Lateral migration of a viscoelastic drop in a Newtonian fluid in a shear flow near a wall

Abstract

Wall induced lateral migration of a viscoelastic (FENE-MCR) drop in a Newtonian fluid is investigated. Just like a Newtonian drop, a viscoelastic drop reaches a quasi-steady state where the lateral velocity only depends on the instantaneous distance from the wall. The drop migration velocity and the deformation scale inversely with the square and the cube of the distance from the wall, respectively. The migration velocity varies non-monotonically with increasing viscoelasticity (increasing Deborah number); initially increasing and then decreasing. An analytical explanation has been given of the effects by computing the migration velocity as arising from an image stresslet field due to the drop. The semi-analytical expression matches well with the simulated migration velocity away from the wall. It contains a viscoelastic stresslet component apart from those arising from interfacial tension and viscosity ratio. The migration dynamics is a result of the competition between the viscous (interfacial tension and viscosity ratio) and the viscoelastic effects. The viscoelastic stresslet contribution towards the migration velocity steadily increases. But the interfacial stresslet—arising purely from the drop shape—first increases and then decreases with rising Deborah number causing the migration velocity to be non-monotonic. The geometric effect of the interfacial stresslet is caused by a corresponding nonmonotonic variation of the drop inclination. High viscosity ratio is briefly considered to show that the drop viscoelasticity could stabilize a drop against breakup, and the increase in migration velocity due to viscoelasticity is larger compared to the viscosity-matched case.

I. INTRODUCTION

Cross-stream migration of drops and particles plays a crucial role in flows of industrial emulsions—oil production,¹ food processing,² injection molding of plastics with fillers^3,4—as well as in biological flows of cells in small capillaries⁵ and microfluidic devices.⁶ In Stokes flow, a neutrally buoyant rigid particle in a viscous liquid does not migrate across streamlines in a wall bounded shear due to symmetry under flow reversal.^7–9 Reversibility can be broken by particle deformability, viscoelasticity, or inertia. In this paper, we numerically investigate a viscoelastic drop migrating under shear near a wall in a viscous liquid, and provide an analytic explanation of the phenomenon.

Drop migration in Newtonian systems has been studied extensively using experimental,^10–17 theoretical,^18–26 and numerical techniques,^27–29 and excellent reviews have been written.^30,31 In contrast, the literature of migration in non-Newtonian systems^32–40 is meager, and most of it describes migration of rigid particles.^41–44 Apart from experimental observations,^10,11,33,40 there has been only one report⁴⁵ of theoretical investigation of viscoelastic effects on drop migration, where the authors performed a rigorous perturbative analysis of the viscoelastic effects of both drop and matrix phases on migration using a second order fluid model. To our knowledge, there was no numerical investigation of drop migration in a viscoelastic system before our recent investigation of a viscous drop migrating in a FENE matrix in shear near a wall—N/V (Newtonian in viscoelastic).⁴⁶ Readers are referred there for a detailed discussion of the migration literature. The migration is caused by a stresslet field induced by the drop in presence of the wall.¹² The matrix viscoelasticity retards drop migration, an effect we clearly show to arise from a non-Newtonian contribution to the stresslet field. It is computed as an integral of the normal stress differences in the flow field. Here, we extend the numerical and theoretical analysis to the V/N case—a viscoelastic drop in a Newtonian fluid.

Note that the perturbative analysis⁴⁵ predicted that viscoelasticity either in drop or matrix phase promotes drop migration in plane shear, in contradiction to recent experimental and numerical findings in the literature which show rigid particles in a sheared viscoelastic medium moving closer to a wall.⁴⁷ In the limit of a high viscosity, a viscous drop behaves similar to a rigid particle. Such contradiction points to the limitation of perturbative analysis and demonstrates a need for full scale numerical simulation. Such simulations, especially in simple canonical systems—e.g., involving one drop in shear,^48–52 extension,⁵³ or gravitational fall⁵⁴ using simplest possible constitutive equations^48–54—are critical for developing physical intuition about viscoelastic effects in multiphase systems. Unlike viscous systems, our understanding of viscoelasticity is severely limited, and yet of great importance for flow modeling and simulation. The difficulty lies in the subtle competition between multiple effects in viscoelastic system often giving rise to baffling experimental observations. For instance, there were contradictory observations about effects of matrix viscoelasticity on drop deformation in the literature.^55–57 Numerical simulation showed that it can both decrease drop migration at small Deborah or Wissenberg numbers by aligning a drop away from the axis of extension, or increase at higher Deborah numbers by local stretching at the drop tips.^49,57 The effects of viscoelasticity in the drop phase is typically smaller than those due to matrix viscoelasticity. However, they also generate several interesting phenomena. Although a viscoelastic drop deforms less in shear compared to a viscous drop due to normal stresses in the circular streamlines,⁴⁸ at higher viscosity ratios, viscoelastic drop deformation can be greater than the viscous case. The phenomenon arises due to a reduction of strain rate at higher viscosity ratios resulting in a reduction in deformation-inhibiting normal stresses, and simultaneous alignment with extension axis that enhances deformation.

