Chunfeng Zhou

Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing

This paper describes a novel numerical algorithm for simulating interfacial dynamics of non-Newtonian fluids. The interface between two immiscible fluids is treated as a thin mixing layer across which physical properties vary steeply but continuously. The property and evolution of the interfacial layer is governed by a phase-field variable ? that obeys a Cahn-Hilliard type of convection-diffusion equation. This circumvents the task of directly tracking the interface, and produces the correct interfacial tension from the free energy stored in the mixing layer. Viscoelasticity and other types of constitutive equations can be incorporated easily into the variational phase-field framework. The greatest challenge of this approach is in resolving the sharp gradients at the interface. This is achieved by using an efficient adaptive meshing scheme governed by the phase-field variable. The finite-element scheme easily accommodates complex flow geometry and the adaptive meshing makes it possible to simulate large-scale two-phase systems of complex fluids. In two-dimensional and axisymmetric three-dimensional implementations, the numerical toolkit is applied here to drop deformation in shear and elongational flows, rise of drops and retraction of drops and torii. Some of these solutions serve as validation of the method and illustrate its key features, while others explore novel physics of viscoelasticity in the deformation and evolution of interfaces.

Sharp-interface limit of the Cahn-Hilliard model for moving contact lines

Diffuse-interface models may be used to compute moving contact lines because the Cahn–Hilliard diffusion regularizes the singularity at the contact line. This paper investigates the basic questions underlying this approach. Through scaling arguments and numerical computations, we demonstrate that the Cahn–Hilliard model approaches a sharp-interface limit when the interfacial thickness is reduced below a threshold while other parameters are fixed. In this limit, the contact line has a diffusion length that is related to the slip length in sharp-interface models. Based on the numerical results, we propose a criterion for attaining the sharp-interface limit in computing moving contact lines.

Formation of simple and compound drops in microfluidic devices

This work is motivated by the recent experimental development of microfluidic flow-focusing devices that produce highly monodisperse simple or compound drops. Using finite elements with adaptive meshing in a diffuse-interface framework, we simulate the breakup of simple and compound jets in coflowing conditions, and explore the flow regimes that prevail in different parameter ranges. Moreover, we investigate the effects of viscoelasticity on interface rupture and drop pinch-off. The formation of simple drops exhibits a dripping regime at relatively low flow rates and a jetting regime at higher flow rates. In both regimes, drops form because of the combined effects of capillary instability and viscous drag. The drop size increases with the flow rate of the inner fluid and decreases with that of the outer fluid. Viscoelasticity in the drop phase increases the drop size in the dripping regime but decreases it in the jetting regime. The formation of compound drops is a delicate process that takes place in a narrow window of flow and rheological parameters. Encapsulation of the inner drop depends critically on coordination of capillary waves on the inner and outer interfaces.

Short note: Spontaneous shrinkage of drops and mass conservation in phase-field simulations

In this note, we examine the implications of Cahn-Hilliard diffusion on mass conservation when using a phase-field model for simulating two-phase flows. Even though the phase-field variable @f is conserved globally, a drop shrinks spontaneously while @f shifts from its expected values in the bulk phases. Those changes are found to be proportional to the interfacial thickness, and we suggest guidelines for minimizing the loss of mass. Moreover, there exists a critical radius below which drops will eventually disappear. With a properly chosen mobility parameter, however, this process will be much slower than the physics of interest and thus has little ill effect on the simulation.

A computational study of the coalescence between a drop and an interface in Newtonian and viscoelastic fluids

A drop falling onto a fluid-fluid interface may not merge with it at once but may undergo a so-called partial coalescence cascade. Experimental observations of this phenomenon have revealed fascinating features of the process for Newtonian as well as polymeric fluids. In this paper, we describe numerical simulations of partial coalescence based on a phase-field method. Results show that partial coalescence occurs for an intermediate range of drop sizes, and proceeds in two stages: capillary waves propagating along the drop and transforming it into a fluid column, and neck formation on the column and pinch-off of the secondary drop. In the first stage, interfacial energy turns into kinetic energy following film rupture, while in the second, the kinetic energy overcomes an energy barrier due to the initial increase in interfacial area during neck formation. A parametric study establishes a criterion for partial coalescence in terms of a maximum Ohnesorge number that applies to a wide range of fluid densitie...

