 2(e)), respectively. Also, Rey- –2 nolds number of exterior fluid is Re≅510× and, the Deborah and capillary numbers are 0.54 and 0.208, respectively. According to the Fig. 2(a), due to domination of surface tension force for small volume, the drop shape is spherical but the shape of the drop is transformed to an oblate spherical shape (Figs. 2(b) and 2(c)) by increasing the Deborah number (resulted from enhancing the drop volume). At higher volumes (Figs. 2(d) and 2(e)), a dim- ple shape at the rear end of the drop appears. By increas- ing the drop volume the dimple grows inward. As can be seen, the presented Oldroyd-B drop shapes have shown suitable agreement with the previous experimental results of Sostarecz and Belmonte (2003). According to Fig. 2(e), for higher values of Deborah number (large drop), the shape obtained from current study shows more adaption with the experimental result than the third order model used by Sostarecz and Belmonte (2003). In Fig. 3, we study the steady shape of the drop for larger volumes. The drop volume is 1.4 ml and Deborah number is De≅1.81 . According to the figure, both analytical solu- tion of present study and Sostarecz and Belmonte (2003) Fig. 2. I: Steady shapes from analytical solution of Sostarez and show a considerable deviation from experimental obser- Belmonte (2003). II: steady shapes obtained of present study vation. Here, the value of the Deborah number is larger than using Oldroyd-B model. III: steady shapes from experimental 1 and because the Deborah number is employed as the per- observations of Sostarez and Belmonte (2003). Polymer solution turbation parameter, the analytical solution fails. of 0.16% xanthan gum by weight 80:20 glycerol/water by vol- ume in 9.8 P oil. The volumes of drops are: a) 0.01 ml, b) 0.12, 5.2. Effects of k and β variation c) 0.21, d) 0.35 and e) 0.52. In this section, we study the effects of k and β variations on the motion and deformation of the steady state drop. One of the dimensionless parameters for viscoelastic drop is ment in the drop volume consequently increases the De λΓ elasto-capillary number Ec ==------. This dimen- Reynolds number due to role of terminal velocity and Ca ()k+1 ηR increment of drop radius. But, the largest Reynolds sionless parameter is only depended on fluid properties for number refers to biggest size of falling drop (R=0.55 a specific geometry (2005). Fig. 4 shows the drop terminal cm). For this radius of drop, the Reynolds number is velocity versus the elasto-capillary number for different very small (Re = 0.05 1). values of the viscosity ratio (β). For the special case of the

98 Korea-Australia Rheology J., Vol. 26, No. 1 (2014) An analytical solution for creeping motion of a viscoelastic drop falling through a Newtonian fluid

Fig. 5. Effect of β variation on the steady state viscoelastic drop falling through a viscous fluid for De ===0.6,Ca 5,k 0 and Fig. 4. Terminal velocity vs. elasto-capillary number for viscoelastic viscosity ratio a) β = 0 , b) β = 0.1 , c) β = 0.2 , d) β = 0.4 , e) –3 drop in exterior fluid at: R=0.4 cm, k=30, ρ˜ –0.290gcmρ = ⋅ β = 0.6 , f) β = 0.8 . η = 1 Pa .s .