Here, we have used a front tracking finite difference method⁵⁸ with a modified version of the finitely extensible nonlinear elastic model due to Chilcott and Rallison (FENE-MCR).⁵⁹ The FENE-MCR model has a single relaxation time, a constant shear viscosity and a positive first normal stress difference—all characteristics of a Boger fluid—and has been used in many viscoelastic studies.^60–65 The mathematical formulation and its numerical implementation are described in Sec. II. Section III discusses the problem setup and convergence. Section IV presents and analyzes the results of the simulation. A theoretical analysis relating the interfacial, viscous, and viscoelastic stresses in the system with the lateral migration velocity is presented following the numerical investigation. Section V summarizes the findings.

II. MATHEMATICAL FORMULATION AND NUMERICAL IMPLEMENTATION

The mathematical formulation and the numerical implementation for Newtonian drops migrating in a viscoelastic shear flow have been described in our recent publication⁴⁶ and the formulation here is similar. Therefore, it is only briefly sketched for completeness. The velocity field u of the droplet matrix system is governed by the incompressible momentum conservation equations

$$\begin{array}{c}\frac{\partial \left(\rho \mathbf{u}\right)}{\partial t}+\nabla ·(\rho \mathbf{uu})=\nabla ·\mathit{\tau }-\underset{\partial \mathrm{B}}{\int }d{\mathbf{x}}_{\mathbf{B}}\kappa \mathbf{n}\Gamma \delta (\mathbf{x}-{\mathbf{x}}_{\mathbf{B}}),\hfill \\ \nabla ·\mathbf{u}=\mathbf{0},\hfill \end{array}$$ (1)

in the entire domain Ω. The total stress τ is decomposed into pressure, polymeric and viscous parts:

$$\mathit{\tau }=-p\mathbf{I}+{\mathbf{T}}^p+{\mathbf{T}}^v,{\mathbf{T}}^v={\mu }_s\mathbf{D},$$ (2)

where p is the pressure, μ_s is the solvent viscosity, and D = (∇u) + (∇u)^T is the deformation rate tensor. The superscript T represents the transpose. T^p is the extra stress (or viscoelastic stress) due to the presence of polymer. Γ is the interfacial tension (constant), ∂B represents the surface of the drop consisting of points x_B, κ is the local curvature, n is the outward normal, and δ(x − x_B) is the three dimensional Dirac delta function. The viscoelastic stress T^p is computed through the conformation tensor A which satisfies the following equation:

$$\frac{\partial \mathbf{A}}{\partial t}+\mathbf{u}·\nabla \mathbf{A}=\nabla \mathbf{u}·\mathbf{A}+\mathbf{A}·{\left(\nabla \mathbf{u}\right)}^T-\frac{f}{\lambda }(\mathbf{A}-\mathbf{I}),$$ (3)

where $f=\frac{L^2}{L^2-tr\left(\mathbf{A}\right)}$, μ_p is the polymeric viscosity, λ is the relaxation time, and L is the finite extensibility.

The relation between the stress T^p and conformation tensor A is

$$\mathbf{A}=\left(\frac{\lambda }{{\mu }_{p}f}\right){\mathbf{T}}^p+\mathbf{I}.$$ (4)

Therefore, the constitutive equation for the stresses becomes

$$\begin{array}{ccc}& & \frac{\partial {\mathbf{T}}^p}{\partial t}+\left\{\mathbf{u}·\nabla {\mathbf{T}}^p-\nabla \mathbf{u}·{\mathbf{T}}^p-{\mathbf{T}}^p·\nabla {\mathbf{u}}^T\right\}+\hfill \\ \\ & & f{\mathbf{T}}^p\left[\frac{\partial }{\partial t}\left(1/f\right)+\mathbf{u}·\nabla \left(1/f\right)\right]+\frac{f}{\lambda }{\mathbf{T}}^p=\frac{f}{\lambda }{\mu }_p\mathbf{D},\hfill \end{array}$$ (5)

$$f=\frac{L^2+\lambda /{\mu }_p\left(\sum T_{ii}^p\right)}{L^2-3}.$$

In the limit of L → ∞, we obtain the Oldroyd-B equation. The terms $f{\mathbf{T}}^p\left[\frac{\partial }{\partial t}\left(1/f\right)+\mathbf{u}·\nabla \left(1/f\right)\right]$ are negligible in our simulations, and by dropping them we arrive at the FENE-MCR equation:

$$\frac{\partial {\mathbf{T}}^p}{\partial t}+\left\{\mathbf{u}·\nabla {\mathbf{T}}^p-\nabla \mathbf{u}·{\mathbf{T}}^p-{\mathbf{T}}^p·\nabla {\mathbf{u}}^T\right\}+\frac{f}{\lambda }{\mathbf{T}}^p=\frac{f}{\lambda }{\mu }_p\mathbf{D}.$$ (6)