Simulation of Neutrophil Deformation and Transport in Capillaries using Newtonian and Viscoelastic Drop Models

It is well known that neutrophils take much longer to traverse the pulmonary capillary bed than erythrocytes, and this is likely due to differences in the structure and rheology of the cells. In this study, we simulate the transit of a neutrophil in a capillary using a Newtonian drop model and a viscoelastic drop model. The cell membrane is represented by an interface with isotropic and constant tension, and the cell motion and deformation are described by a phase-field method. The governing equations are solved using finite elements in an axisymmetric geometry, and the thin interfaces are resolved by mesh adaptivity. With a fixed pressure drop, the entry of a cell into a capillary consists of several stages in which the flow rate varies in distinct manners. The entrance time is consistent with experimental measurements. It decreases with the pressure drop, increases with the cell viscosity and generally decreases with the relaxation time of a viscoelastic cytoplasm. The capillary geometry has a strong effect on the entry and transit of a neutrophil. The entrance time increases sharply when the capillary diameter decreases or when the capillary is constricted by a pinch.

The rise of Newtonian drops in a nematic liquid crystal

We simulate the rise of Newtonian drops in a nematic liquid crystal parallel to the far-field molecular orientation. The moving interface is computed in a diffuseinterface framework, and the anisotropic rheology of the liquid crystal is represented by the Leslie–Ericksen theory, regularized to permit topological defects. Results reveal interesting coupling between the flow field and the orientational field surrounding the drop, especially the defect configuration. The flow generally sweeps the point and ring defects downstream, and may transform a ring defect into a point defect. The stability of these defects and their transformation are depicted in a phase diagram in terms of the Ericksen number and the ratio between surface anchoring and bulk elastic energies. The nematic orientation affects the flow field in return. Drops with planar anchoring on the surface rise faster than those with homeotropic anchoring, and the former features a vortex ring in the wake. These are attributed to the viscous anisotropy of the nematic. With homeotropic anchoring, the drop rising velocity experiences an overshoot, owing to the transformation of the initial surface ring defect to a satellite point defect. With both types of anchoring, the drag coefficient of the drop decreases with increasing Ericksen number as the flow-alignment of the nematic orientation reduces the effective viscosity of the liquid crystal.

Dynamic Simulation of Droplet Interaction and Self-Assembly in a Nematic Liquid Crystal

We use dynamic simulations to explore the pairwise interaction and multiparticle assembly of droplets suspended in a nematic liquid crystal. The computation is based on a regularized Leslie-Ericksen theory that allows orientational defects. The homeotropic anchoring on the drop surface is of sufficient strength as to produce a satellite point defect near the droplet. Based on the position of the defects relative to the host droplet and the far-field molecular orientation, we have identified five types of pairwise attractive and repulsive forces. In particular, long-range attraction between two droplets with their line of centers along the far-field orientation decays as R-4, with R being the center-to-center separation. This agrees with prior static calculations and a phenomenological model that treats the attraction as that between two dipoles. For interaction in shorter ranges, our simulations agree qualitatively with experimental measurements and static calculations. However, there is considerable quantitative discrepancy among the few existing studies and our simulation. We suggest that this is partly due to the dynamic nature of the process, which has never been taken into account in prior calculations. Multidrop simulations show the formation of linear chains through pairwise interactions between nearby droplets. Parallel chains repel or attract each other depending on the relative orientation of the drop-to-defect vector. These are consistent with experimental observations of chain formation and two-dimensional self-assembly in bulk nematics and smectic-C films.

Elastic encapsulation in bicomponent stratified flow of viscoelastic fluids

The second normal stress difference N2 experienced by non-Newtonian fluids flowing in a pipe may give rise to secondary flows in the transverse direction. As a result, one component tends to encapsulate the other in stratified flows. In multilayer coextrusion, such secondary flows tend to distort the interface and affect layer uniformity. This paper presents numerical simulations of the elastically driven encapsulation in two-component stratified viscoelastic fluids. The simulations are based on a phase-field theoretical model and use finite elements with adaptive meshing to resolve the moving interfaces. The results suggest two mechanisms for elastic encapsulation: One due to the mismatch of N2 between the components and the other due to noncircular geometry of the cross section. In circular pipes, the more elastic fluid tends to protrude into the other component in the center of the pipe and become encapsulated. This produces the curtate cycloid interface shape commonly seen in experiments. If the cross...

Heart-shaped bubbles rising in anisotropic liquids

This Letter reports on numerical simulations motivated by experimental observations of an unusual inverted-heart shape for bubbles rising in an anisotropic micellar solution. We explain the bubble shape by assuming that the micelles are aligned into a nematic phase, whose anchoring energy on the bubble competes with the interfacial tension and the bulk elasticity of the nematic to modify the interfacial curvature. Numerical results show that bubbles with sufficiently strong planar anchoring rising in a vertically aligned nematic indeed assume the observed shape. The parameter values required are compared with the experimental materials and conditions.