Effects of matrix viscoelasticity on the lateral migration of a deformable drop in a wall-bounded shear

Abstract

The dynamics of a drop deforming, orienting and moving in a shear flow of a viscoelastic liquid near a wall is numerically investigated using a front-tracking finitedifference method and a semi-analytic theory. The viscoelasticity is modelled using the modified FENE-CR constitutive equation. In a Newtonian system, deformation in a drop breaks the reversal symmetry of the system resulting in a migration away from the wall. This study shows that the matrix elasticity reduces the migration velocity, the reduction scaling approximately linearly with viscoelasticity (product of the Deborah number De and the ratio of polymer viscosity to total viscosity β). Similar to a Newtonian system, for small Deborah numbers, the dynamics quickly reaches a quasisteady state where deformation, inclination, as well as migration and slip velocities become independent of the initial drop-wall separation. They all approximately scale inversely with the square of the instantaneous separation except for deformation which scales inversely with the cube of separation. The deformation shows a non-monotonic variation with increasing viscoelasticity similar to the case of a drop in an unbounded shear and is found to influence little the change in migration. Two competing effects due to matrix viscoelasticity on drop migration are identified. The first stems from the reduced inclination angle of the drop with increasing viscoelasticity that tries to enhance migration velocity. However, it is overcome by the second effect inhibiting migration that results from the normal stress differences from the curved streamlines around the drop; they are more curved on the side away from the wall compared with those in the gap between the wall and the drop, an effect that is also present for a rigid particle. A perturbative theory of migration is developed for small ratio of the drop size to its separation from the wall that clearly shows the migration to be caused by the image stresslet field due to the drop in presence of the wall. The theory delineates the two competing viscoelastic effects, their relative magnitudes, and predicts migration that matches well with the simulation. Using the simulation results and the stresslet theory, we develop an algebraic expression for the quasi-steady migration velocity as a function of Ca, De and β. The transient dynamics of the migrating drop is seen to be governed by the finite time needed for development of the viscoelastic stresses. For larger capillary numbers, in both Newtonian and viscoelastic matrices, a viscous drop fails to reach a quasi-steady state independent of initial drop-wall separation. Matrix viscoelasticity tends to prevent drop breakup. Drops that † Email address for correspondence: sarkar@gwu.edu Near-wall migration of a viscous drop in a sheared viscoelastic matrix 319 break up in a Newtonian matrix are stabilized in a viscoelastic matrix if it is initially far away from the wall. Initial proximity to the wall enhances deformation and aids in drop breakup.

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

“…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.…”

“…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.…”

“…Other details of the problem setup can be found in our previous publication. 46 Nondimensionalizing the problem using a andγ −1 we obtain the non-dimensional parameters Reynolds number Re = ρ m a 2γ / μ m , capillary number Ca = μ m aγ / , Deborah number De = λγ , 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.…”

“…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.…”

A front-tracking method for computational modeling of viscoelastic two-phase flow systems

“…The method was extended to treat viscoelastic interfacial flows first by Sarkar and Schowalter [48] and has been successfully used to study viscoelastic drop dynamics in shear and buoyancy-driven flows [1,[33][34][35]. Sarkar and Schowalter [48] used the semi-analytical method that is limited to low or moderate Weissenberg numbers as mentioned earlier.…”