Numerical simulations of the complex flows of complex (viscoelastic) fluids are investigated. The primary fluid investigated in this thesis is human blood, a complex fluid which can be modelled via viscoelastic constitutive models. The most commonly used constitutive models for viscoelastic fluids include the OldroydB, Giesekus, Johnson-Segalman, Finitely Extensible Non-Linear Elastic (FENE), Phan-Thein-Tanner (PTT) models etc. Our Numerical approach is based on the finite volume methods implemented on the OpenFOAM platform. We employ the Giesekus, Oldroyd-B, and Generalized Oldroyd-B viscoelastic constitutive models in this thesis, depending on the underlying context. Numerical validation of our results is conducted via the most used benchmark flow problems for viscoelastic fluid flow. The robust and efficient numerical methodologies are then deployed to investigate the flow characteristics, and hence illustrate various novel behavior, for blood flow in stenosed and bifurcated arteries. The present work took advantage of the availability of a reasonable set of viscoelastic constitutive model solvers within OpenFOAM, specifically the viscoelasticFluidFoam solver which we modified and developed to suit our focused needs for blood flow computations. The modified computational algorithms were successfully validated against well-known benchmark flow problems in the literature. Noting that the Giesekus viscoelastic constitutive model is a generalization of both the Oldroyd-B and Generalized Oldroyd-B models, the validation of results is carried out via the Giesekus model enabling us to develop a general-purpose code capable of simulating several viscoelastic constitutive models. The main results were otherwise presented for the Oldroyd-B and Generalized Oldroyd-B models as these are the most applicable to blood flow modelling. The results demonstrate that the velocity spurt through the stenosis is directly proportional to the constriction caused by the stenosis. The higher the blockage from the constriction, the higher the corresponding velocity spurt through the constriction. This velocity behavior, as the constriction blockage increases, correspondingly increase the wall shear stresses. High wall shear stresses significantly increase the possibility of rupture of the stenosis/blockage. This can lead to catastrophic consequences in the usual case where the stenosis is caused by tumor growth.

Our aim was to aid the viscoelasticity into the model for thermal convection by developing our own code in Fortran 90 and to study the role of viscoelasticity in this model. We should have included the viscoelasticity by Maxwell model; however, due to numerical instability we changed it for Oldroyd-B model. We were adding the terms of objective derivative into our code step by step and we were observing how they influence the behaviour of thermal convection. Partial time derivative and advective terms were included in whole complexity, the corrotational terms need more numerical testing. Our work suggest that the influence of viscoelasticity on thermal convection is noticeable. Powered by TCPDF (www.tcpdf.org)

In this work, we develop a method for solving viscoelastic fluid flows using the Navier-Stokes equations coupled with the Oldroyd-B model. We solve the Navier-Stokes equations in skew-symmetric form using the mixed finite element method, and we solve the Oldroyd-B model using the discontinuous Galerkin method. The Crank-Nicolson scheme is used for the temporal discretization of the Navier-Stokes equations in order to achieve a second-order accuracy in time, while the optimal third-order total-variation diminishing Runge-Kutta scheme is used for the temporal discretization of the Oldroyd-B equation. The overall accuracy in time is therefore limited to second-order due to the Crank-Nicolson scheme; however, a third-order Runge-Kutta scheme is implemented for greater stability over lower order Runge-Kutta schemes. We test our numerical method using the 2D cavity flow benchmark problem and compare results generated with those found in literature while discussing the influence of mesh refinement on suppressing oscillations in the polymer stress. / Master of Science / Viscoelastic fluids are a type of non-Newtonian fluid of great importance to the study of fluid flows. Such fluids exhibit both viscous and elastic behaviors. We develop a numerical method to solve the partial differential equations governing viscoelastic fluid flows using various finite element methods. Our method is then validated using previous numerical results in literature.

The main focus of this work is on the non-Newtonian effects on the inertial instabilities in shear flows. Both inelastic (Carreau) and elastic models (Oldroyd-B and FENE-P) have been employed to examine the main features of the non-Newtonian fluids; shear-thinning, shear-thickening and elasticity. Several classical configurations have been considered; flow past a circular cylinder, in a lid-driven cavity and in a channel. We have used a wide range of tools for linear stability analysis, modal, non-modal, energy and sensitivity analysis, to determine the instability mechanisms of the non-Newtonian flows and compare them with those of the Newtonian flows. Direct numerical simulations have been also used to prove the results obtained by the linear stability analysis. Significant modifications/alterations in the instability of the different flows have been observed under the action of the non-Newtonian effects. In general, shear-thinning/shear-thickening effects destabilize/stabilize the flow around the cylinder and in a lid driven cavity. Viscoelastic effects both stabilize and destabilize the channel flow depending on the ratio between the viscoelastic and flow time scales. The instability mechanism is just slightly modified in the cylinder flow whereas new instability mechanisms arise in the lid-driven cavity flow. We observe that the non-Newtonian effect can alter the inertial flow at both baseflow and perturbation level (e.g. Carreau fluid past a cylinder or in a lid driven cavity) or it may just affect the perturbations (e.g. Oldroyd-B fluid in channel). In all the flow cases studied, the modifications in the instability dynamics are shown to be strongly connected to the contribution of the different terms in the perturbation kinetic energy budget. / <p>QC 20140113</p>