Newtonian drop (Ec = 0 or β = 0), the terminal velocity for any viscosity ratio is equal. Therefore, terminal velocity for Newtonian creeping drop falling in viscous fluid is 2 2()ρ˜ –ρ ()k+1 gR U∞ = ------. Fig. 4 shows that the terminal velocity 3η()3k+2 is not varying appreciably as a function of elasto-capillary number and viscosity ratio (β). In this Figure, variation of the terminal velocity is within ~2% (from 0.995 to 1.02). Fig. 5 illustrates the effects of the viscosity ratio on the shape of the drop at constant Deborah and capillary num- bers. For β= 0 (Newtonian drop), the shape of the drop is Fig. 6. Effect of k variation on the drop shape at De ==0.7,Ca 0.5 in spherical form Hadamard (1911) and Rybczynski (1911). β = 0.8 . By increasing the viscosity ratio, viscosity of the polymeric contribution of the drop increases as well as the viscoelastic normal stresses, which are dominating on the surface ten- to the dramatically enhancing of at the rear end of the sion force and change the drop shape to an oblate. Fig. 6 drop. In the second row of figure, the Deborah number is shows the effect of k variation (drop zero-shear rate vis- constant while the capillary number is increased which cosity to viscosity of Newtonian fluid) on the deformation shows the similar effect on the dimple growing. of falling drop. The results indicate that except in a special The radial- normal component of the viscoelastic stress case of k=0.25, the other cases show small effect of k in the at the interface of the drop is shown in Fig. 8. This com- drop shape, which is related to the fractional dependent of ponent of the viscoelastic stress causes that the drop the viscoelastic stresses (Eqs. (A14), A(22) and A(30)) on loses its spherical shape and takes a dimple at the rear this term. In other words, the effect of k is decreased notice- end side. It is evident that the maximum value of is ably in viscoelastic stresses at k 1. occurred at θ = 0 (at the rear end of the drop). By increasing the Deborah number, the radial-normal com- 5.3. Effect of De and Ca ponent of the viscoelastic stress is increased. Fig. 9 illus- In Fig. 7, several drop shapes for various combinations trates the distribution of versus the viscosity ratio (β). of the Deborah and capillary numbers are presented. In the According to the figure, is enhanced by increasing the first row of figure, the capillary number is considered con- viscosity ratio (increasing the polymeric contribution in stant while the Deborah number (elastic behavior of fluid) the drop). In this figure, the maximum value of is is growing. The drop loses its spherical shape and takes a related to the drop with De ====0.4,Ca 0.4,k 20,β 0.8 dimple at the rear end side of the drop at Ca = 0.65 and which is around 25% larger than that of the drop with De≥0.6 . In addition, increasing the Deborah number De ==0.4,Ca 0.4, k ==20,β 0.1 . results in dimple growth inward, which could be attributed The streamlines of interior flow for Newtonian and

Korea-Australia Rheology J., Vol. 26, No. 1 (2014) 99 B.Z. Vamerzani, M. Norouzi and B. Firoozabadi

Fig. 7. Steady state shapes of viscoelastic drop at different cap- illary and Deborah numbers at k ==50,β 0.8 . Fig. 9. Normal component of viscoelastic stress ( ) at free sur- face aginst the viscosity ratio for De ===0.4,Ca 0.4,k 20 .

Newtonian liquid around the viscoelastic drop is shown in Fig. 11. By increasing the Deborah number in a con- stant capillary number, the maximum velocity of exterior flow is reduced. For better presentation, streamlines of the exterior flow, the range of radius has, 1+ζ<

100 Korea-Australia Rheology J., Vol. 26, No. 1 (2014) An analytical solution for creeping motion of a viscoelastic drop falling through a Newtonian fluid

Fig. 10. Streamlines of interior flow in different capillary and Deborah numbers at k = 30 and β = 0.8 .

Fig. 11. Streamlines of exterior flow in different Ca and Deborah numbers at k = 30 and β = 0.8 .

Korea-Australia Rheology J., Vol. 26, No. 1 (2014) 101 B.Z. Vamerzani, M. Norouzi and B. Firoozabadi study, we showed that the maximum radial-normal com- ponent of the viscoelastic stress is occurred at the rear end of the drop and the normal component is enhanced by increas- ing the Deborah number. In this investigation.

Appendix (A13) The details of perturbation solution for velocity and pressure fields are presented in this section. The zero order terms

(A1)

(A2)

(A3)

(A4) (A14)

(A5)

The terms in order of DeCa

(A15)

(A6) (A16)

1 The terms in order of De (A17)

(A7)

(A8) (A18)

(A9) (A19) (A10)

(A11) (A20)

(A12)

102 Korea-Australia Rheology J., Vol. 26, No. 1 (2014) An analytical solution for creeping motion of a viscoelastic drop falling through a Newtonian fluid

(A28)

(A21)

(A29)

(A22) The terms in order of De2

(A23)

(A24)

(A25)

(A26)

(A30) (A27)

Korea-Australia Rheology J., Vol. 26, No. 1 (2014) 103 B.Z. Vamerzani, M. Norouzi and B. Firoozabadi

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