Pengtao Yue

A diffuse-interface method for simulating two-phase flows of complex fluids

Two-phase systems of microstructured complex fluids are an important class of engineering materials. Their flow behaviour is interesting because of the coupling among three disparate length scales: molecular or supra-molecular conformation inside each component, mesoscopic interfacial morphology and macroscopic hydrodynamics. In this paper, we propose a diffuse-interface approach to simulating the flow of such materials. The diffuse-interface model circumvents certain numerical difficulties in tracking the interface in the classical sharp-interface description. More importantly, our energy-based variational formalism makes it possible to incorporate complex rheology easily, as long as it is due to the evolution of a microstructure describable by a free energy. Thus, complex rheology and interfacial dynamics are treated in a unified framework. An additional advantage of our model is that the energy law of the system guarantees the existence of a solution. We will outline the general approach for any two-phase complex fluids, and then present, as an example, a detailed formulation for an emulsion of nematic drops in a Newtonian matrix. Using spectral discretizations, we compute shear-induced deformation, head-on collision and coalescence of drops where the matrix and drop phases are Newtonian or viscoelastic Oldroyd-B fluids. Numerical results are compared with previous studies as a validation of the theoretical model and numerical code. Finally, we simulate the retraction of an extended nematic drop in a Newtonian matrix as a method for measuring interfacial tension.

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.

Viscoelastic effects on drop deformation in steady shear

This paper applies a diffuse-interface model to simulate the deformation of single drops in steady shear flows when one of the components is viscoelastic, represented by an Oldroyd-B model. In Newtonian fluids, drop deformation is dominated by the competition between interfacial tension and viscous forces due to flow. A fundamental question is how viscoelasticity in the drop or matrix phase influences drop deformation in shear. To answer this question, one has to deal with the dual complexity of nonNewtonian rheology and interfacial dynamics. Recently, we developed a diffuse-interface formulation that incorporates complex rheology and interfacial dynamics in a unified framework. Using a two-dimensional spectral implementation, our simulations show that, in agreement with observations, a viscoelastic drop deforms less than a comparable Newtonian drop. When the matrix is viscoelastic, however, the drop deformation is suppressed when the Deborah number De is small, but increases with De for larger De. This non-monotonic dependence on matrix viscoelasticity resolves an apparent contradiction in previous experiments. By analysing the flow and stress fields near the interface, we trace the effects to the normal stress in the viscoelastic phase and its modification of the flow field. These results, along with prior experimental observations, form a coherent picture of viscoelastic effects on steady-state drop deformation in shear.

Flow patterns in the sedimentation of an elliptical particle

We study the dynamics of a single two-dimensional elliptical particle sedimenting in a Newtonian fluid using numerical simulations. The main emphasis in this work is to study the effect of boundaries on the flow patterns observed during sedimentation. The simulations were performed using a multi-block lattice Boltzmann method as well as a finite-element technique and the results are shown to be consistent. We have conducted a detailed study on the effects of density ratio, aspect ratio and the channel blockage ratio on the flow patterns during sedimentation. As the channel blockage ratio is varied, our results show that there are five distinct modes of sedimentation: oscillating, tumbling along the wall, vertical sedimentation, horizontal sedimentation and an inclined mode where the particle sediments with a non-trivial orientation to the vertical. The inclined mode is shown to form a smooth bridge between the vertical and horizontal modes of sedimentation. For narrow channels, the mode of sedimentation is found to be sensitively dependent on the initial orientation of the particle. We present a phase diagram showing the transitions between the various modes of sedimentation with changing blockage ratio of the channel.

An Energetic Variational Formulation with Phase Field Methods for Interfacial Dynamics of Complex Fluids: Advantages and Challenges

The use of a phase field to describe interfacial phenomena has a long and fruitful tradition. There are two key ingredients to the method: the transformation of Lagrangian description of geometric motions to Eulerian description framework, and the employment of the energetic variational procedure to derive the coupled systems. Several groups have used this theoretical framework to approximate Navier-Stokes systems for two-phase flows. Recently, we have adapted the method to simulate interfacial dynamics in blends of microstructured complex fluids. This review has two objectives. The first is to give a more or less self-contained exposition of the method. We will briefly review the literature, present the governing equations and discuss a suitable numerical schemes, such as spectral methods. The second objective is to elucidate the subtleties of the model that need to be handled properly for certain applications. These points, rarely discussed in the literature, are essential for a realistic representation of the physics and a successful numerical implementation. The advantages and limitations of the method will be illustrated by numerical examples. We hope that this review will encourage readers whose applications may potentially benefit from a similar approach to explore it further.

Wall energy relaxation in the Cahn–Hilliard model for moving contact lines

The Cahn–Hilliard model uses diffusion between fluid components to regularize the stress singularity at a moving contact line. In addition, it represents the dynamics of the near-wall layer by the relaxation of a wall energy. The first part of the paper elucidates the role of the wall relaxation in a flowing system, with two main results. First, we show that wall energy relaxation produces a dynamic contact angle that deviates from the static one, and derive an analytical formula for the deviation. Second, we demonstrate that wall relaxation competes with Cahn–Hilliard diffusion in defining the apparent contact angle, the former tending to “rotate” the interface at the contact line while the latter to “bend” it in the bulk. Thus, varying the two in coordination may compensate each other to produce the same macroscopic solution that is insensitive to the microscopic dynamics of the contact line. The second part of the paper exploits this competition to develop a computational strategy for simulating realis...

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...