In strongly sheared emulsions, experiments (Galloway and Macosko 2002) have shown that systems consisting of one continuous (matrix) and one dispersed (drops) phase may undergo a coalescence cascade leading to a system in which both phases are continuous, (sponge-like). Such configurations may have desirable mechanical and electrical properties and thus have wide ranging applications.
Using a new and improved diffuse-inteface method (accurate
surface tension force formulation, volume-preservation, and
efficient nonlinear multigrid solver) developed by Kim and
Lowengrub 2004, we perform numerical simulations of
cocontinuous blends and determine the conditions for
formation. We also characterize their rheology.
[MB.002] Rounded ends and near cusps: liquid surfaces in a viscous straining flow
Marko Kleine Berkenbusch, Wendy Zhang (University of Chicago)
Emulsification studies show that droplets immersed in an
external straining flow are easily pulled apart. In
situations of comparable viscosity of the inner and outer
liquid, the only stable steady-state shapes are slightly
deformed drops with rounded ends, with no particular
small-scale structure to them. Cohen et al., on the other
hand, found in their selective withdrawal experiments that
the interface between two horizontal immiscible fluid layers
can evolve into a steady-state structure with an extremely
sharp, cusp-like tip when a convergent flow is imposed by
withdrawing the upper fluid. Although viscous stresses
balance against surface tension effects in both situations,
the interfaces show qualitatively different behaviors. In
order to understand the reasons for this difference, we
study a model in which a pendant drop at the end of an
infinite nozzle is deformed by an external straining flow.
the interface deformation, and its dependence on nozzle
boundary conditions, are calculated via a boundary intergral
simulation.
[MB.003] Oscillating extensional rheology of emulsion of drops at finite Reynolds number: negative elasticity
Xiaoyi Li, Kausik Sarkar (Mechanical Engineering, University of Delaware, Newark, DE 19716)
We numerically investigate the rheological response of an
emulsion in an oscillating extensional flow at finite
Reynolds number. The rheology of an emulsion of drops is
dependent on the morphology, viz. size, shape and
orientation of drops. Till date, research on drop
deformation and rheology has mostly been restricted to
steady linear flows. We use Front-tracking method to compute
deformation of a drop, and obtain excess stress by
Batchelor’s formula. Drop-drop interactions are neglected.
We investigate the relation between excess interfacial
stress and imposed strain rate, varying interfacial tension,
flow frequency, and inertia. The stress-strain relation is a
function of the phase between the drop deformation and the
imposed flow. At low Reynolds number, the simulation
recovers the linear oscillatory rheology (loss and storage
moduli) of Oldroyd and Bousmina. At low surface tension, the
excess stress is predominantly elastic, while at high
surface tension it is viscous. Increased drop inertia leads
to resonance and complex phase in drop deformation. The
resulting excess interfacial stress displays a non-monotonic
variation with frequency and obtains a negative elastic
modulus at low frequency.
[MB.004] Motion of a Deformed Sphere with Slip in Creeping Flows
Andre Benard, Liping Jia (Mechanical Engineering, Michigan State University), Charles Petty (Chemical Engineering, Michigan State University)
An analytical solution for the motion of a slightly deformed sphere in creeping flows with the assumption of slip on the particle surface is presented. Explicit expressions are obtained for the hydrodynamic force and torque exerted by the fluid on the deformed sphere. A perturbation method, based on previous work done by Brenner [1964] and Lamb[1945], is used to solve for the motion of a fluid influenced by the presence of a deformed sphere. Slip is assumed at the surface of the particle. Hydrodynamic force and torque exerted by the fluid on the deformed sphere are expressed explicitly for a translational and rotational deformed sphere. The equation governing the motion and orientation of a spheroid induced by homogenous flows is also presented. This evolution equation for the orientation of the spheroid is similar to the equation derived by Jeffery [1922]. Solutions of this equation show that the period of rotation of the particle with slip is longer than for the same particle without slip. Furthermore, when the slip coefficient is sufficiently low, the particle rotates to a fixed angle that corresponds to a quasi-steady state in the flow.
REFERENCES Brenner, H. 1964 The Stokes resistance of a
slightly deformed sphere. Chemical Engineering Science 19,
519-539 Jeffery, G.B.1922 The motion of ellipsoidal
particles immersed in a viscous fluid. Proc. Soc. Lond.
Math., 102, 161-179 Lamb, H. 1945 Hydrodynamics, sixth
version, Dover, New York, U.S.A
[MB.005] Simulation of Drop Dynamics in Complex Fluid: Shear-induced Deformation
Pengtao Yue, James J. Feng (Department of Chemical and Biological Engineering and Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z4, Canada), Chun Liu (, Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA), Jie Shen (Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA)
This talk reports our recent simulations of drop deformation
in steady and startup shear flows. At least one of the
components is viscoelastic. The steady deformation depends
on the viscoelasticity of the components in a complex way.
If the drop phase is viscoelastic, then drop deformation
decreases with increasing Deborah number. If the matrix is
viscoelastic, however, the drop deformation decreases with
De for small De, but increases with De for larger De. This
reconciles conflicting experimental reports in the
literature. In shear startup, drop deformation undergoes an
overshoot and undershoot scenario, consistent with
experimental observations. By analyzing the flow and
nonlinear stress fields at the interface, we explain the
above results by the rheology of the fluids.
