Foam fractionation is an economical and environmentally friendly separation method for surface active material using a rising column of foam. The system of foam fractionation column with reflux is selected since such a system can improve the enrichment of the product collected from the top of the column. Due to the reflux, it is assumed that there is more surface active material (surfactant and/or protein) in the Plateau border than that in the foam lamella, so that the Plateau border acts as a surfactant/protein reservoir. The aim of this thesis is to investigate the adsorption and transport of surface active material such as surfactant and/or protein onto the surface of a lamella in a foam fractionation column with reflux using mathematical simulation. There are two steps involved in adsorption of surface active material onto a bubble surface within foam, which are diffusion from the bulk solution into the subsurface, a layer next to the interface, followed by adsorption of that material from the subsurface onto the interface. The diffusion follows the Fick's second law, while the adsorption may follow the Henry, Langmuir or Frumkin isotherms, depending on the properties of the surface active material. The adsorption of mixed protein-surfactant follows the Frumkin isotherm. When there is a competition between protein and surfactant, the protein arrives onto the interface at a later time due to a slower diffusion rate and it displaces the surfactant molecules already on the surface since protein has a higher affinity for that surface than surfactant. The surfactant transport from a Plateau border onto a foam lamella is determined by the interaction of forces applied on the lamella surface, such as film drainage, due to the pressure gradient between the lamella and the Plateau border, the Marangoni effect, due to the gradient of surface tension, and surface viscosity, as a reaction to surface motion. In this thesis, there are two different models of film drainage. One approach uses assumption of a film with a mobile interface and the other model assumes a film with a rigid interface. In the absence of surface viscosity, the Marangoni effect dominates the film drainage resulting in accumulation of surfactant on the surface of the foam lamella in the case of a lamella with a rigid interface. In the case of a film with a mobile interface, the film drainage dominates the Marangoni effect and surfactant is washed away from the surface of the lamella. When the drainage is very fast, such as that which is achieved by a film with a mobile interface, the film could be predicted to attain the thickness of a common black film, well within the residence time in a foam fractionation column, at which point the film stops draining and surfactant starts to accumulate on the lamella surface. The desirable condition in operation of a foam fractionation column however is when the Marangoni effect dominates the film drainage and surfactant accumulates on the surface of a foam lamella such as the one achieved by a film with a rigid interface. In the presence of surface viscosity and the absence of film drainage, the surface viscous forces oppose the Marangoni effect and reduce the amount of surfactant transport onto the foam lamella. A larger surface viscosity results in less surfactant transport onto the foam lamella. In addition, the characteristic time scale required for surfactant transport is shorter with a shorter film length.

De nombreux systèmes fluides dans les domaines de la biologie ou encore de la cosmétique sont limités par une interface dont les propriétés mécaniques régissent la stabilité. En particulier, les objets tels que des gouttes, vésicules ou polymersomes se déforment dans un écoulement simple et mènent à une grande richesse de dynamiques spatio-temporelles contrôlées par la nature des matériaux qui composent l'interface. Les travaux présentés concernent l'étude numérique de la déformation de ces objets dans un écoulement de Stokes, en particulier dans des situations où les viscosités de l'interface jouent un rôle important. Un code de calcul couplant intégrales de frontières et éléments finis a été utilisé afin de décrire la physique interfaciale et étudier leur comportement une fois plongés dans un écoulement. Ces travaux ont permis d'étudier l'influence des viscosités interfaciales sur la dynamique d'une goutte dans un écoulement extensionnel plan, leur influence sur sa dynamique de déformation et sur les conditions de rupture de celle-ci. Les études réalisées sur une vésicule fortement dégonflée et plongée dans un écoulement cisaillé ont caractérisé la bifurcation entre les deux familles de forme existantes dans ces conditions. Ces formes ayant une influence sur la dynamique de la vésicule dans l'écoulement, celle-ci a été étudiée dans le cadre d'un écoulement infini puis proche d'une paroi parallèle à l'écoulement. Enfin, de premiers résultats sur la dynamique d'un polymersome dans un écoulement cisaillé permettent de construire un diagramme de phase illustrant les différents comportement de cet objet en fonction de la viscosité de la membrane et du taux de cisaillement / There are many fluid systems in the biology, food industry, pharmacology or cosmestics fields that are bound by an interface which mechanical properties rule the system stability. Objects like droplets, vesicles or polymersomes change their shape in a simple flow which lead to a wealth of space and time dynamics. These properties are controlled by the nature of the interface material. The aim of this work is the numerical study of the deformation of droplets, vesicles and polymersomes in a Stokes flow, especially when the interfacial viscosities play an important role. A numerical computation code coupling boundary integrals and finite elements was used to describe the interfacial physics of these objects and study their behaviour when immerged in a flow. Multiple resolution strategies where developped to this end in order to optimize the numerical computation in the cas of an interface with viscosities.Using this work, the influence of interfacial viscosities on the dynamics of a droplet in an extensional flow is studied : in particular, their influence on the stretching dynamics of a droplet and its break up conditions was characterized. The study of a vesicle, droplet bounded by a lipid bilayer, strongly deflated and immerged in a shear flow detailed the bifurcation between two shape types existing for this system. These shapes have an influence on the vesicle dynamics under flow, which is studied for an unbounded flow and a near-wall flow. Finally, we show first results about the dynamics of a polymersome in a shear flow. We used them to build a phase diagram for the behaviour of this object depending on the membrane viscosity and the shear rate