Using the elastic and viscous stress splitting method proposed and developed by us,⁵³ the viscoelastic stress for FENE-MCR equation is given by

$$\begin{array}{ccc}\hfill {\left({\mathbf{T}}^p\right)}^{n+1}& =& \left[{\left({\mathbf{T}}^p\right)}^n-{\left({\mu }_p\mathbf{D}\right)}^n\right]e^{-(f/\lambda )\Delta t}+{\left({\mu }_p\mathbf{D}\right)}^n\hfill \\ \\ & -& \frac{\lambda }{f}{\left[\mathbf{u}·\nabla {\mathbf{T}}^p-\nabla \mathbf{u}·{\mathbf{T}}^p-{\mathbf{T}}^p·\nabla {\mathbf{u}}^T\right]}^n\left[1-e^{-(f/\lambda )\Delta t}\right].\hfill \end{array}$$ (7)

Note that Eq. (7) appeared in our recent article⁴⁶ with a slight typographical error. A front-tracking finite difference method is used to simulate the drop dynamics.

III. PROBLEM SETUP AND CONVERGENCE

As in our previous study,⁴⁶ the problem is started by placing a spherical drop of radius a in a three dimensional rectangular computational domain at t = 0 at a distance h_i from the bottom wall of the domain. The domain is periodic in the flow (x) and the vorticity (z) directions with domain sizes in those directions L_x = 10a and L_z = 5a, respectively. The size (L_y) in the gradient direction (y) is 10a with walls at the top and the bottom boundary. The lower wall is stationary and the upper wall is impulsively started at t = 0 with velocity U creating a shear rate of $\dot{\gamma }=U/L_y$. Other details of the problem setup can be found in our previous publication.⁴⁶ Nondimensionalizing the problem using a and ${\dot{\gamma }}^{-1}$ we obtain the non-dimensional parameters Reynolds number $\mathrm{Re}={\rho }_m{a}^2\dot{\gamma }/{\mu }_m$, capillary number $Ca={\mu }_{m}a\dot{\gamma }/\Gamma$, Deborah number $De=\lambda \dot{\gamma }$, viscosity ratio λ_μ = μ_d / μ_m, density ratio λ_ρ = ρ_d / ρ_m, and β = μ_pd/μ_d—the ratio of the polymeric to the total drop viscosity. The total viscosity of the drop is given as μ_d = μ_sd + μ_pd, sum of the solvent and polymeric viscosities. We have fixed the density ratio at 1 and L at 20. The values of β and viscosity ratio are 0.5 and 1, except where we have studied their effects. The code is explicit which restricts us to a small non-zero Reynolds number, for which we have chosen a value of 0.03. In our recent paper,⁴⁶ simulation at Re = 0.03 has shown an excellent match with Stokes flow analytical results and boundary element (BEM) simulations.

The convergence of the viscoelastic algorithms (Oldroyd-B and FENE-MCR) has been established in our previous publications^{46,48–52,54} for several problems related to drop dynamics. The convergence of drop deformation parameter D = (L − B)/(L + B) (assuming the drop to be an ellipsoid, L and B are the major and the minor axes) and lateral migration velocity is plotted in Fig. 1. N is the number of grid points in the x and y directions; z direction has w N/2 grid points. Inset shows the quadratic convergence of the algorithm. We choose 96 × 96 × 48 discretization in the flow, gradient and vorticity directions, respectively, with error in velocity less than 4.5%. The effects of finite domain lengths have been carefully investigated by changing the domain size with minimal effects from the periodic images of the drop in the x and the z directions. We also verified that the effect of the upper wall (at a distance of L_y = 10a from the lower one) has very little effects on the migration velocity for cases considered here (all are restricted to h < 2.5a). Note that recent simulations explored effects of domain confinement on particle motion in a viscoelastic medium.⁶⁶

FIG. 1.

Quadratic convergence of the viscoelastic algorithm. N is the number of grid points in the x and y directions. Deformation (top) and velocity (bottom) are plotted for several grid discretizations for De = 1.5, Ca = 0.1 at h/a = 1.5.

