James J. Feng

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.

Direct Simulation of Initial Value Problems for the Motion of Solid Bodies in a Newtonian Fluid Part 1. Sedimentation

This paper reports the result of direct simulations of fluid–particle motions in two dimensions. We solve the initial value problem for the sedimentation of circular and elliptical particles in a vertical channel. The fluid motion is computed from the Navier–Stokes equations for moderate Reynolds numbers in the hundreds. The particles are moved according to the equations of motion of a rigid body under the action of gravity and hydrodynamic forces arising from the motion of the fluid. The solutions are as exact as our finite-element calculations will allow. As the Reynolds number is increased to 600, a circular particle can be said to experience five different regimes of motion: steady motion with and without overshoot and weak, strong and irregular oscillations. An elliptic particle always turn its long axis perpendicular to the fall, and drifts to the centreline of the channel during sedimentation. Steady drift, damped oscillation and periodic oscillation of the particle are observed for different ranges of the Reynolds number. For two particles which interact while settling, a steady staggered structure, a periodic wake-action regime and an active drafting–kissing–tumbling scenario are realized at increasing Reynolds numbers. The non-linear effects of particle–fluid, particle–wall and interparticle interactions are analysed, and the mechanisms controlling the simulated flows are shown to be lubrication, turning couples on long bodies, steady and unsteady wakes and wake interactions. The results are compared to experimental and theoretical results previously published.

Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 2. Couette and Poiseuille flows

This paper reports the results of a two-dimensional finite element simulation of the motion of a circular particle in a Couette and a Poiseuille flow. The size of the particle and the Reynolds number are large enough to include fully nonlinear inertial effects and wall effects. Both neutrally buoyant and non-neutrally buoyant particles are studied, and the results are compared with pertinent experimental data and perturbation theories. A neutrally buoyant particle is shown to migrate to the centreline in a Couette flow, and exhibits the Segre-Silberberg effect in a Poiseuille flow. Non-neutrally buoyant particles have more complicated patterns of migration, depending upon the density difference between the fluid and the particle. The driving forces of the migration have been identified as a wall repulsion due to lubrication, an inertial lift related to shear slip, a lift due to particle rotation and, in the case of Poiseuille flow, a lift caused by the velocity profile curvature. These forces are analysed by examining the distributions of pressure and shear stress on the particle. The stagnation pressure on the particle surface are particularly important in determining the direction of migration.

The stretching of an electrified non-Newtonian jet: A model for electrospinning

Electrospinning uses an external electrostatic field to accelerate and stretch a charged polymer jet, and may produce ultrafine “nanofibers.” Many polymers have been successfully electrospun in the laboratory. Recently Hohman et al. [Phys. Fluids, 13, 2201 (2001)] proposed an electrohydrodynamic model for electrospinning Newtonian jets. A problem arises, however, with the boundary condition at the nozzle. Unless the initial surface charge density is zero or very small, the jet bulges out upon exiting the nozzle in a “ballooning instability,” which never occurs in reality. In this paper, we will first describe a slightly different Newtonian model that avoids the instability. Well-behaved solutions are produced that are insensitive to the initial charge density, except inside a tiny “boundary layer” at the nozzle. Then a non-Newtonian viscosity function is introduced into the model and the effects of extension thinning and thickening are explored. Results show two distinct regimes of stretching. For a “mild...

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.

Stretching of a straight electrically charged viscoelastic jet

A charged polymer jet may be accelerated and stretched by an external electric field, and this process is relevant to electrospinning for making nanofibers. The stretching of an electrified jet is governed by the interplay among electrostatics, fluid mechanics and rheology, and the role of viscoelasticity has not been systematically explored before. This paper presents a slender-body theory for the stretching of a straight charged jet of Giesekus fluid. Results show strain-hardening as the most influential rheological property. It causes the tensile force to rise at the start, which enhances stretching of the jet. Further downstream, however, the higher elongational viscosity tends to suppress jet stretching. In the end, strain-hardening leads to thicker fibers. This confirms the main result of a previous study using empirical rheological models. The behavior of the electrically driven jet forms an interesting contrast to that in conventional fiber spinning.

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.

Lubrication theory in highly compressible porous media: the mechanics of skiing, from red cells to humans

A generalized lubrication theory that is applicable to highly deformable porous layers is developed using an effective-medium approach (Brinkman equation). This theory is valid in the limit where the structure is so compressible that the normal forces generated by elastic compression of the fibres comprising the solid phase are negligible compared to the pressure forces generated within the porous layer. We assume that the deformation of the solid phase is primarily due to boundary compression as opposed to the motion of the fluid phase. A generalized Reynolds equation is derived in which the spatial variation of the Darcy permeability parameter, α = H /√ K p , due to the matrix compression is determined by new local hydrodynamic solutions for the flow through a simplified periodic fibre model for the deformed matrix. Here H is the undeformed layer thickness and K p the Darcy permeability. This simplified model assumes that the fibres compress linearly with the deformed gap height in the vertical direction, but the fibre spacing in the horizontal plane remains unchanged. The model is thus able to capture the essential nonlinearity that results from large-amplitude deformations of the matrix layer. The new theory shows that there is an unexpected striking similarity between the gliding motion of a red cell moving over the endothelial glycocalyx that lines our microvessels and a human skier or snowboarder skiing on compressed powder. In both cases one observes an order-of-magnitude compression of the matrix layer when the motion is arrested and predicts values of α that are of order 100. In this large-α limit one finds that the pressure and lift forces generated within the compressed matrix are four orders-of-magnitude greater than classical lubrication theory. In the case of the red cell these repulsive forces may explain why red cells do not experience constant adhesive molecular interactions with the endothelial plasmalemma, whereas in the case of the skier or snowboarder the theory explains why a 70 kg human can glide through compressed powder without sinking to the base as would occur if the motion is arrested. The principal difference between the tightly fitting red cell and the snowboarder is the lateral leakage of the excess pressure at the edges of the snowboard which greatly diminishes the lift force. A simplified axisymmetric model is presented for the red cell to explain the striking pop out phenomenon in which a red cell that starts from rest will quickly lift off the surface and then glide near the edge of the glycocalyx and also for the unexpectedly large apparent viscosity measured by Pries et al . (1994) in vivo .

Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method

We study the retraction and pinch-off of a liquid filament and the formation of drops by using an energetic variational phase field model, which describes the motion of mixtures of two incompressible fluids. An efficient and accurate numerical scheme is presented and implemented for the coupled nonlinear systems of Navier-Stokes type linear momentum equations and volume preserving Allen-Cahn type phase equations. Detailed numerical simulations for a Newtonian fluid filament falling into another ambient Newtonian fluid are carried out. The dynamical scaling behavior and the pinch-off behavior, as well as the formation of the consequent satellite droplets are investigated.