We consider the flow of nonlinear Maxwell fluids in the unsteady quasistatic case, where the effect of inertia is neglected. We study the well-posedness of the resulting PDE initial-boundary value problem. This well-posedness depends on the unique solvability of an elliptic boundary value problem. We first present results for the 3D case, locally and globally in time, with sufficiently small initial data, and for a simple shear flow problem, locally in time with arbitrary initial data; after that we extend our results to some 3D flow problems, locally in time, with large initial data. Additionally, we present results for models of White-Metzner type in 3D flow, locally and globally in time, with sufficiently small initial data. We solve our problem using an iteration between elliptic and hyperbolic linear subproblems. The limit of the iteration provides the solution of our original problem. / Ph. D.

Viscoelastic fluids, or Non-Newtonian fluids, are those that do not have a linear algebraic relation between the velocity field and the stresses arising in the media. Such fluids exhibit properties of both solids and liquids, and therefore cannot be modeled with methods of elasticity or Newtonian fluid mechanics. The popular models of viscoelasticity differ from each other only by the differential equation that describes the constitutive law for the fluid. Also, the media can have several relaxation modes, such as fluid mixes. This means that the stresses are determined as the sum of the stresses for each individual relaxation mode, which are described by corresponding differential equations evolving independently. The question of controllability of the equations that describe the evolution of viscoelastic fluids is largely open. The presence of the non-algebraic constitutive relation makes the analysis unfeasible in general setup. The presence of several relaxation modes makes the problem even more complicated. Another issue is the necessity of controlling the stresses, since they are not determined by the momentary velocity field, thus they need to be included as the controlled states. In this work we are concentrating on the controllability of the stresses arising in the viscoelastic fluid that has its motion constrained to be of the shearing type. This restriction allows us to concentrate on the stresses only and assign the shearing rate to be the control. We consider only the Upper Convected Maxwell fluid which has several relaxation modes present. The results demonstrate that contrary to the one relaxation mode case the normal stresses cannot be driven arbitrary close to the exponentially decaying regime, unless the shearing stresses satisfy certain requirements, while the shear stresses remain exactly controllable. / Ph. D.

Contraction flow of viscoelastic fluids has been a benchmark problem in non-Newtonian fluid mechanics because it mimics flows occurring in a number of industrial applications. It is also of considerable interest to academia to gain fundamental understanding of factors that affect the evolution of vortices and a complete understanding of the dynamics for a simple polymeric fluid has not been achieved. In this two part study we investigate the effect of pre deformation of a Boger fluid in a contraction geometry and the flow of surfactants in a parallel contraction geometry. Entry flow of a polymeric fluid results in the formation of upstream vortices,the presence of recirculation zones may lead to a nonuniform residence time and hence inferior quality products. In this work we study the effect of pre-stretching dilute flexible chain polymers by placing a cylinder in front of a contraction in a microfluidic device.This deformation applied to the polymer is remembered before it completely relaxes, this memory effect changes the rheological properties during the fading period of the deformation history. Applying pre-deformation gives rise to new type of vortex evolution that is different from the standard contraction case. Semi-dilute surfactant solutions that exhibit shear thickening nature can be potentially used in enhanced oil recovery to increase the sweep efficiency. Two parallel microfluidic contractions of different cross sectional area are used to investigate the rheological effect on the mass flux of the two channels. Shear thickening micellar solutions were found to increase the mass flux through the small channel compared to a newtonian fluid. This effect was observed only for a small range of flow rates. As flow rates increased inlet instabilities were observed that evolved into a chaotic behavior upon further increase in the net flow rate.

No description available.