IV. RESULTS

In the Stokes limit, the steady lateral migration $U_{lat}/\dot{\gamma }a$ is known to scale as ∝ Ca(a/h)². The proportionality constant was obtained by various investigators through perturbation analysis as well as BEM simulation—0.6,⁴⁵ 0.583,¹² 0.44–0.49,²⁸ and 0.41.²⁴ In our recent publication,⁴⁶ we showed the same approximate scaling (with constant ∼0.48) for migration velocity, matching very well with the BEM simulation.²⁸ The deformation of a drop as a function of distance was compared with the theoretical expression of Shapira and Haber.²³ For higher viscosity systems, we compared our results with BEM simulations of Uijttewaal and Nijhof.²⁷

A. Effects of Deborah number, β, and capillary number

To study the effects of De and β, we have fixed the capillary number at Ca = 0.1. In Fig. 2(a), we plot the migration velocity as a function of instantaneous distance from the wall for several initial drop positions both for the Newtonian and a viscoelastic (De = 1) cases. We notice that a viscoelastic drop migrates quicker and possesses higher velocity than a Newtonian drop. Note that this is in contrast to the case of a viscous drop in a viscoelastic matrix (N/V) where matrix viscoelasticity retards drop migration.⁴⁶ Also note that the viscoelastic effects are much less pronounced in this (V/N) case than in the N/V case, as the viscoelasticity here is limited inside the confined space of a drop where the magnitude of shear remains small giving rise to small viscoelastic stresses. In free shear as well, the deviation in drop dynamics in the N/V case deviates only by a small amount from the purely Newtonian (N/N) case.^48,51,67 Similar to the purely Newtonian²⁸ as well as the N/V system,⁴⁶ the viscoelastic drop after a transient period settles in a quasi-steady state of migration where the drop velocity depends only on the instantaneous separation from the wall h independent of the initial separation h_i; curves for different initial positions collapse on a single curve. Here onward, we concentrate on the quasi-steady dynamics until in Subsection III C, where we investigate transient effects for large deformation. The inset of Fig. 2(a) shows the approximate scaling ∼(a/h)² similar to the N/N and the N/V cases. It also shows that drop viscoelasticity initially increases migration velocity and then decreases at higher De. The non-monotonicity is clearly observed in Fig. 2(b) where we have plotted quasi-steady lateral migration velocities as a function of De for several instantaneous wall to drop distances all for Ca = 0.1.

FIG. 2.

(a) Lateral migration velocity vs. the distance of the drop from the wall for varying De and initial drop height h_i/a for Ca = 0.1. Inset plots quasi-steady migration velocity against inverse square of the instantaneous wall distance for different Deborah numbers. The symbols are data from the simulations and the lines are linear fits. (b) Quasi-steady lateral migration velocity vs. De for different wall-drop distances for Ca = 0.1.

In Fig. 3, we plot the vertical component of the viscoelastic force (∫_∂B(n · T^p)da) where n is the outward normal to the drop surface ∂B. Fig. 3(a) plots it vs. the instantaneous distance from the wall for three different initial heights and two Deborah numbers. The force curve for a particular Deborah number is independent of the initial drop height and depends only on the instantaneous position. Fig. 3(b) plotting viscoelastic forces vs. De for three different instantaneous heights shows that the viscoelastic force is nonmonotonic which in turn causes the velocity to be nonmonotonic as seen in Fig. 2(b).

FIG. 3.

(a) Quasi-steady viscoelastic force on the drop vs. the distance of the drop from the wall for varying De and initial drop height (h_i/a) for Ca = 0.1. (b) Quasi-steady viscoelastic net force on the drop vs. De for different drop distances.

In Fig. 4(a), we plot deformation for varying De. In free shear, with increasing De, the drop deformation decreases.⁴⁸ The decrease in deformation in presence of viscoelasticity is due to the presence of inhibiting normal stresses at the tip of the drop.⁴⁸ At higher viscosity ratios the deformation is non-monotonic.⁵¹ Shapira and Haber showed that in Stokes flow deformation varies as D ∼ (a/h)³.²³ The same scaling is seen in our simulation. Figure 4(b) shows that the inclination angle of the deformed drop increases with De as in the free shear case and it scales as ∼(a/h)². Inset of Figure 4(b) plots the angle of inclination at h/a = 1.4 and 1.75 to show that it varies nonmonotonically with De. We will show that the angle of inclination, a geometric effect, plays critical role in determining the variation of the migration velocity.

FIG. 4.

(a) Quasi-steady deformation vs. the inverse of the cube of the drop-to-wall distance for varying De for Ca = 0.1. The symbols are data from the simulations and the lines are the linear fits. (b) Inclination angle against (a/h)² for the same Ca and De. Inset shows inclination relative to the Newtonian value versus De at two drop-wall separations.

In Fig. 5(a), we investigate the effects of polymer viscosity in the drop fluid by varying β. The migration velocity as a function of De at h/a = 1.75 shows that migration velocity increases with increasing β. Fig. 5(b) shows that $U_{lat}/\dot{\gamma }a\propto Ca$ similar to the Newtonian case for different De values. By plotting migration velocity normalized by its Newtonian value in the inset, one sees that as Ca increases, the viscoelastic effects diminish, as was also seen in N/V case.⁴⁶

FIG. 5.