Escoamentos sem inércia de fluido elasto-viscoplástico, dentro de uma cavidade, são numericamente analisados. As soluções visam compreender a influência dos efeitos viscosos e elásticos na topologia de superfícies de escoamento. Assumindo-se que o colapso da microestrutura do material é instantâneo, o modelo mecânico é constituído pelas equações governantes de massa e momentum para fluidos incompressíveis, associado a uma equação hiperbólica para o tensor tensão extra, baseado na equação do modelo Oldroyd-B (Nassar et al, 2011). A principal característica do modelo é considerar a viscosidade e o tempo de relaxação como função da taxa de deformação, permitindo a pseudoplasticidade de viscosidade e restringindo os efeitos elásticos para as regiões não deformadas do material. As simulações numéricas são realizadas através do método de Galerkin mínimos quadrados a três campos: tensor tensão extra, pressão e velocidade. Os resultados mostram que as superfícies de escoamento do material são fortemente influenciadas pela ação combinada entre os efeitos elásticos e viscosos, estando em conformidade com a recente visualização experimental dos fluxos elasto-viscoplásticos. / Elasto viscoplastic uid ows without inertia, within a cavity, are numerically analyzed. The solutions aim to understand the in uence of viscous and elastic e ects on the topology of yield surfaces. Assuming that the collapse of the material microstructure is instantaneous, the mechanical model consists of the governing equations of mass and momentum for incompressible uids, associated with a hyperbolic equation for the extra stress tensor, based on the equation of the Oldroyd-B model (Nassar et al, 2011). The main feature of model is to consider the viscosity and the relaxation time as a function of shear rate, allowing the shear-thinning of viscosity and restricting elastic e ects for regions not deformed material. Numerical simulations are performed by the method of Galerkin Least Squares to three elds: extra stress tensor, pressure and velocity. The results show that the yield surfaces of material are strongly in uenced by the combined action between the elastic and viscous e ects, complying with the recent experimental visualization of elasto-viscoplastic ows.

Submitted by RODRIGO KOITI ISHIZAKA (rodrigo_ishizaka@hotmail.com) on 2018-04-09T20:10:27Z No. of bitstreams: 1 Dissertação Final PDF.pdf: 10461391 bytes, checksum: a78974d2db3d166540e03d4cee7def37 (MD5) / Rejected by ALESSANDRA KUBA OSHIRO ASSUNÇÃO (alessandra@fct.unesp.br), reason: Solicitamos que realize correções na submissão seguindo as orientações abaixo: - Número de página das referências no sumário está incorreto (consta p.86 quando é p.87); - No sumário consta Apêndices, porém no trabalho não. É finalizado nas referências. Agradecemos a compreensão. on 2018-04-10T12:44:45Z (GMT) / Submitted by RODRIGO KOITI ISHIZAKA (rodrigo_ishizaka@hotmail.com) on 2018-04-10T16:09:35Z No. of bitstreams: 1 Dissertação Final PDF.pdf: 10461372 bytes, checksum: 679a6e28fa80deb9079156b9b6b70c36 (MD5) / Approved for entry into archive by ALESSANDRA KUBA OSHIRO ASSUNÇÃO (alessandra@fct.unesp.br) on 2018-04-10T20:02:48Z (GMT) No. of bitstreams: 1 ishizaka_rk_me_prud.pdf: 10461372 bytes, checksum: 679a6e28fa80deb9079156b9b6b70c36 (MD5) / Made available in DSpace on 2018-04-10T20:02:48Z (GMT). No. of bitstreams: 1 ishizaka_rk_me_prud.pdf: 10461372 bytes, checksum: 679a6e28fa80deb9079156b9b6b70c36 (MD5) Previous issue date: 2018-02-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O presente trabalho de mestrado consiste em apresentar uma modelagem para escoamentos isotérmicos viscoelásticos em que a viscosidade varia de acordo com a pressão e estudos numéricos para alguns problemas bidimensionais, como escoamentos entre placas paralelas e expansão planar 1:4. Para este trabalho é utilizado o modelo viscoelástico Oldroyd-B e uma modelagem linear da viscosidade com relação a pressão. O método numérico desenvolvido é baseado no método da projeção para desacoplar velocidade e pressão nas equações de Navier-Stokes e depois calcula os tensores pela equação constitutiva com a informação da viscosidade variando com a pressão. As equações são discretizadas em uma malha deslocada pelo método de diferenças finitas. Os resultados numéricos são obtidos das simulações de escoamentos em um canal e expansão planar 1:4 bidimensional, cujo foco é destacar algumas diferenças entre o modelo Oldroyd-B Standard e o modelo Oldroyd-B sob influência da viscosidade que varia linearmente com a pressão. Este trabalho de mestrado também tem por objetivo resolver problemas, com fluidos viscoelásticos, em que haja regiões com variações da pressão e estudar estes resultados numéricos comparando-os com o modelo Oldroyd-B Standard. / The present master's work consists in presenting a model for viscoelastic isothermal flows with pressure-dependent viscosity and numerical studies for some two-dimensional problems, such as channel flow and planar expansion 1:4. In this work will be considered the Oldroyd-B viscoelastic model and a linear modeling viscosity with a linear relationship between viscosity and pressure. The numerical method developed is based on the projection method to decouple velocity and pressure in the Navier-Stokes equations and then calculates the tensor from the constitutive equation taking account the pressure-dependent viscosity. The equations are discretized on a staggered grid by using the finite diference method. The numerical results are obtained for two problems: two-dimensional channel and planar expansion 1:4 flows, whose focus is to highlight some diferences between the standard Oldroyd-B model and the Oldroyd-B model under the influence of viscosity that varies linearly with pressure. This master's work aims to solve problems of viscoelastic fluids in which there exist regions where pressure varies and to study these numerical results comparing them with the standard Oldroyd-B model.