[pt] O processo de revestimento por rotação é caracterizado pelo uso de cilindros girantes para controlar a espessura e aplicar uma fina camada de líquido em um substrato em movimento. A não ser a baixas velocidades dos cilindros, o escoamento bi-dimensional na região de formação dos filmes sobre cada cilindro é instável e o padrão observado experimentalmente consiste em um escoamento tri-dimensional e periódico na direção transversal ao substrato. Esta instabilidade pode limitar a velocidade máxima do processo se a camada líquida depositada sobre o substrato tem que ser uniforme. Para líquidos Newtonianos, a estabilidade deste escoamento é determinada pela competição de forças viscosas e capilares: a instabilidade ocorre acima de um número de capilaridade máximo. Apesar da maioria dos líquidos utilizados em processos de revestimento serem não Newtonianos, as análises disponíveis deste escoamento se limitam a estudos de líquidos Newtonianos. O comportamento não Newtoniano do líquido pode alterar completamente a natureza do escoamento perto da superfície livre; quando pequenas quantidades de polímeros flexíveis de alto peso molecular estão presentes, a instabilidade na direção transversal ocorre a velocidades muito mais baixas, quando comparado ao caso Newtoniano. Os mecanismos responsáveis pela instabilidade a baixas velocidades ainda não são completamente compreendidos. Este escoamento viscoelástico com superfície livre é analisado neste trabalho através de duas equações constitutivas diferenciais, o modelo de Oldroyld-B e o modelo de FENE-P. As equações de conservação de massa, quantidade de movimentos acopladas com os modelos constitutivos, e as equações não-lineares de mapeamento que transformam o problema de superfície livre em um problema de valor de contorno foram resolvidas pelo método de elementos finitos DEVSS-G/SUPG. O sistema de equações algébricas não linear foi resolvido pelo método de Newton com continuação por pseudo-comprimento de arco. Os resultados mostram como o campo de tensão muda com o aumento do número de Weissenberg (elasticidade do líquido), levando a formação de uma camada limite de tensão elástica na superfície livre e tensões elásticas compressivas na direção transversal, que podem explicar o aparecimento da instabilidade a baixas velocidades. Este trabalho também apresenta a formulação de estabilidade linear para escoamentos viscoelásticos com superfícies livres. O modelo dá origem a um problema de auto-valor generalizado, que foi resolvido pelo método de GMRES (ARPACK). Os auto-valores dominantes da matriz Jacobiana indicam a estabilidade do escoamento. Esta formulação foi testada em três escoamentos distintos: escoamento em uma cavidade de tampa móvel, piscina de líquido estática e um escoamento de Couette (simples de cisalhamento). / [en] Roll coating is distinguished by the use of one or more gaps between rotating cylinders to meter and apply a liquid layer to a substrate. Except at low speed, the film splitting flow that occurs in forward roll coating is three-dimensional and results in more or less regular stripes in the machine direction. This instability can limit the speed of the process if a smooth film is required as a final product. For Newtonian liquids, the stability of the film-split flow is determined by the competition of capillary forces and viscous forces: the onset of meniscus nonuniformity is market by a critical value of the capillary number. Although most of the liquids coated industrially are polymeric solutions and dispersions, that are not Newtonian, most of previous theoretical analyses of film splitting flows dealt only with Newtonian liquids. Non-Newtonian behavior can drastically change the nature of the flow near the free surface; when minute amounts of flexible polymer are present, the onset of the three-dimensional instability occurs at much lower speeds than in the Newtonian case. the mechanisms responsible for the early onset of this flow instability is not well understood. This free surface coating flow is analyzed here with differential constitutive models, the Oldroyld-B and the FENE-P equations. The continuity, momentum equations coupled with the constitutive models, and the non-linear mapping equations that transform the free boundary problem into a fixed boundary problem are solved by Newton s method with pseudo-arc-length continuation. The results show how the stress field changes with Weisenberg number, leading to the formation of an elastic boundary layer near the free surface and compressive elastic stresses in the crss-flow direction that may explain the onset of the ribbing instability at the smaller Capillary numbers when viscoelastic liquids are used. This work also presents the formulation for linear stanility analysis of viscoelastic free surface flows. The model leads to a generalized eigenproblem that is solved here using the Arnoldi s method with the software (ARPACK). The leading eigenvalues of the Jacobian Matrix indicate the stability of the flow. The formulation is tested in three different flows: lid-driven cavity, static liquid pool and a couette flow.

