Axisymmetric Extensional Flow Research Articles

Numerical investigation of the effect of surface viscosity on droplet breakup and relaxation under axisymmetric extensional flow

In this study, we perform boundary-integral simulations to investigate the role of interfacial viscosity in the deformation and breakup of a single droplet suspended in an axisymmetric extensional flow under the Stokes flow regime. We model the insoluble surfactant monolayer using the Boussinesq–Scriven constitutive relationship for a Newtonian interface. We compare the deformation and breakup results from our boundary-element simulations with results from small deformation perturbation theories. We observe that the surface shear/dilational viscosity increases/decreases the critical capillary number beyond which the droplet becomes unstable and breaks apart by reducing/increasing the droplet deformation at a given capillary number compared with a clean droplet. We present the relative importance of surface shear and dilational viscosity on droplet stability based on their measured values reported in experimental studies on surfactants, lipid bilayers and proteins. In the second half of the paper, we incorporate the effect of surfactant transport by solving the time-dependent convection–diffusion equation and consider a nonlinear equation of state (Langmuir adsorption isotherm) to correlate the interfacial tension with the changes in surfactant concentration. We explore the coupled influence of pressure-dependent surface viscosity and Marangoni stresses on droplet deformation and breakup. In the case of a droplet with pressure-dependent surface shear viscosity, we find that a droplet with pressure-thinning/thickening surfactant is less/more deformed than a droplet with pressure-independent surfactant. We conclude by discussing the combined impact of surface viscosity and surfactant transport on the relaxation of an initially extended droplet in a quiescent external fluid.

On the diffusion from a slender drop in an extensional flow: Inertial effects at zero Peclet numbers

The transfer of mass between a slender drop and a liquid in an axisymmetric extensional flow, at zero Peclet numbers, is theoretically studied. We allow the external flow to have a small (but not zero) amount of inertia. The problem is governed by three dimensionless parameters: the capillary number (Ca>>1), the viscosity ratio (λ<<1) and the external Reynolds number (Re<<1). It is shown that the shape of the drop is, at first approximation, a slender spindle suggesting the usage of the bispherical coordinate system. Applying the method presented by Szegö and Pyne, for the electrostatic capacity of a spindle, an analytical solution containing conal functions is obtained. The results show that, as the capillary number or the Reynolds number increase, the drop becomes thinner and longer, the surface area increases, resulting in larger mass transfer rates, with the capillary number being the most influential parameter.

Mass transfer around slender drops in an extensional flow: Inertial effects at large peclet numbers

AbstractThe exchange of mass between a slender drop and a viscous liquid in an axisymmetric extensional flow and at large Peclet numbers is examined. Four cases are discussed: an inviscid drop and a viscous drop, under both creeping flow conditions and when the external flow has a weak amount of inertia. Solutions are presented as a function of the capillary number (Ca ≫ 1), the viscosity ratio (κ ≪ 1), the external Reynolds number (Re ≪ 1), and the Peclet number (Pe ≫ 1). The results suggest that the steady mass transfer rate, to or from the slender drop, is proportional to the elongation rate to the (relative) high power of 3/2. In general, as inertia increases, the drop becomes thinner and longer, and the average tangential surface velocity and the surface area increase, resulting in larger mass transfer rates, especially near the breakup point.

Asymmetric breakup of a droplet in an axisymmetric extensional flow

The asymmetric breakups of a droplet in an axisymmetric cross-like microfluidic device are investigated by using a three-dimensional volume of fluid (VOF) multiphase numerical model. Two kinds of asymmetries (droplet location deviation from the symmetric geometry center and different flow rates at two symmetric outlets) generate asymmetric flow fields near the droplet, which results in the asymmetric breakup of the latter. Four typical breakup regimes (no breakup, one-side breakup, retraction breakup and direct breakup) have been observed. Two regime maps are plotted to describe the transition from one regime to another for the two types of different asymmetries, respectively. A power law model, which is based on the three critical factors (the capillary number, the asymmetry of flow fields and the initial volume ratio), is employed to predict the volume ratio of the two unequal daughter droplets generated in the direct breakup. The influences of capillary numbers and the asymmetries have been studied systematically in this paper. The larger the asymmetry is, the bigger the one-side breakup zone is. The larger the capillary number is, the more possible the breakup is in the direct breakup zone. When the radius of the initial droplet is 20μm, the critical capillary numbers are 0.122, 0.128, 0.145, 0.165, 0.192 and 0.226 for flow asymmetry factor AS=0.05, 0.1, 0.2, 0.3, 0.4 and 0.5, respectively, in the flow system whose asymmetry is generated by location deviations. In the flow system whose asymmetry is generated by two different flow rates at two outlets, the critical capillary numbers are 0.121, 0.133, 0.145, 0.156 and 0.167 for AS=1/21, 3/23, 1/5, 7/27 and 9/29, respectively.

