Identification of deformable droplets from boundary measurements: the case of non-stationary Stokes problem

In this paper we derive the asymptotic expansion of the velocity field of a small deformable droplet immersed in an incompressible Newtonian fluid. Using an appropriate physical scaling of the surface tension with respect to the droplet volume, we show that the first order of the asymptotic can be expressed in terms of the velocity field in the absence of the droplet and a new kind of moment tensor, called the curvature moment tensor. Our asymptotic formula extends those already derived for rigid droplets and aimed to obtain simplified macroscale properties of a dilute suspension composed of identical droplets dispersed in an incompressible Newtonian fluid from knowledge of its microscopic properties. We finally determine explicitly the curvature moment tensor for ellipses and ellipsoids.

The extensional deformation of a viscoelastic droplet suspended in a viscoelastic medium was investigated, both theoretically and experimentally. A theoretical analysis was carried out on the deformation of a droplet suspended in a steady extensional flow field, where both fluids may be represented by the second‐order fluid model. The study took into account the effects of the elasticity, viscosity, and interfacial tension of the fluids concerned. The shape of a deformed droplet is determined by solving the equations of motion for both fluids (droplet phase and medium). A perturbation technique was employed and the iteration method was used to determine the shape of the droplet undergoing extensional deformation. Series solutions of stream function inside and outside the droplet, pressure distribution around the droplet, and the deformation of the droplet were obtained. For the experimental study, a transparent flow channel consisting of a conical section and a straight cylindrical tube was constructed. Along its central axis, the conical section provides extensional flow. Droplets of known volume were injected in the conical section at the centerline of the flow channel, through which a viscoelastic fluid was flowing at a constant flow rate. The deformation patterns of droplets were recorded on both movie and still films as they were traveling along the centerline of the conical section. Tracer particles were used to determine the axial velocity profiles. The suspending liquids were aqueous solutions of polyacrylamide at various concentrations (viscoelastic fluids) and corn syrup (Newtonian fluid). For viscoelastic droplets, polyisobutylene dissolved in decalin at various concentrations was used, and for Newtonian droplets, Indopols and benzene were used. It was observed that the droplets, initially spherical, were slightly deformed in the upper section of the cone and greatly elongated at the entrance region of the cylindrical tube. The viscoelastic droplets were less deformable than the Newtonian droplets, and highly viscoelastic media gave rise to large deformations of droplets. The deformability of droplets was analyzed based on the flow conditions and the rheological properties of the fluids concerned. A comparison is made between the theoretically predicted and experimentally observed shapes of droplets in the conical section, only where extension rate is constant.

This study typically emphasizes analyzing the geometrical singularities of weak solutions of the mixed boundary value problem for the stationary Stokes and Navier-Stokes system in two-dimensional non-smooth domains with corner points and points at which the type of boundary conditions change. The existence of these points on the boundary generally generates local singularities in the solution. We will see the impact of the geometrical singularities of the boundary or the mixed boundary conditions on the qualitative properties of the solution including its regularity. The solvability of the underlying boundary value problem is analyzed in weighted Sobolev spaces and the regularity theorems are formulated in the context of these spaces. To compute the singular terms for various boundary conditions, the generalized form of the boundary eigenvalue problem for the stationary Stokes system is derived. The emerging eigenvalues and eigenfunctions produce singular terms, which permits us to evaluate the optimal regularity of the corresponding weak solution of the Stokes system. Additionally, the obtained results for the Stokes system are further extended for the non-linear Navier-Stokes system.

We consider the Cahn–Hilliard/Navier–Stokes system with non–degenerate mobility in the space–periodic case, describing the flow of two viscous immiscible and incompressible Newtonian fluids with matched densities. We identify sufficient conditions on the velocity field for weak solutions to satisfy an energy identity, improving previous results on the literature.

In this paper, we provide a rigorous derivation of an asymptotic formula for the perturbation of eigenvalues associated to the Stokes eigenvalue problem with Dirichlet conditions and in the presence of small deformable inclusions. Taking advantage of the small sizes of the inclusions immersed in an incompressible Newtonian fluid having kinematic viscosity different from the background one, we show that our asymptotic formula can be expressed in terms of the eigenvalue in the absence of the inclusions and in terms of the viscous moment tensor (VMT).

The present paper is dedicated to the investigation of the 3D dynamics of two viscous immiscible liquids in an unbounded domain at low Reynolds numbers. The numerical technique is based on the boundary element method. To accelerate the calculations and increase the problem scale the parallelization of computations on graphic processors (GPU) using CUDA technology is used. The inclination angle and the deformation of droplets in a shear flow at various parameters are studied. The obtained results are compared with the experimental data represented in the literature, numerical results and the small deformations theory. The calculation of rheological characteristics for dilute emulsions in shear flow is carried out for different viscosity ratios of internal and external liquids.