(a) Lateral migration velocity plotted for varying De for different values of β at h/a = 1.75. (b) Lateral migration velocity as a function of Ca for different Deborah numbers at h/a = 1.7. Inset shows the variation of lateral velocities against De for several Ca when the velocities are normalized by their respective Newtonian value at h/a = 1.7.

B. A far-field theory of viscoelastic drop migration

In our recent paper on the N/V migration case,⁴⁶ we developed a far-field theory for drop migration. Note that the only analytical theory available for effects of viscoelasticity on drop migration is due to Chan and Leal,⁴⁵ where the authors used a algebraically demanding perturbation analysis to the problem. The analysis although rigorous does not elucidate the underlying physics. The far field theory is based on an earlier idea proposed by Smart and Leighton¹² that the migration arises due to an image stresslet field induced by the drop in presence of the wall. In our previous paper, we further developed the idea extending it to the case of a viscoelastic matrix.⁴⁶ We clearly showed that the stresslet had three contributions due to the interfacial tension, viscosity ratio, and matrix viscoelasticity. With increasing viscoelasticity, in a viscosity-matched system, the interfacial contribution increases due to increasing inclination angle. But it is outweighed by the direct reduction in the non-Newtonian part due to matrix viscoelasticity resulting in an overall decrease in migration velocity. We also recently developed the same theoretical analysis for migration of a capsule enclosed by an elastic membrane.⁶⁸ Here we apply the technique to the viscoelastic drop case. The derivation is similar to the viscoelastic matrix case but differs in an important way—the argument for the Taylor series expansion of the single and double layer Green's functions are far more straight forward here in V/N case than in the N/V case. Here we only briefly sketch the analysis omitting the details presented previously.⁴⁶

We rewrite the governing equations (the equations used for the front tracking simulation) in the limit of zero Reynolds number as

$$\begin{array}{c}-\nabla p+{\mu }_m{\nabla }^2\mathbf{u}=0,\hfill \\ -\nabla \tilde{p}+{\mu }_d{\nabla }^2\tilde{\mathbf{u}}=-\nabla ·({\tilde{\mathbf{T}}}^p-{\mu }_{pd}\tilde{\mathbf{D}})=-\nabla ·{\tilde{\mathbf{T}}}^{NN},\hfill \end{array}$$ (8)

in the matrix and the drop phase, respectively. Variables with a tilde represent field variables inside the drop. The redefined non-Newtonian stress ${\tilde{\mathbf{T}}}^{NN}={\tilde{\mathbf{T}}}^p-{\mu }_{pd}\tilde{\mathbf{D}}$ gives rise to a force term in the Stokes equation in the drop phase. Following the usual manipulation,⁴⁶ one can write the solution outside the drop using a Green's function formulation

$$\begin{array}{ccc}\hfill u_j\left(\mathbf{x}\right)& =& u_j^{\infty }-\frac{1}{8\pi {\mu }_m}\underset{A_d}{\int }{f}_i\left(\mathbf{y}\right)G_{ij}(\mathbf{x},\mathbf{y})dA\left(\mathbf{y}\right)+\frac{1}{8\pi }\underset{A_d}{\int }{u}_i\left(\mathbf{y}\right)M_{ijk}(\mathbf{x},\mathbf{y})n_k\left(\mathbf{y}\right)dA\left(\mathbf{y}\right),\hfill \\ \\ & & G_{ij}(\mathbf{x},\mathbf{y})=G_{ij}^{FS}(\mathbf{x},\mathbf{y})+G_{ij}^w(\mathbf{x},\mathbf{y}),M_{ijk}(\mathbf{x},\mathbf{y})=M_{ijk}^{FS}(\mathbf{x},\mathbf{y})+M_{ijk}^w(\mathbf{x},\mathbf{y}),\hfill \end{array}$$ (9)

A_d is the drop surface. f_i is the traction on the boundary. We use a proper Green's function that adds a contribution $G_{ij}^w(\mathbf{x},\mathbf{y})$ to the free space Green's function $G_{ij}^{FS}(\mathbf{x},\mathbf{y})$, so that G_ij(x, y) = 0 on the wall.⁶⁹ M_ijk(x, y) is the stress due to this Green's function. This special property of Green's function along with the no-slip condition eliminates the surface integral over the wall. Using the second part of the governing Eq. (8) inside the drop phase, we can write a similar equation for the velocity field ${\tilde{u}}_j$ inside the drop (normal are opposite to the outside field) but evaluating at a point x outside the drop