Experimental Characterization of Breakage Rate of Colloidal Aggregates in Axisymmetric Extensional Flow

Aggregates prepared under fully destabilized conditions by the action of Brownian motion were exposed to an extensional flow generated at the entrance of a sudden contraction. Two noninvasive techniques were used to monitor their breakup process [i.e. light scattering and three-dimensional (3D) particle tracking velocimetry (3D-PTV)]. While the first one can be used to measure the size and the morphology of formed fragments after the breakage event, the latter is capable of resolving trajectories of individual aggregates up to the breakage point as well as the trajectories of formed fragments. Furthermore, measured velocity gradients were used to determine the local hydrodynamic conditions at the breakage point. All this information was combined to experimentally determine for the first time the breakage rate of individual aggregates, given in the form of a size reduction rate K(R), as a function of the applied strain rate, as well as the properties of the formed fragments (i.e., the number of formed fragments and the size ratio between the largest fragment and the original aggregate). It was found that K(R) scales with the applied strain rate according to a power law with the slope being dependent on the initial fractal dimension only, while the obtained data indicates a linear dependency of K(R) with the initial aggregate size. Furthermore, the probability distribution function (PDF) of the number of formed fragments and the PDF of the size ratio between the largest fragment and the original aggregate indicate that breakage will result with high probability (75%) in the formation of two to three fragments with a rather asymmetric ratio of sizes of about 0.8. The obtained results are well in agreement with the results from the numerical simulations published in the literature.

Asymmetry of flow fields and asymmetric breakup of a droplet

A general understanding of asymmetric breakups of a droplet due to the asymmetry of flow fields is developed. Through a volume of fluid method, asymmetric breakups of a droplet suspended in an axisymmetric extensional flow are investigated. A regime map is depicted to show four typical hydrodynamic regimes (no breakup, one-side breakup, retraction breakup and direct breakup). Three major factors critical to an asymmetric breakup (the capillary number Ca determining the cutting speed, the asymmetry As of the flow field causing the droplet translation and the initial volume ratio r0) are presented. Based on this analysis, a simple model is proposed to predict the volume ratio rs of two unequal daughter droplets. Differing from the symmetric breakup, the asymmetry effects are studied systematically, which could be extended to direct the study of diverse asymmetric situations.

Bubble Deformation and Growth Inside Viscoelastic Filaments Undergoing Very Large Extensions

Motivated by the probe experiment for characterizing the adhesion strength of polymeric materials, we studied the axisymmetric extensional flow of a viscoelastic liquid filament that contains one or three initially spherical bubbles along its axis of symmetry. The filament is confined between two parallel and coaxial disks of the same radius and its initially cylindrical outer surface is surrounded by air. The flow is induced by the axial extension of the upper disk under constant velocity, while the lower disk remains stationary. The rheology of the liquid is described by the exponential Phan–Thien and Tanner viscoelastic model. A quasi-elliptic transformation combined with domain decomposition and local mesh refinement is employed to discretize the domain. The mixed finite element Galerkin method for the mass and momentum balances combined with the EVSS-G method and SUPG weighting for the constitutive equation are used to obtain their weak forms. All these are necessary for the successful simulation of ...

Elastic capsule deformation in general irrotational linear flows

Knowledge of the response of elastic capsules to an imposed fluid flow is necessary for predicting the deformation and motion of biological cells and synthetic capsules in microfluidic devices and in microcirculation. Capsules have been studied in shear, planar extensional and axisymmetric extensional flows. Here, the flow gradient matrix of a general irrotational linear flow is characterized by two parameters, its strain rate, defined as the maximum of the principal strain rates, and a new term, q, the difference of the two lesser principal strain rates, scaled by the maximum principal strain rate; this characterization is valid for ellipsoids in irrotational linear flow, and gives good results for spheres in general linear flows at low capillary numbers. We demonstrate that deformable non-spherical particles align with the principal axes of an imposed irrotational flow. Thus, it is most practical to model deformation of non-spherical particles already aligned with the flow, rather than considering each arbitrary orientation. Capsule deformation was modeled for a sphere, a prolate spheroid and an oblate spheroid, subjected to combinations of uniaxial, biaxial and planar extensional flows; modeling was performed using the immersed boundary method. The time response of each capsule to each flow was found, as were the steady-state deformation factor, mean strain energy and surface area. For a given capillary number, planar flows led to more deformation than uniaxial or biaxial extensional flows. Capsule behavior in all cases was bounded by the response of capsules to uniaxial, biaxial and planar extensional flows.