Escoamentos sem inércia de fluido elasto-viscoplástico, dentro de uma cavidade, são numericamente analisados. As soluções visam compreender a influência dos efeitos viscosos e elásticos na topologia de superfícies de escoamento. Assumindo-se que o colapso da microestrutura do material é instantâneo, o modelo mecânico é constituído pelas equações governantes de massa e momentum para fluidos incompressíveis, associado a uma equação hiperbólica para o tensor tensão extra, baseado na equação do modelo Oldroyd-B (Nassar et al, 2011). A principal característica do modelo é considerar a viscosidade e o tempo de relaxação como função da taxa de deformação, permitindo a pseudoplasticidade de viscosidade e restringindo os efeitos elásticos para as regiões não deformadas do material. As simulações numéricas são realizadas através do método de Galerkin mínimos quadrados a três campos: tensor tensão extra, pressão e velocidade. Os resultados mostram que as superfícies de escoamento do material são fortemente influenciadas pela ação combinada entre os efeitos elásticos e viscosos, estando em conformidade com a recente visualização experimental dos fluxos elasto-viscoplásticos. / Elasto viscoplastic uid ows without inertia, within a cavity, are numerically analyzed. The solutions aim to understand the in uence of viscous and elastic e ects on the topology of yield surfaces. Assuming that the collapse of the material microstructure is instantaneous, the mechanical model consists of the governing equations of mass and momentum for incompressible uids, associated with a hyperbolic equation for the extra stress tensor, based on the equation of the Oldroyd-B model (Nassar et al, 2011). The main feature of model is to consider the viscosity and the relaxation time as a function of shear rate, allowing the shear-thinning of viscosity and restricting elastic e ects for regions not deformed material. Numerical simulations are performed by the method of Galerkin Least Squares to three elds: extra stress tensor, pressure and velocity. The results show that the yield surfaces of material are strongly in uenced by the combined action between the elastic and viscous e ects, complying with the recent experimental visualization of elasto-viscoplastic ows.

Escoamentos sem inércia de fluido elasto-viscoplástico, dentro de uma cavidade, são numericamente analisados. As soluções visam compreender a influência dos efeitos viscosos e elásticos na topologia de superfícies de escoamento. Assumindo-se que o colapso da microestrutura do material é instantâneo, o modelo mecânico é constituído pelas equações governantes de massa e momentum para fluidos incompressíveis, associado a uma equação hiperbólica para o tensor tensão extra, baseado na equação do modelo Oldroyd-B (Nassar et al, 2011). A principal característica do modelo é considerar a viscosidade e o tempo de relaxação como função da taxa de deformação, permitindo a pseudoplasticidade de viscosidade e restringindo os efeitos elásticos para as regiões não deformadas do material. As simulações numéricas são realizadas através do método de Galerkin mínimos quadrados a três campos: tensor tensão extra, pressão e velocidade. Os resultados mostram que as superfícies de escoamento do material são fortemente influenciadas pela ação combinada entre os efeitos elásticos e viscosos, estando em conformidade com a recente visualização experimental dos fluxos elasto-viscoplásticos. / Elasto viscoplastic uid ows without inertia, within a cavity, are numerically analyzed. The solutions aim to understand the in uence of viscous and elastic e ects on the topology of yield surfaces. Assuming that the collapse of the material microstructure is instantaneous, the mechanical model consists of the governing equations of mass and momentum for incompressible uids, associated with a hyperbolic equation for the extra stress tensor, based on the equation of the Oldroyd-B model (Nassar et al, 2011). The main feature of model is to consider the viscosity and the relaxation time as a function of shear rate, allowing the shear-thinning of viscosity and restricting elastic e ects for regions not deformed material. Numerical simulations are performed by the method of Galerkin Least Squares to three elds: extra stress tensor, pressure and velocity. The results show that the yield surfaces of material are strongly in uenced by the combined action between the elastic and viscous e ects, complying with the recent experimental visualization of elasto-viscoplastic ows.