$$\begin{array}{ccc}\hfill 0& =& \frac{1}{8\pi {\mu }_d}\underset{A_d}{\int }{\tilde{f}}_i\left(\mathbf{y}\right)G_{ij}(\mathbf{x},\mathbf{y})dA\left(\mathbf{y}\right)-\frac{1}{8\pi }\underset{A_d}{\int }{\tilde{u}}_i\left(\mathbf{y}\right)M_{ijk}(\mathbf{x},\mathbf{y})n_k\left(\mathbf{y}\right)dA\left(\mathbf{y}\right)\hfill \\ & & -\frac{1}{8\pi {\mu }_d}\underset{V_d}{\int }{\tilde{T}}_{ik}^{NN}\left(\mathbf{y}\right)G_{ij}(\mathbf{x},\mathbf{y})dV\left(\mathbf{y}\right).\hfill \end{array}$$ (10)

Here V_d is the volume of the drop and ${\tilde{f}}_i=({\tilde{T}}_{ij}^v+{\mu }_{pd}{\tilde{D}}_{ij}+{\tilde{T}}_{ij}^{NN})n_j$ is the total traction at the surface. Note that an integration by parts on the original volume integral term has been performed to convert the divergence term to arrive at (10). From (9) and (10), one obtains

$$\begin{array}{ccc}\hfill u_j\left(\mathbf{x}\right)& =& u_j^{\infty }-\frac{1}{8\pi {\mu }_m}\underset{A_d}{\int }\Delta f_i\left(\mathbf{y}\right)G_{ij}(\mathbf{x},\mathbf{y})dA\left(\mathbf{y}\right)+\frac{(1-{\lambda }_{\mu })}{8\pi }\underset{A_d}{\int }{u}_i\left(\mathbf{y}\right)M_{ijk}(\mathbf{x},\mathbf{y})n_k\left(\mathbf{y}\right)dA\left(\mathbf{y}\right)\hfill \\ & & -\frac{1}{8\pi {\mu }_m}\underset{V_d}{\int }{\tilde{T}}_{ik}^{NN}\left(\mathbf{y}\right)G_{ij}(\mathbf{x},\mathbf{y})dV\left(\mathbf{y}\right).\hfill \end{array}$$ (11)

Here $f_i-{\tilde{f}}_i\genfrac{}{}{0pt}{}{\Delta }{=}\Delta f=\Gamma (\nabla ·\mathbf{n})\mathbf{n}\mathrm{on}{A}^d$. In the far-field, we use a one-term Taylor-series expansion around the center of the drop y_c,

$$\begin{array}{ccc}\hfill G_{ij}(\mathbf{x},\mathbf{y})& =& G_{ij}(\mathbf{x},{\mathbf{y}}_c)+\frac{\partial G_{ij}(\mathbf{x},{\mathbf{y}}_c)}{\partial y_{ck}}(y_k-y_{ck})+O{\left(\frac{a}{\left|\mathbf{y}-{\mathbf{y}}_c\right|}\right)}^3,\hfill \\ \\ \hfill M_{ijk}(\mathbf{x},\mathbf{y})& =& M_{ijk}(\mathbf{x},{\mathbf{y}}_c)+O{\left(\frac{a}{\left|\mathbf{y}-{\mathbf{y}}_c\right|}\right)}^3.\hfill \end{array}$$ (12)

For a force-free drop ($\underset{A_d}{\int }\Delta f_j\left(\mathbf{y}\right)dA\left(\mathbf{y}\right)=0$) using incompressibility ($\underset{A_d}{\int }{u}_k\left(\mathbf{y}\right)n_k\left(\mathbf{y}\right)dA\left(\mathbf{y}\right)=0$), we obtain

$$\begin{array}{ccc}\hfill u_j\left(\mathbf{x}\right)& =& u_j^{\infty }\left(\mathbf{x}\right)-\frac{1}{8\pi {\mu }_m}\frac{\partial G_{ij}(\mathbf{x},{\mathbf{y}}_c)}{\partial y_{ck}}\times \left\{\Gamma \underset{A_d}{\int }\left(\frac{{\delta }_{ik}}{3}-n_i{n}_k\right)dA\left(\mathbf{y}\right)-{\mu }_m\left(1-{\lambda }_{\mu }\right)\right.\hfill \\ & & \left.\times \underset{A_d}{\int }(u_i{n}_k+u_k{n}_i)\left(\mathbf{y}\right)dA\left(\mathbf{y}\right)+\underset{V_d}{\int }{T}_{ik}^{NN\prime }\left(\mathbf{y}\right)dV\left(\mathbf{y}\right)\right\}\hfill \\ & =& u_j^{\infty }\left(\mathbf{x}\right)-\frac{1}{8\pi {\mu }_m}\frac{\partial G_{ij}(\mathbf{x},{\mathbf{y}}_c)}{\partial y_{ck}}\left(S_{ik}^{\mathrm{int}\prime }+S_{ik}^{vrat\prime }+S_{ik}^{NN\prime }\right)\hfill \\ & =& u_j^{\infty }\left(\mathbf{x}\right)-\frac{1}{8\pi {\mu }_m}\frac{\partial G_{ij}(\mathbf{x},{\mathbf{y}}_c)}{\partial y_{ck}}\left(S_{ik}^{\mathrm{int}}+S_{ik}^{vrat}+S_{ik}^{NN}\right),\hfill \end{array}$$ (13)