Many geological flows involve turbulence, wherein the velocity field involves complex, fluctuating motions superimposed on a mean motion. Flows in natural river channels are virtually always turbulent. Magma flow in dikes and sills, and lava flows, can be turbulent. Atmospheric flows involving eolian transport are turbulent. The complex, convective overturning of fluid in a magma chamber or geyser is a form of turbulence. Thus, a description of the basic qualities of these complex flows is essential for understanding many geological flow phenomena. Turbulent flows generally are associated with large Reynolds numbers. Recall from Chapter 5 that the Reynolds number Re is a measure of the ratio of inertial to viscous forces acting on a fluid element, . . . Re = ρUL/μ . . . . . . (14.1) . . . where the characteristic velocity U and length L are defined in terms of the particular flow system. Thus, turbulence is typically associated, for given fluid density ρ and viscosity μ, with high-speed flows (although we must be careful in applying this generality to thermally driven convective motions; see Chapter 16). A simple, visual illustration of this occurs when smoke rises from a cigar within otherwise calm, surrounding air. The smoke acts as a flow tracer. Smoke molecules at the cigar tip start from rest, since they are initially attached to the cigar. Upward fluid motion, as traced by the smoke, initially is of low speed, and viscous forces have a relatively important influence on its behavior. The flow is laminar; smoke streaklines are smooth and locally parallel. But as the flow accelerates upward, it typically reaches a point where viscous forces are no longer sufficient to damp out destabilizing effects of growing inertial forces, and the flow becomes turbulent, manifest as whirling, swirling fluid motions (see Tolkien [1937]). Throughout this chapter we will consider only incompressible Newtonian fluids. Unfortunately, the complexity of turbulent fluid motions precludes directly using the Navier–Stokes equations to describe them. Instead, we will adopt a procedure whereby the Navier–Stokes equations are recast in terms of temporally averaged or spatially averaged values of velocity and pressure, and fluctuations about these averages.

Water accumulation in fuel cell anode and cathode channels can lead to hydrogen starvation and therefore, severe electrode degradation, and significantly fuel cell performance deterioration. For this reason, liquid water transport in micro-channels remains an active area of research in fuel cells as well as many other research areas [1, 2]. Two-phase flow in micro-channels is governed by surface tension and viscous effects and involves the interaction of air, water and the solid substrate. To date most numerical studies have been performed using volume of fluid (VOF) [3] or level set (LS) [4] implementations in commercial software, however these methods are usually explicit thereby limiting the maximum time step that can be used. Investigations on new methods to solve two-phase flows in micro-channels is key to develop alternative methods that allow for faster simulation time and allow to study physical process that remain a challenge, such as interface conditions between the channel and porous media in fuel cells [5]. In this work, a novel formulation based on a Lagrangian-Eulerian formulation is presented and experimentally validated [6]. The governing equations for both air and water are the Navier-Stokes equations. Air is represented using a fixed mesh, whereas a moving mesh is used to discretize the water domain. This formulation is particularly advantageous to the problem at hand, since it allows for exact tracking the air-water interface. An implicit term is used to represent the surface tension effects, allowing us to use time steps greater than those from explicit formulations [7]. For the validation of the model, several experiments have been performed in a transparent microchannel. Droplet deformation and shedding on three substrates, i.e, Kapton, PTFE and a gas diffusion layer (Toray H60 10%PTFE), is studied both numerically and experimentally. The goal of the experiments is to reproduce different conditions for injected water in a microchannel. Kapton and PTFE are smooth surfaces, the former being hydrophilic and the latter hydrophobic, whereas the GDL is a hydrophobic rough substrate and is used to emulate the conditions in a fuel cell channel. Two cameras are used to capture the emergence of water into the channel. The first camera is used to obtain images of the droplet’s deformation from a lateral point of view, allowing us to measure the advancing and receding contact angles. The second camera obtains images along the channel, and therefore deformation effects of the droplet on the direction perpendicular to the airflow can also be quantified. The presented model can predict droplet emergence, deformation and posterior detachment. Numerical results are consistent with the experimental data. For instance, the advancing contact angle remains approximately constant in rough surfaces, whereas the receding contact angle decreases, showing a slight increase prior to droplet detachment. Results obtained with the current model are also compared to VOF results previously reported in literature and large discrepancies with the evolution of droplet deformation are observed. The model is shown to be able to predict the conditions that lead to droplet, slug and film flow in fuel cell channels.