[MB.006] Self-Assembly of Isotropic Drops in a Nematic Matrix Simulated by a Diffuse-Interface Method
Oleksandr Barannyk, Pengtao Yue, James J. Feng (Department of Chemical and Biological Engineering and Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z4, Canada), Chun Liu (Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA), Jie Shen (Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA)
Recent experiments have reported self-assembly of isotropic
droplets in an anisotropic nematic suspending fluid. The
most commonly seen patterns are lines of droplets parallel
to the undisturbed matrix orientation. We have devised a
diffuse-interface model to represent this two-phase system
of complex fluids. A mixing energy is used to model the
interfacial tension so that the interfaces no longer need to
be tracked explicitly. Meanwhile, the liquid crystalline
microstructure is represented by a regularized Frank elastic
energy that is incorporated into a general variational
framework. This talk describes dynamic simulations that
explore the interaction of droplets and their assembly into
regular patterns. With strong homeotropic anchoring on the
surface, each drop nucleates a satellite point defect, and
neighboring drops attract along their axis to form lines.
The final pattern resembles experimental observations.
[MB.007] Analysis of Chaotic Mixing in Droplet Microfluidics
Ali Nadim (Keck Graduate Institute and Claremont Graduate University), Reza Miraghaie (Mechanical and Aerospace Engineering, UCLA)
Mixing inside a 2D circular drop driven by non-uniform
surface tension along its boundary is studied numerically.
The internal flow undergoes substantial change as the
surface tension gradient on the boundary is switched
periodically. Under Stokes flow conditions relevant to
microfluidics, the stream-function of the flow can be
obtained analytically, leading to a nonlinear system of
ordinary differential equations for the pathlines of passive
tracers in the drop. This system exhibits chaotic mixing for
certain ranges of the switching period. In addition, the
type and order of switching phenomena are seen to be
effective parameters for controlling the mixing.
Poincaré maps for different switching periods are
presented, demonstrating fixed points, attractors and basins
of attraction inside the drop. Also, the center of mass and
first and second moments of a swarm of passive particles,
initially concentrated in a small region in the drop, are
calculated to quantify the degree and quality of mixing. A
threshold is found for the switching period, above which
mixing is nearly complete.
[MB.008] A viscoelastic VOF-PROST code for the study of drop deformation
Y. Renardy, M. Renardy (Dept of Mathematics, Virginia Tech), D.B. Khismatullin (Dept of Biomed. Eng., Duke University), T. Chinyoka (Dept. of Mathematics, Virginia Tech)
A volume of fluid method is developed with a parabolic
representation of the interface for the surface tension
force (VOF-PROST). This three-dimensional transient code is
extended to treat viscoelastic liquids with the Oldroyd-B
constitutive equation. Simulations of deformation for a
Newtonian drop in a viscoelastic medium under shear are
reported.
[MB.009] Effects of surfactant on drop dynamics in Stokes flow.
Petia Vlahovska (Division of Engineering, Brown University), Jerzy Blawzdziwewicz (Departmet of Mechanical Engineering, Yale University), Michael Loewenberg (Department of Chemical Engineering, Yale University)
Surfactants modify interfacial properties and significantly
affect drop behavior in flow. We study dynamics a drop,
which is covered with a dilute monolayer of insoluble
surfactant, in linear viscous flows, both unbounded and in
the presence of a wall. The problem is challenging because
of the nonlinear coupling of drop deformation, surfactant
redistribution, and bulk flows. Analytical expansions
applicable where the surfactant distribution and drop shape
are only slightly perturbed from their equilibrium state are
developed. Numerical results, including three-dimensional
boundary integral simulations are used to determine the
range of convergence for the expansions. The theoretical
analyses quantify the interplay between the restoring
mechanisms (Marangoni-- and shape-- relaxation, and drop
rotation by the imposed flow) and the distortion of
surfactant distribution and drop shape induced by the
straining component of the flow. Studies of the hydrodynamic
interactions of a surfactant-covered drop with a wall show
that nonuniform surfactant distribution causes a spherical
drop to migrate away from the wall. Calculations of the
stress in a dilute emulsion of surfactant-covered drops
reveal rich rheological behavior.
[MB.010] The Effects of Shape Distortion and Charge Convection on the Settling Velocity of Drops in Electric Fields
Xiumei Xu, G.M. Homsy (Department of Mechanical and Environmental Engineering, University of California-Santa Barbara)
The deformation and settling velocity of a translating dielectric liquid drop in a uniform electric field are investigated theoretically at low Reynolds number. Perturbation methods are employed for shape distortion and charge convection respectively, and the settling velocity is calculated by combining the corrections from the two contributions linearly at the lowest order. The result for the settling velocity reads U_S=U_H(1+C_1*aE^2+C_2*E^2), where U_H is the Hadamard-Rybczynski velocity. C_1 and C_2 are dimensional coefficients describing the effects of shape distortion and charge convection respectively. Here a is the drop radius and E the electric field gradient. For most fluid pairs in the experimental papers by Torza et al.(1971) and Vizika amp; Saville(1992), the coefficients C_1 and C_2 have opposite sign, which indicates the existence of critical drop radius determining whether the settling velocity is greater or less than U_H. The radius is typically small, in the range of 10^-9 to 10^-3 m. Experimental attempts to verify its existence are described.