Neste trabalho apresentamos um estudo assintótico e numérico de escoamentos viscoelásticos com singularidades de tensão. Estas singularidades surgem como consequência de uma mudança abrupta nas condições de contorno, como no caso do stick-slip, ou devido a presença de quinas na geometria do problema, como no escoamento da contração. Para o problema stick-slip definimos o comportamento assintótico do fluido Oldroyd-B sobre um campo de velocidade Newtoniano. Esta análise foi feita com o método assintótico das expansões, que pode ser estendida para outros tipos de fluidos. O estudo assintótico do stick-slip com o modelo Oldroyd-B revelou que as equações deste modelo não estão bem definidas para este problema, pois este fluido estende o valor singular ao longo de toda a superfície livre, gerando resultados sem sentido físico. Além disso, os resultados assintóticos dos problemas stick-slip e da contração 4:1 foram verificados numericamente através da integração das equações constitutivas ao longo de linhas de corrente. Vale destacar que, além da tradicional formulação Cartesiana do tensor (CSF), também utilizamos a formulação natural do tensor (NSF), que tem a vantagem de capturar de modo mais acurado os resultados próximos às singularidades. Além do mais, desenvolvemos um método numérico para resolver as equações de Navier-Stokes combinadas com as equações constitutivas das formulações CSF e NSF para os modelos PTT e Giesekus nos dois problemas estudados. Vale ressaltar que, não há na literatura resultados numéricos, para o caso transiente, com a formulação NSF para os modelos PTT e Giesekus. Por fim, verificamos numericamente o comportamento assintótico das tensões próximo as singularidades, bem como a configuração das camadas limites para os problemas mencionados. / In this work we present an asymptotic and numerical study of viscoelastic flows with stress singularities. These singularities arise as a consequence of an abrupt change in the boundary conditions, as in the case of the stick-slip flow, or due to the presence of corners in the geometry of the problem, as in the contraction flow. For the stick-slip problem, we define the asymptotic behavior of the Oldroyd-B fluid over a Newtonian velocity field. This analysis was done with the method of matched asymptotic expansions, which can be extended to other types of fluids. The asymptotic study of the stick-slip flow for the Oldroyd-B model revealed that the equations of this model are not well defined for this problem, because this fluid extends the singularity throughout the free surface, generating results with no physical meaning. Besides that, the asymptotic results of the stick-slip and 4:1 contraction problems were verified numerically by integrating the constitutive equations along streamlines. It is worth mentioning that we performed asymptotic and numerical studies with the natural stress formulation (NSF) in addition to the Cartesian stress formulation (CSF). The NSF can capture the numerical results in a more accurate manner near singularities. Furthermore, we developed a numerical method to solve the Navier-Stokes equations combined with the constitutive equations of the CSF and NSF formulations for the PTT and Giesekus in the two problems studied. It is worth noting that there is no numerical results, for the transient case, with the NSF formulation for the PTT and Giesekus. Finally, we verified numerically the asymptotic behavior of stresses close to the singularities, as well as the configuration of the boundary layers for the problems mentioned above.