where

$$\begin{array}{ccc}& & S_{ik}^{\mathrm{int}\prime }=\Gamma \int \left(\frac{{\delta }_{ik}}{3}-n_i{n}_k\right)dA\left(\mathbf{y}\right),\hfill \\ & & S_{ik}^{vrat\prime }=-{\mu }_m\left(1-{\lambda }_{\mu }\right)\underset{A_d}{\int }(u_i{n}_k+u_k{n}_i)\left(\mathbf{y}\right)dA\left(\mathbf{y}\right),\hfill \\ & & S_{ik}^{NN\prime }=\underset{V_d}{\int }{T}_{ik}^{NN\prime }\left(\mathbf{y}\right)d\hfill \end{array}$$ (14)

are the contributions to the stresslet due to the interfacial tension, viscosity ratio, and the non-Newtonian effects. These terms without primes in the last expression in (13) represent their traceless forms (due to incompressibility, ∂G_ik(x, y_c)/∂y_ck = 0; the trace of the stresslet does not contribute). An identity due to Rosenkilde⁷⁰ is used to transform the interfacial part $S_{ik}^{\mathrm{int}}$ to the interface tensor $\int \left({\delta }_{ik}/3-n_i{n}_k\right)dA\left(\mathbf{y}\right)$ first so defined by Batchelor.⁷¹

Using an expression for the image propagator near a rigid wall¹² with normal n, one obtains from (13),

$$u_j^{\mathrm{drift}}{n}_j=-\frac{1}{8\pi {\mu }_m}\left(\frac{9}{8h^2}\right)\left(S_{ik}^{\mathrm{int}}+S_{ik}^{\mathrm{vrat}}+S_{ik}^{NN}\right)n_i{n}_k,\begin{array}{cc}& \end{array}{\left(\frac{a}{h}\right)}^2\ll 1.$$ (15)

For the case here with the wall at x₂ = y = 0, the migration velocity is

$$U_{lat}=-\frac{1}{8\pi {\mu }_m}\left(\frac{9}{8h^2}\right)\left(S_{22}^{\mathrm{int}}+S_{22}^{\mathrm{vrat}}+S_{22}^{NN}\right).$$ (16)

Note that the non-Newtonian part $S_{22}^{NN}$ can be shown to be arising from the difference between the first and the second normal stress differences:⁴⁶

$$S_{22}^{NN}=\underset{V_d}{\int }\left(T_{22}^{N{N}^{\prime }}-\frac{T_{22}^{N{N}^{\prime }}+T_{11}^{N{N}^{\prime }}+T_{33}^{N{N}^{\prime }}}{3}\right)dV=\underset{V_d}{\int }\left(\frac{N_1^{N{N}^{\prime }}-N_2^{N{N}^{\prime }}}{3}\right)dV.$$ (17)

In (16), the second term is absent for a viscosity matched system (λ_μ = 1). For the N/V case,⁴⁶ we showed the theory to be only valid away from the wall. We therefore choose a distance of h = 2.45a where Newtonian comparison works well for examining the effects of the viscoelasticity.

In Fig. 6(a), we plot $-S_{22}^{\mathrm{int}}$, which is the only contribution for a Newtonian system, showing that it varies nonmonotonically with increasing viscoelasticity—first increases and then decreases—similar to the migration velocity. $-S_{22}^{\mathrm{int}}$ is a purely geometric quantity and is determined by the instantaneous drop shape. Increasing deformation and decreasing angle of inclination increase it. We saw in Figure 4 that with increasing De, the deformation decreases monotonically but inclination angle shows a nonmonotonic variation with De first decreasing and then increasing. Here, we again show the nonmonotonicity of the angle for several Ca values in Fig. 6(b). The non-Newtonian part $-S_{22}^{NN}$ shown in Fig. 6(c) adds a positive contribution to the migration velocity. Note that for the N/V case, $-S_{22}^{NN}$ was negative which thereby retarded migration away from the wall.⁴⁶ Finally, in Fig. 6(d), we compare the lateral velocities for viscoelastic cases between the theory and simulations for three values of Ca. The theory matches well with the simulation, capturing the nonmonotonic variation, the slight difference arising from the finite Reynolds number (Re = 0.03) of our simulation. In the N/V case investigated previously, the inclination angle decreases with increasing De; the geometric stresslet $-S_{22}^{\mathrm{int}}$ due to interfacial tension increases, but the overall velocity variation with De is dictated by the non-Newtonian contribution $-S_{22}^{NN}$ that decreases with De. Here the nonmonotonic velocity variation with De is dictated by the interfacial part $-S_{22}^{\mathrm{int}}$ caused by the variation in the angle of inclination.