Effects of a surfactant monolayer on the measurement of equilibrium interfacial tension of a drop in extensional flow

The effect of surfactant monolayer concentration on the measurement of interfacial surface tension using transient drop deformation methods is studied using the Boundary Integral Method. Emulsion droplets with a surfactant monolayer modeled with the Langmuir equation of state initially in equilibrium are suddenly subjected to axisymmetric extensional flows until a steady state deformation is reached. The external flow is then removed and the retraction of the drops to a spherical equilibrium shape in a quiescent state is simulated. The transient response of the drop to the imposed flow is analyzed to obtain a characteristic response time, τ s ∗ . Neglecting the initial and final stages, the retraction process can be closely approximated by an exponential decay with a characteristic time, τ r ∗ . The strength of the external flow on each model drop is increased in order to investigate the coupled effect of deformation and surfactant distribution on the characteristic relaxation time. Different model drops are considered by varying the internal viscosity and the equilibrium surfactant concentrations from a surfactant free state (clean) to high concentrations approaching the maximum packing limit. The characteristic times obtained from the simulated drop dynamics both in extension and retraction are used to determine an apparent surface tension employing linear theory. In extension the apparent surface tension under predicts the prescribed equilibrium surface tension. The error increases monotonically with the equilibrium surfactant concentration and diverges as the maximum packing limit is approached. In retraction the apparent surface tension under predicts the prescribed equilibrium surface tension depends non-monotonically on the equilibrium surfactant concentration. The error is highest for moderate surfactant concentrations and decreases as the maximum packing limit is approached. It was found that the difference between the prescribed surface tension and the apparent surface tension increased as the viscosity ratio decreased. Differences as large as 40% were seen between the prescribed surface tension and the apparent surface tension predicted by the linear theory.

Deformation and breakup of a non-Newtonian slender drop in an extensional flow: inertial effects and stability

We consider the deformation and breakup of a non-Newtonian slender drop in a Newtonian liquid, subject to an axisymmetric extensional flow, and the influence of inertia in the continuous phase. The non-Newtonian fluid inside the drop is described by the simple power-law model and the unsteady deformation of the drop is represented by a single partial differential equation. The steady-state problem is governed by four parameters: the capillary number; the viscosity ratio; the external Reynolds number; and the exponent characterizing the power-law model for the non-Newtonian drop. For Newtonian drops, as inertia increases, drop breakup is facilitated. However, for shear thinning drops, the influence of increasing inertia results first in preventing and then in facilitating drop breakup. Multiple stationary solutions were also found and a stability analysis has been performed in order to distinguish between stable and unstable stationary states.

Flow enhanced diffusion-limited polymerization of rodlike molecules

Polymerization of rigid rodlike molecules with reactive end groups requires near parallel orientation of the molecules. The reaction is diffusion limited because of the low mobility of the molecules in the later stages of the reaction. Experimental studies have shown that flow-induced molecular orientation enhances the rate of polymerization [Agarwal and Khakhar, Nature 360, 53 (1992)]. Here a theoretical study of the polymerization under axisymmetric extensional flow is carried out to obtain the effective reaction rate constant (keff) for the reaction. Computations show that an increase in the intrinsic rate constant (kh) results in a decrease in the relative rate constant krel=keff/kh. Reduction in the rotational diffusivity (Dr) results in a significant reduction in krel; however, the variation of the translational diffusion coefficient perpendicular to the rod axis D⊥ has only a small effect for D⊥/D‖≪1, where D‖ is the diffusivity parallel to the rod axis. The imposition of flow increases the effective rate constant, however, the variation of krel with other parameters remains qualitatively similar at different Peclet numbers (Pe=ε/Dr, where ε is the extensional rate). To simulate the variation of the rate constant during polymerization, computations are carried out for different rod lengths using correlations to estimate rod diffusivities. Results indicate that krel initially decreases and then increases after a certain critical degree of polymerization, which reduces with increase in Peclet number. For sufficiently high extensional rates (ε∼200 s−1) the rate constant becomes higher than the intrinsic value (krel&amp;gt;1).