The surface oscillations in a deformable wall are known to induce an instability in the adjacent flow even in the absence of inertia. This instability, if understood properly, can be exploited to generate a well-mixed flow pattern with improved transport coefficients in microfluidic systems, wherein the benefits of inertial instabilities can not be realised. In order to utilise the wall deformability in micro-devices as well as other biotechnological applications, the quantitative knowledge of the critical parameter for the on-set of instability and the nature of bifurcation in the region of transition point are essential. With this objective, a major portion of this thesis deals with the stability analysis of flow past a flexible surface. For Newtonian flow over a deformable solid medium, the analyses of hydrodynamic stability in two flow regimes are presented: the viscous mode instability in the limit of zero Reynolds number, and the wall mode instability in the limit of high Reynolds number. The flexible solid in both analyses is described as a neo-Hookean solid continuum of finite thickness. The previous work on viscous instability using the same solid model ignored the viscous dissipation in the solid. In the present study, a purely elastic neo-Hookean model is augmented to incorporate the viscous stresses accounting for the dissipative mechanism in an aqueous gel-like solid medium. The linear stability analysis for this neo-Hookean viscoelastic solid shows a dramatic influence of solid viscosity on the stability behaviour. The important parameter here is where ηr is the solid viscosity relative to the fluid viscosity and H is the solid-to-fluid thickness ratio. While the effect solid viscosity is stabilizing for a further increase in viscosity in the regime reduces the critical shear rate for transition, indicating a destabilizing influence of solid viscosity. The weakly nonlinear analysis indicates that the bifurcation is subcritical for most values of H when ηr =0. However, for non-zero solid viscosity, the analysis reveals a range of ηr for which the nature of bifurcation is supercritical. The results are in contrast to the behaviour for the Hookean (linear) elastic solid, for which the effect of solid viscosity is always stabilising and the bifurcation is subcritical for all values of H and ηr. For the wall mode instability, critical parameters for the linear and the neo-Hookean elastic solid are found to be very close. The weakly nonlinear analysis of the wall mode instability shows that the instability is driven to a supercritically stable branch, indicating the possibility of a stable complex flow pattern which is ) correction to the base flow. The amplitude of the supercritically bifurcated equilibrium state, A1e, is derived in the vicinity of the critical point, and its scaling with the flow Reynolds number is obtained. The nonlinear analysis is also carried out using the asymptotic analysis in small parameter Re−1/3. The asymptotic results are found to be in good agreement with the numerical solutions for For a polymeric flow over a deformable solid medium, the viscous instability is analysed by extending the viscous mode for the Newtonian fluid to the fluid with finite elasticity. The viscoelastic fluid is described by an Oldroyd-B model which introduces two additional parameters: the Weissenberg number, W , and β, the ratio of solvent-to-solution viscosity. The polymer viscosity parameter β is an indirect measure of polymer concentration with the extreme cases of β =1 representing the Newtonian fluid and β =0the upper convected Maxwell fluid. The analysis considers both the linearly elastic and the neo-Hookean models to describe the deformable solid. The analysis reveals the presence of two classes of modes: the finite wavelength modes and the shortwave modes. The behaviour of the finite wavelength modes is similar for both the models of solid medium. The effect of increasing fluid Weissenberg number and also increasing polymer concentration (achieved by reducing β below 1) on the finite wavelength instability is stabilising. The viscous instability ceases to exist for W larger than a certain maximum value Wmax. The behaviour of the shortwave mode is remarkably different for both the models of solid. Using the shortwave asymptotic, the differences are elucidated and it is shown that the shortwave instabilities in both the models are qualitatively different modes. For a linear elastic solid model, the shortwave mode is attributed to the normal-stresses in polymeric fluid with high Weissenberg number. This mode does not exist for the Newtonian flow and is a downstream travelling disturbance wave. On the other hand, the shortwave mode for the neo-Hookean model is attributed to the normal-stress difference in the elastic solid. Hence, this mode does exist for the Newtonian fluid and is an upstream travelling disturbance wave. The role of polymer concentration in the criticality of finite wavelength and shortwave modes is examined for a wide range of Weissenberg number. The results are condensed in a map showing the stability boundaries in parametric space covering β, W and H. The weakly nonlinear analysis reveals that the bifurcation of linear instability is subcritical when there is no dissipation in the solid. The nature of bifurcation, however, changes to supercritical when the viscous effects in the solid are taken into account. The final problem of this thesis deals with the flow past a rigid surface. Here, the stability of base profile in a plane Couette flow of dilute polymeric fluid is studied at moderate Reynolds number. Three variants of Oldroyd-B model have been analysed, viz. the classical Oldroyd-B model, the diffusive Oldroyd-B model, and the non-homogeneous Oldroyd-B model. The Newtonian wall modes are modified marginally for the polymeric fluid described by the classical Oldroyd-B model. The Oldroyd-B model with artificial diffusivity introduces the additional ‘diffusive modes’ which scale with P´eclet number. The diffusive modes become the slowest decaying modes, in comparison to the wall modes, for large wavenumber disturbances. For these two models, the polymeric flow is linearly stable. Using the equilibrium flow method, wherein the nonlinear flow is assumed to be at the transition point, the finite amplitude disturbances are analysed, and the threshold energy necessary for subcritical transition is estimated. The third variant of Oldroyd-B model accounts for non-homogeneous polymer concentration coupled with the stress field. This model exhibits an instability in the linear analysis. The ‘concentration mode’ becomes unstable when the fluid Weissenberg number exceeds a certain transition value. This instability is driven by the stress-induced fluctuations in polymer number density.