<p>Droplet dispersions with adsorbed exotic surface active species (proteins, fatty alcohol, fatty acids, solid particulates, lipids, or polymers) find an immense number of applications in the field of engineering and bioscience. Interfacial rheology plays an essential role in the dynamics of many of these systems, yet little is understood about how these effects alter droplet dynamics. Most surfactants studied historically have been simple enough that the droplet dynamics can be described by Marangoni effects (surfactant concentration gradients), surface dilution, and adsorption/desorption kinetics without including the intrinsic surface rheology. One of the challenges in examining droplet systems with complex interfaces is that the intrinsic rheological effects are strongly coupled with surfactant transport effects (surface convection, diffusion, dilution and adsorption/desorption). The surface rheology can impact the ability of surfactant to transport along the surface, while surfactant transport can alter the surface rheology by changing the surface concentration. In this work, we develop axisymmetric boundary-integral simulations that allow us to quantitatively explore the combined effect of intrinsic surface rheology and surfactant transport on droplet dynamics in the Stokes flow limit. We assume that the droplet interface is predominantly viscous and that the Boussinesq Scriven constitutive relationship describes the properties of the viscous membrane. The key questions that we address in this work are:</p> <p><br></p> <ul> <li>How do viscous membranes impact droplet deformation, breakup and relaxation?      </li> </ul> <p>     When a droplet is placed under external flow, it can either attain a stable shape under flow or stretch indefinitely above a critical flow rate and break apart. In this topic, we first discuss the breakup conditions for a droplet suspended in an unbounded immiscible fluid under a general linear flow field using perturbation theories for surface viscosity in the limit of small droplet deformation. We neglect the inhomogeneity in surfactant concentration and surface tension for this part. We find that the surface shear/dilational viscosity increases/decreases the critical capillary number for droplet breakup compared to a clean droplet at the same capillary number and droplet viscosity ratio value. In the second part of this topic, we solve the problem using boundary integral simulations for the case of axisymmetric extensional flow. Numerically solving this problem allows us to examine the effect of Marangoni stresses, pressure thickening/thinning surface viscosities, and stronger flows. We compare the droplet breakup results from our simulations to results from second-order perturbation theories. We present the physical mechanism behind our observations using traction arguments from interfacial viscosities. We conclude this topic by examining the combined role of surface viscosity and surfactant transport on the relaxation of an initially extended droplet in a quiescent external fluid.</p> <p><br></p> <ul> <li>How do viscous membranes alter droplet sedimentation?</li> </ul> <p>      When an initially deformed droplet sediment under gravity, it can either revert to a spherical shape or undergo instability where the droplet develops a long tail or cavity at its rear end. Here, we use numerical simulations to discuss how interfacial viscosity alters the breakup criterion and the formation of threads/cavities under gravity. We examine the combined influence of intrinsic surface viscosity and surfactant transport on droplet stability by assuming a linear dependence of surface tension on surfactant concentration and an exponential dependence of interfacial viscosities on surface pressure. We find that surface shear viscosity inhibits the tail/cavity growth at the droplet’s rear end and increases the critical capillary number compared to a clean droplet. In contrast, surface dilational viscosity promotes tail/cavity growth and lowers the critical capillary number compared to a clean droplet.</p> <p><br></p> <ul> <li>How do viscous membranes affect droplet coalescence?</li> </ul> <p>      When two droplets approach under external flow, a thin film is formed between the two droplets. Here, we develop numerical simulations to model the full coalescence process from the collision of two droplets under uniaxial compressional flow to the point where the film approaches rupture. We investigate the role of interfacial viscosity on the film profiles and drainage time. We observe that both surface shear and dilational viscosity significantly delay the film drainage time relative to a clean droplet. Interestingly, we find that the film drainage behaviour of a droplet with surface viscosity is not altered by the relative ratio of shear to dilational viscosity but rather depends on the sum of shear and dilational Boussinesq numbers. This is in contrast to the effect of surface viscosity observed in the previous processes (droplet breakup and sedimentation), where surface shear viscosity increases the critical capillary number compared to a clean droplet, while surface dilatational viscosity has the opposite effect.</p>

[pt] Diversos processos apresentam escoamentos com superfícies livres. Alguns desses processos vão além dos problemas de engenharia, incluindo questões cotidianas como gotas de chuva caindo do céu, água fluindo pelo rio ou através de uma torneira. Na indústria, o processo de extrusão e revestimento são dois exemplos de processos que são fortemente afetados pelo comportamento da interface. O modelo de interface livre mais comumente utilizado foi desenvolvido no século XIX e descreve como isotrópico o comportamento das interfaces e dependente de um único parâmetro, denominado tensão interfacial. Desde então, os avanços na área de reologia interfacial vêm mostrando que os fenômenos interfaciais são mais complexos e precisam de mais informações para serem modelados. Nesta linha de pensamento, este trabalho analisa o efeito da viscosidade interfacial na dinâmica do processo de extrusão e revestimento por slot, no qual o conjunto de equações diferenciais que governam o problema é resolvido pelo método dos elementos finitos. / [en] Several processes present free surfaces flows. Some of those processes go beyond engineering problems, including everyday issues like raindrops falling from the sky, water flowing down the river or through a faucet. In industry, extrusion and coating process are two examples of processes that are strongly affected by the behavior of the interface. The most commonly used free interface model was developed in the 19th century and describes as isotropic the behavior of interfaces and dependent on asingle parameter called interfacial tension. Since then, advances in the areaof interfacial rheology have been showing that the interfacial phenomena are more complex and accurate of more information to be modeled. In this line of thinking, this work analyzes the effect of interface viscosity on the dynamics of extrusion and slot coating process, in which the set of differential equations that governs the problem is solved by Finite Element method.

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