Calculation of droplet deformation at intermediate Reynolds number using a Volume of Fluid technique

The deformation of a droplet or bubble in simple extensional flow fields is investigated at intermediate Reynolds number using a Volume-of-Fluid (VOF) numerical algorithm (MAC2) developed by Rudman. The results of an evaluation study of the MAC2 algorithm for such flows are presented. The final steady shapes of droplets in axisymmetric extensional flow fields determined by MAC2 are compared with the published numerical results of Ramaswamy and Leal. This work forms the initial stage of a more comprehensive study of droplet deformation in time-dependent shear fields experienced by drops flowing through devices (e.g.\ homogenisers) designed for droplet breakup.

Marangoni effects on drop deformation in an extensional flow: The role of surfactant physical chemistry. I. Insoluble surfactants

The shape of a drop centered in an axisymmetric extensional flow is determined by the viscous stresses that deform the drop and surface tension γ′ that resists the deformation. The ratio of these stresses is given by the capillary number, Ca. When Ca is small enough, the drop attains a steady shape. However, above a threshold value, Cacr, the drop elongates continuously, and no steady shape is attained. When surfactants are present on the drop interface, the surface tension is determined by the surface concentration profile, which varies throughout the deformation process. Initially, the surface tension is given by γeq′, in equilibrium with the uniform surface concentration Γeq′. When the flow is initiated, surfactant is swept toward the drop tips, reducing the surface tension there, and altering the interfacial stress balance tangentially through Marangoni stresses and normally through the Laplace pressure. In this paper, the effects of an insoluble surfactant on drop deformation are studied. In previous work, either a surface equation of state for the surface tension γ′ that is linear in the surface concentration Γ′ was used, an approximation that is valid only for dilute Γ′, or Γ′ sufficiently dilute for the linear approximation to be valid were studied. In this paper, a nonlinear surface equation of state that accounts for surface saturation and nonideal interactions among the surfactant molecules is adopted. The linear framework results are recovered for Γ′ that are sufficiently dilute. As Γ′ is increased, the effects of saturation and surfactant interactions are probed at constant initial Γeq′ and at constant initial γeq′. Finally, the case of strong intersurfactant cohesion is treated with a first-order surface phase transformation model. At moderate surface concentrations, these nonlinear phenomena strongly alter the steady drop deformations and Cacr relative to the uniform surface tension and linear equation of state results.

Mass transfer between a slender bubble and a viscous liquid in axisymmetric extensional flow

Mass transfer between a slender bubble and a viscous Newtonian liquid in axisymmetric extensional and creeping flow has been studied. The exact steady state solution, at large Peclet numbers, has been obtained using the method presented by Levich for uniform flow around a spherical drop and also by the useful equation derived by Lochiel and Calderbank for axisymmetric bodies of revolution.

Breakup of liquid threads in linear flows

We study, theoretically, the surface-tension-driven breakup of a long filament of fluid in a general linear flow, v = L·x. By analyzing the problem in a moving frame and assuming a circular cross section we find that the flow around the filament is an axisymmetric extensional flow with a time-dependent strength, which can be calculated from the rate of rotation of the filament and a contribution to the axial velocity which varies with the azimuthal angle. The analysis of the axisymmetric time-dependent case does not appear to be overly restrictive: the asymmetric variation may be small even in the case of a simple shear flow, in which the asymmetry is the greatest among all possible linear flows, depending on the initial orientation of the filament. We present calculations for two special cases: hyperbolic extensional flow and simple shear flow. The results indicate that under similar conditions, the drop fragments produced on breakup in simple shear flow are larger than those in hyperbolic extensional flow. The predictions of the theory also compare reasonably well with some previous experimental data in hyperbolic extensional flow and simple shear flow.

The kinetics of flowing dispersions: VIII. Doublets of rigid spheres (theoretical)

A general method of calculating forces, torques, and translational and rotational velocities of rigid, neutral, equal-sized spheres in a viscous fluid undergoing uniform shear flows is presented. The method is based on the matrix formulation of the hydrodynamic resistances by Brenner and O'Neill, and it is illustrated for simple shear and planar and axisymmetric extensional flows, for which the trajectories of pairs of sphere centers are calculated. It is shown that, in simple shear, trajectories are either open or closed; in extensional flows, all trajectories must be open. This has interesting implications in the dispersion of aggregates by shear. Although the translational and rotational behavior of interacting spheres is characterized by the type of flow, the behavior of the spheres in each flow is calculable from flow-independent quantities. Special emphasis is placed on the collision behavior of doublets. Initially separate spheres can never touch, but they can come into very close proximity, especially during equatorial encounters. In simple shear, translating spheres rotate about their own axes, which, in a doublet, causes one sphere to roll relative to the other except when they touch or are rigidly connected. The rotations of free and connected doublets and dumbbells are related to those of prolate spheroids.