Numerical study of deformation and breakup of a multi-core compound droplet in simple shear flow

The problem of stabilizing a non-stationary time-delay control system is solved. An algorithm is formulated for constructing controls of a linear non-stationary system of differential equations with variable delays. The method is illustrated by the construction of a stabilizing control for linear non-stationary second-order delay differential systems.

A study is made of a stationary Stokes's system in a periodically perforated domain with boundary conditions of mixed type, which describes the motion of a viscous incompressible fluid in a porous medium in the presence of friction between the fluid and the walls of the pores. The relation between the leading terms of the asymptotic expansions with respect to for the fluid velocity and the pressure is obtained, where is the parameter characterizing the fineness of the porous structure.

In the past several years, we have extensively studied the problems of dendrite growth with no convection, from a pure melt, as well as from a binary mixture. An interfacial wave theory has been established for selecting the tip-velocity and determining the formation of micro-structure at the later stage of growth. The theoretical predictions agree with the available experimental data very well (see Figure 1). In the present work, we turn to investigate the effect of convection in melt. Assume that a single dendrite growing into an undercooled pure melt in the negative z-axis direction with a constant average velocity U. At the far field, a uniform external flow against the dendrite with the velocity (U ∞)D may be applied. We assume that the mass density of the liquid phase is ρ, while the mass density of the solid phase is ρs. Due to the external flow and/or the change of density in solidification, a convective motion in melt is produced. The fluid motion will affect the heat transport process and change the temperature distribution. We consider the melt as an incompressible Newtonian fluid. Then system involves the following parameters: $$ {T_\infty };\,\,\varepsilon \, = \,{{\sqrt \Gamma } \over {\eta _0^2}};\,\,\alpha \, = \,{{\rho s - \rho } \over \rho };\,{U_\infty };\,\,Pr = {\nu \over {{\kappa _T}}}{\rm{.}} $$ (1.1)

The n -observability problem is considered for linear dynamical systems. The discussion of the problem is divided into two sections: the first covers linear nonstationary continuous time systems and the second covers discrete time systems. A system is called n -observable at time t' , if the state can be reconstructed from n samples of the output history and from the input over this time interval. The first result gives a necessary and sufficient condition for a linear nonstationary continuous system to be n -observable at time t' . When this condition is satisfied, a formula is given by which the state at t' can be calculated from a knowledge of the system's output at n points in its history and from the system's input over this period of history. Although the theorem does provide a test for n -observability at t' , this test is not easily applied. To remedy this, further results relate this test to a more easily applied test for observability. These theorems connect the necessary and sufficient condition for n - observability with Kalman's necessary and sufficient condition for observability. The corresponding results for linear stationary continuous systems are given. For discrete time systems, the notation is purposely kept the same as for continuous time systems. In this way, the similarity of the relations of discrete to continuous systems is more obvious. The n -observability problem for discrete nonstationary systems is more obvious. The n -observability problem for discrete non-stationary systems is discussed, and a necessary and sufficient condition for n -observability is derived. Again, the corresponding results for stationary systems are given. Finally, the use of state reconstruction in the synthesis of feedback controllers is discussed. This discussion is limited to linear stationary continuous time system. An example is included to clarify the details.

The Brownian motion of a nanoparticle in an incompressible Newtonian fluid (quiescent or fully developed Poiseuille flow) has been investigated with an arbitrary Lagrangian-Eulerian based finite element method. Results for the motion in a compressible fluid medium are estimated. Thermal fluctuations from the fluid are implemented using a fluctuating hydrodynamics approach. The instantaneous flow around the particle and the particle motion are fully resolved. Carriers of two different sizes with three different densities have been investigated (nearly neutrally buoyant). The numerical results show that (a) the calculated temperature of the nearly neutrally buoyant Brownian particle in a quiescent fluid satisfies the equipartition theorem; (b) the translational and rotational decay of the velocity autocorrelation functions result in algebraic tails, over long time; (c) the translational and rotational mean square displacements of the particle obeys Stokes-Einstein and Stokes-Einstein-Debye relations, respectively. Larger the particle, longer the time taken to attain this limit; and (d) the parallel and perpendicular diffusivities of the particle closer to the wall are consistent with the analytical results, where available.

This paper analyzes a free boundary problem that was proposed in [F. F. Weller, J. Math. Biol. , 61 (2010), pp. 805--818] as a model for the growth of platelet aggregates (thrombi) in the course of primary hemostasis. The fluid motion is described by the Navier--Stokes system, and the velocity of the free boundary separating fluid and thrombus is related to the gradient of the platelet density. Since thrombus growth disturbs the flow and thus alters the transport of platelets to the thrombus surface, the fluid dynamic equations, the species equations, and the growth processes are fully coupled. For this coupled system, the existence of a smooth solution locally in time is proven.