O processo de moldagem por injeção de peças de paredes finas apresenta diferenças significativas em relação a processos convencionais de moldagem por injeção de termoplásticos. Processos de paredes finas são caracterizados pelo preenchimento de cavidades com espessuras inferiores a 1 mm utilizando velocidades de preenchimento elevadas. Estes dois fatores afetam o desenvolvimento de tensões induzidas pelo escoamento durante o preenchimento da cavidade, sendo que não foram encontrados na literatura estudos de determinação de tensões induzidas pelo escoamento em cavidades de paredes finas. Neste trabalho são apresentados resultados de determinação de tensões induzidas pelo escoamento em cavidades de paredes finas em comparação com resultados de simulação com cavidades características de processos convencionais. Para o cálculo das tensões induzidas pelo escoamento foi utilizado o modelo incompressível de Leonov. Para isso foi desenvolvida uma metodologia numérica para tratar problemas não isotérmicos utilizando o pacote CFD OpenFOAM. Foi utilizada a abordagem não acoplada, ou seja, o comportamento viscoelástico não é considerado na obtenção dos campos de velocidade, pressão e temperatura. Esta metodologia foi utilizada na simulação do preenchimento de cavidades bidimensionais. Os resultados indicaram que as tensões são influenciadas fortemente pela espessura da cavidade enquanto a velocidade de escoamento causou pequena variação das tensões induzidas pelo escoamento. Este trabalho mostrou que as tensões induzidas pelo escoamento não podem ser desconsideradas na produção de peças moldadas por injeção pelo processo de paredes finas. / Thin wall injection molding process of thermoplastics has significant differences compared to conventional injection molding process. Thin wall processes are characterized by cavities thinner than 1 mm and very high injection velocities. Although these two factors are expected to increase the flow induced stresses development during cavities filling, no previous study on this subject has been found in literature. In the present work, flow induced stresses under thin wall injection molding conditions are calculated using a viscoelastic model and compared to the results obtained under conventional injection conditions. In order to do this, a numerical methodology, based on the solver viscoelasticInterFoam, was developed in the OpenFOAM package to deal with the non-isothermal flow occurring during the mold filling stage. A non-coupled approach was used to calculate the stress field, i.e., the viscoelastic behavior was not considered in the determination of velocity, pressure and temperature fields. This methodology was used in the analysis of the filling stage of two-dimensional cavities. The results indicated that the cavity thickness has more influence on the flow induced stresses than the injection velocity. The obtained results also indicate that the flow induced stresses cannot be neglected in thin wall injection molding processes.

