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In this thesis we describe the development of a boundary integral method for the simulation of non-Newtonian drops and vesicles subjected to a viscous flow. A vesicle consists of a viscous drop encapsulated by a lipid bilayer and is modelled as a two-layer drop, of which the outer layer is viscoelastic. As the typical Reynolds number of the flow pr...
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Imaging the anelastic deformation within the crust and lithosphere using surface geophysical data remains a significant challenge in part due to the wide range of physical processes operating at different depths and to various levels of localization that they embody. Models of Earth's elastic properties from seismological imaging combined with geod...
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... Boundary integral methods have been successfully used recently for simulations of complex multiphase flows: drop deformation and breakup; 1-5 drop-to-drop interaction; [6][7][8] suspension of liquid drops in viscous flow; 9-11 deformation of a liquid drop adhering to a solid surface and free-surface viscoelastic flows. [12][13][14] The main disadvantage of the boundary integral method is the singularity of the free-space Green's kernels. 15 In most of the studies special attention is paid to the accurate calculation of the boundary integrals around the singular point. ...
A boundary integral method for D simulation of multiphase Stokes flows. The method is based on a non-singular contour-integral representation of the single and double layer potentials, which improves the stability and accuracy. To demonstrate the advantages of the presented method the following problems are considered: Drop formation and breakup; Drop-to-drop interaction and related film drainage; Foam-drop formation and its dynamic in simple shear flow; Dynamic contact line problems, where both types of three-phase contact lines (liquid-fluid-liquid and liquid-fluid-solid) are considered.
... Even so, relatively few prior numerical works have been applied to study the dynamics of viscoelastic drops. The boundary element method (BEM) was used to study the transient deformation of polymer drops in a Newtonian matrix (suspending fluid) in uniaxial extensional flow by Toose and co-workers [12][13][14]. A special finite difference method by Ramaswamy and Leal was used to obtain steady-state solutions for viscoelastic drops in a Newtonian matrix [15] and for Newtonian drops in a viscoelastic matrix [16]. ...
... This suggests the upwinding needed with the less refined mesh can be avoided altogether by additional mesh refinement and that the amount of dissipation or smoothing associated with upwinding is kept to a tolerable level. Our computational code has also been validated more extensively for drop extension by comparison with the analytical solutions of Delaby et al. [7][8][9] and the boundary element code of Toose and co-workers [12][13][14]; see [19] for extensive details. ...
We present results from modeling the deformation of a viscoelastic drop suspended in another viscoelastic fluid subjected to uniaxial extensional flow using the DEVSSG-FEM. Viscoelasticity is implemented using the Oldroyd-B constitutive relation for both the drop and surrounding matrix fluids. To allow efficient solution of the discretized problem, we employ an implicit temporal integration scheme with an accelerated quasi-Newton method. Important viscoelastic effects for both drop deformation during extensional flow and drop retraction following cessation of flow are elucidated. Viscoelastic drops in a Newtonian matrix lengthen less at steady state extension than Newtonian drops because of the accommodation of stress by elasticity. However, the stored elastic effects cause rapid tip retraction during the recovery of polymeric drops. Drops stretched in a viscoelastic exterior flow are enhanced in length compared to those in a Newtonian matrix because of first normal stresses from the matrix. During recovery, drops in a viscoelastic matrix can exhibit significant lengthening upon cessation of extensional flows, causing additional strain before retraction. This behavior is strongly dependent on the details of the exterior flow.
... We choose the time step in such a way that the maximum movement of a grid point as a result of the time integration is smaller than the distance between two neighboring grid points. We use the formula given by Toose in [92] ∆t ≤ min i ...
In a wet-chemical etching process, the resulting etched shape is smaller than
the originally designed shape at the mask. This is caused by the fact that, as soon as material next to the mask is dissolved, material under the mask will be dissolved too. This is the so-called undercut effect. During an etching process, eddies arise inside the etch-hole cavity, which decrease the etch-speed. These eddies may lead to an increase of the undercut. In order to improve the production process, it is necessary to have insight into these phenomena. Therefore, in this thesis the simulation of convection-driven wet-chemical etching is studied. It is known that the boundary of an etch hole moves slowly during a wet-chemical etching process. Therefore, the velocity and concentration fields at each time step can be computed as if the boundary were stationary. Since a typical etch hole is small, the corresponding Reynolds number based on the associated length-scale is sufficiently small to allow the use of the Stokes equations in the description of the flow. If we assume that
the fluid has a uniform density in the etch hole, the flow problem can be solved independently of the concentration problem. In a convection-driven etching process the fluid flow and the eddy structure inside an etch-hole geometry play an important role. Therefore a large part of the thesis is devoted to the flow problem.
An ellipsoidal model was constructed to describe the morphological evolution and Theological properties of a mixture of two immiscible viscoelastic components. The phenomenological parameters in the model were determined by comparing the interfacial velocity gradient with the viscoelastic theory for small deformation. The model was then applied to various model blend systems to predict the steady deformation, transient deformation, and relaxation after cessation of step shear flow. The model predictions were also compared with some available numerical simulations. The predictions on droplet deformation both in steady and transient regimes as well as the relaxation process were found to be in good agreement with the experimental results in literatures. The rheological properties predicted by the present model were also found to agree with the experimental results.
Synopsis The deformation of a viscoelastic drop suspended in a homogeneous viscoelastic matrix is investigated. The drop and the matrix fluids are assumed to obey linear viscoelastic constitutive equations. Small-deformation analysis is carried out on the basis of the general solution for creeping flow around a spherical drop. The corresponding stress is calculated by extending Batchelor's approach to viscoelastic media for both small and large deformations. Good agreement was found between the model predictions and experimental results available in the literature. © 2004 The Society of Rheology.
A modified boundary element method (BEM) and the DEVSS-G finite element method (FEM) are applied to model the deformation of a polymeric drop suspended in another fluid subjected to start-up uniaxial extensional flow. The effects of viscoelasticity, via the Oldroyd-B differential model, are considered for the drop phase using both FEM and BEM and for both the drop and matrix phases using FEM. Where possible, results are compared with the linear deformation theory. Consistent predictions are obtained among the BEM, FEM, and linear theory for purely Newtonian systems and between FEM and linear theory for fully viscoelastic systems. FEM and BEM predictions for viscoelastic drops in a Newtonian matrix agree very well at short times but differ at longer times, with worst agreement occurring as critical flow strength is approached. This suggests that the dominant computational advantages held by the BEM over the FEM for this and similar problems may diminish or even disappear when the issue of accuracy is appropriately considered. Fully viscoelastic problems, which are only feasible using the FEM formulation, shed new insight on the role of viscoelasticity of the matrix fluid in drop deformation. Copyright © 2001 John Wiley & Sons, Ltd.
Dilute polymer blends and immiscible liquid emulsions are characterized by a globular morphology. The dynamics of a single
drop subjected to an imposed flow field has been considered to be a valuable model system to get information on dilute blends.
This problem has been studied either theoretically by developing exact theories for small drop deformations or by developing
simplified models often based on phenomenological assumptions. In this paper, a critical overview of the available models
for the dynamics of a single drop is presented, discussing four different systems, namely the Newtonian system, where a single
Newtonian drop is immersed in an infinite Newtonian matrix; the non-Newtonian system, where at least one of the components,
the drop fluid or the matrix one, is non-Newtonian; the confined Newtonian system, where the matrix is confined and wall effects
alter the drop dynamics; and the confined non-Newtonian system.
KeywordsSingle drop-Drop deformation-Non-Newtonian-Confined systems-Wall effects-Models