FIG. 6.

(a) Interfacial part of the stresslet as a function of De for two values of Ca at h/a = 2.45. (b) Inclination angle for the same cases; inset shows the inclination normalized by its Newtonian value. (c) Non-Newtonian part of the stresslet for the same cases. (d) Migration velocity from the far-field theory (Th) is compared with those from the simulations (Sim) for varying De at three Ca values.

C. Effects of viscosity ratio and super-critical Ca for breakup

In this section, we very briefly study the effects of viscosity ratio to show how it can significantly affect the dynamics especially for those capillary numbers where viscosity matched system gives rise to large deformation and eventual breakup. In Fig. 7(a), we plot the lateral migration velocities for varying De for Ca = 0.5 at a high viscosity ratio λ_μ = 10. This is above the critical capillary number for breakup for viscosity matched Newtonian system. Yet the drop shapes here remain bounded due to the stabilizing effects of the drop phase viscoelasticity and the higher viscosity ratio. We notice that the migration velocity increases monotonically with De, unlike that of a viscosity matched system where the velocity is non-monotonic (Fig. 2(b)). Inset plots the instantaneous velocities (normalized by the Newtonian value) as a function of De, at h/a = 1.75. Also, note that the increase in velocity relative to the Newtonian case is much larger than the viscosity matched system—more than 100% compared to 20% in Fig. 5(a) β = 0.5 case. The evolution of drop deformation plotted in Fig 7(b) shows that increasing viscoelasticity decreases the initial overshoot in deformation. Fig. 7(c) shows that at this high viscosity ratio of 10, migration velocity varies nonmonotonicaly with Ca for all De values unlike the viscosity matched case. Such non-monotonicity for high viscosity ratio cases was also observed in a Newtonian system.²⁷

FIG. 7.

Evolution of (a) lateral migration and (b) deformation for varying De at Ca = 0.5 and λ_μ = 10. Inset of (a) plots the instantaneous normalized velocity (with respect to the Newtonian value) vs. De at h/a = 1.75. (c) Migration velocity as a function of Ca for different De values at h/a = 1.75 and λ_μ = 10.

In Fig. 8(a), we plot the time evolution of the lateral velocity for Newtonian and several Deborah number cases for the same Ca = 0.5 and an intermediate viscosity ratio λ_μ = 3.5. Corresponding deformation is plotted in Fig. 8(b). At this intermediate viscosity ratio, for the Newtonian case and De = 0.5, drops experience breakup, and therefore do not reach a quasi-steady state. On the other hand, higher Deborah number cases De = 1.5 and 2.5 result in bounded shapes. For the lower two Deborah cases, the sudden increase in the velocity after reaching a minimum is because of the necking that forms at large deformation. Similar behavior was also observed for N/V systems.⁴⁶ In Fig. 8(c) drop shapes are shown for De = 0 and 0.5 before and after the velocity minimum (marked in Fig. 8(a)). After the minimum, the drop shapes are more of two droplets connected by a thread like structure.

FIG. 8.

Evolution of (a) lateral migration, (b) deformation for varying De at Ca = 0.5 and λ_μ = 3.5. (c) Drop shapes for De = 0 and 0.5 before and after the minima; time instants are shown and also marked in part (a).

V. CONCLUSIONS

The dynamics of a migrating viscoelastic (FENE-MCR) drop in a Newtonian liquid subjected to a wall bounded shear is numerically investigated. Similar to a purely Newtonian system, the drop settles down to a quasi-steady motion where the dynamics is independent of the initial position and the velocity approximately scales with capillary number and the inverse square of the separation from the wall. With increasing Deborah number, the velocity initially increases, but eventually decreases at high values of Deborah or Wisssenberg number. Using a Green's function formulation of the problem, we have developed a far-field analytical expression of the migration velocity. It describes the migration as caused by the stresslet field due to the drop in presence of the wall. The analytical expression contains a distinct component to the stresslet field contributed by the differences between the first and the second normal stress differences inside the drop. The theory matches with the simulated migration velocity capturing the nonmonotonic trend. The nonmonotonicity is caused by the variation in drop inclination angle. Viscoelastic effects on migration are larger at high viscosity ratios; it can prevent drop break up for drops that would break in viscosity-matched system. It can also generate nonmonotonic variation of velocity with capillary number which has also been noticed for Newtonian systems.