O processo de moldagem por injeção de peças de paredes finas apresenta diferenças significativas em relação a processos convencionais de moldagem por injeção de termoplásticos. Processos de paredes finas são caracterizados pelo preenchimento de cavidades com espessuras inferiores a 1 mm utilizando velocidades de preenchimento elevadas. Estes dois fatores afetam o desenvolvimento de tensões induzidas pelo escoamento durante o preenchimento da cavidade, sendo que não foram encontrados na literatura estudos de determinação de tensões induzidas pelo escoamento em cavidades de paredes finas. Neste trabalho são apresentados resultados de determinação de tensões induzidas pelo escoamento em cavidades de paredes finas em comparação com resultados de simulação com cavidades características de processos convencionais. Para o cálculo das tensões induzidas pelo escoamento foi utilizado o modelo incompressível de Leonov. Para isso foi desenvolvida uma metodologia numérica para tratar problemas não isotérmicos utilizando o pacote CFD OpenFOAM. Foi utilizada a abordagem não acoplada, ou seja, o comportamento viscoelástico não é considerado na obtenção dos campos de velocidade, pressão e temperatura. Esta metodologia foi utilizada na simulação do preenchimento de cavidades bidimensionais. Os resultados indicaram que as tensões são influenciadas fortemente pela espessura da cavidade enquanto a velocidade de escoamento causou pequena variação das tensões induzidas pelo escoamento. Este trabalho mostrou que as tensões induzidas pelo escoamento não podem ser desconsideradas na produção de peças moldadas por injeção pelo processo de paredes finas. / Thin wall injection molding process of thermoplastics has significant differences compared to conventional injection molding process. Thin wall processes are characterized by cavities thinner than 1 mm and very high injection velocities. Although these two factors are expected to increase the flow induced stresses development during cavities filling, no previous study on this subject has been found in literature. In the present work, flow induced stresses under thin wall injection molding conditions are calculated using a viscoelastic model and compared to the results obtained under conventional injection conditions. In order to do this, a numerical methodology, based on the solver viscoelasticInterFoam, was developed in the OpenFOAM package to deal with the non-isothermal flow occurring during the mold filling stage. A non-coupled approach was used to calculate the stress field, i.e., the viscoelastic behavior was not considered in the determination of velocity, pressure and temperature fields. This methodology was used in the analysis of the filling stage of two-dimensional cavities. The results indicated that the cavity thickness has more influence on the flow induced stresses than the injection velocity. The obtained results also indicate that the flow induced stresses cannot be neglected in thin wall injection molding processes.

O processo de moldagem por injeção de peças de paredes finas apresenta diferenças significativas em relação a processos convencionais de moldagem por injeção de termoplásticos. Processos de paredes finas são caracterizados pelo preenchimento de cavidades com espessuras inferiores a 1 mm utilizando velocidades de preenchimento elevadas. Estes dois fatores afetam o desenvolvimento de tensões induzidas pelo escoamento durante o preenchimento da cavidade, sendo que não foram encontrados na literatura estudos de determinação de tensões induzidas pelo escoamento em cavidades de paredes finas. Neste trabalho são apresentados resultados de determinação de tensões induzidas pelo escoamento em cavidades de paredes finas em comparação com resultados de simulação com cavidades características de processos convencionais. Para o cálculo das tensões induzidas pelo escoamento foi utilizado o modelo incompressível de Leonov. Para isso foi desenvolvida uma metodologia numérica para tratar problemas não isotérmicos utilizando o pacote CFD OpenFOAM. Foi utilizada a abordagem não acoplada, ou seja, o comportamento viscoelástico não é considerado na obtenção dos campos de velocidade, pressão e temperatura. Esta metodologia foi utilizada na simulação do preenchimento de cavidades bidimensionais. Os resultados indicaram que as tensões são influenciadas fortemente pela espessura da cavidade enquanto a velocidade de escoamento causou pequena variação das tensões induzidas pelo escoamento. Este trabalho mostrou que as tensões induzidas pelo escoamento não podem ser desconsideradas na produção de peças moldadas por injeção pelo processo de paredes finas. / Thin wall injection molding process of thermoplastics has significant differences compared to conventional injection molding process. Thin wall processes are characterized by cavities thinner than 1 mm and very high injection velocities. Although these two factors are expected to increase the flow induced stresses development during cavities filling, no previous study on this subject has been found in literature. In the present work, flow induced stresses under thin wall injection molding conditions are calculated using a viscoelastic model and compared to the results obtained under conventional injection conditions. In order to do this, a numerical methodology, based on the solver viscoelasticInterFoam, was developed in the OpenFOAM package to deal with the non-isothermal flow occurring during the mold filling stage. A non-coupled approach was used to calculate the stress field, i.e., the viscoelastic behavior was not considered in the determination of velocity, pressure and temperature fields. This methodology was used in the analysis of the filling stage of two-dimensional cavities. The results indicated that the cavity thickness has more influence on the flow induced stresses than the injection velocity. The obtained results also indicate that the flow induced stresses cannot be neglected in thin wall injection molding processes.