We prove a global-in-time and large-data existence of a suitable weak solution to a system of partial differential equations describing an unsteady flow of homogeneous incom- pressible viscoelastic rate-type fluid. The material parameters are continuous functions of temperature and, in particular, the dependence of the shear modulus is assumed to be linear. It is shown that studied models obey the fundamental laws of thermodynamics. The key step towards the existence proof is derivation of the balance of entropy. This in- equality is paramount in the analysis and as its consequence, we obtain sufficient a priori estimates, positivity of temperature and also regularity of the elastic deformation. The second part of the thesis deals with the existence analysis for the isothermal case, however using a completely different method, which is of independent interest. 1

[pt] Filmes finos de líquidos estão presentes em uma gama de aplicações industriais, como processos de atomização e revestimento de substrato. O processo de quebra pode ser divido em duas etapas: o estágio de ruptura, e o estágio de retração. O primeiro, movido pelas forças de van der Waals, ocorre quando uma pequena perturbação cresce e provoca o aparecimento de um pequeno furo no filme. O segundo, movido por forças capilares, provoca o crescimento desse furo levando à desintegração do filme de líquido. A estabilidade de uma cortina de líquido depende das características da perturbação, da espessura do filme e das propriedades do fluido. Análises experimentais mostraram que uma cortina super fina pode ser obtida pela utilização de fluidos viscoelásticos. Os mecanismos físicos associados à esta estabilidade, contudo, não são totalmente compreendidos. Este trabalho apresenta um estudo numérico e teórico dos efeitos das propriedades viscoelásticas na estabilidade de uma cortina de fluido, englobando ambos os estágio do processo. As análises numéricas foram desenvolvidas através da expansão assintótica das variáveis do escoamento com aplicação de um esquema de integração no tempo totalmente implícito. A partir da análise teórica da dinâmica de ruptura foi possível obter um critério de estabilidade linear para perturbações planares e axissimétricas em fluidos Newtonianos e não-Newtonianos. O tempo de ruptura e a velocidade de retração do filme foram calculados numericamente como função das propriedades viscoelásticas do líquido. Resultados mostraram que as forças elásticas atuam de forma a dificultar o processo de quebra e retração. Análises da evolução da espessura mostraram que as propriedades reológicas do fluído também interferem no formato que o filme de fluido assume durante o processo de retração. Para regimes de baixa viscosidade, as forças elásticas atuaram evitando a formação de ondas capilares observadas em fluidos Newtonianos. / [en] Thin free liquid sheets are ubiquitous in many industrial processes, such as atomization and curtain coating. Liquid sheets are susceptible to instabilities at the interface, which can grow, triggering a breakup process. This process can be divided into two different stages: the rupture stage and retraction. The first, driven by van der Waals force, occurs when a small instability grows until it pinches-off the sheet. The second, driven by capillary forces, induces the growth of the hole caused by the pinch-off, leading to the full disintegration of the liquid sheet. The stability of a liquid sheet depends on disturbance characteristics, sheet thickness, and fluid properties. Experimental analyses have shown that thinner stable liquid curtain can be obtained with viscoelastic liquids. The underlyning physical mechanisms associated with increased stability are, however, not fully understood. This work presents a theoretical and numerical analysis of the effect of viscoelasticity on the stability of a thin liquid sheet during both stages of the breakup process. We first analyze the rupture dynamics, deriving linear stability criteria for both planar and axisymmetric perturbations of Newtonian and Oldroyd-B liquids. The time evolution of planar and axisymmetric perturbations in an Oldroyd-B liquid sheet is evaluated using the asymptotic expansion of the flow variables and a fully-implicit time integration scheme. The rupture time and retraction velocity are calculated as a function of the viscoelastic properties. The results show that the liquid rheological behavior does not influence the linear stability criterion. Nevertheless, it has a strong effect on the growth rate of the disturbance and retraction velocity, increasing, thus, the breakup time. The results show that elastic forces act to hinder the rupture and retraction stages. Analysis of the temporal evolution of the thickness profile reveals that liquid rheological behavior also affects the shape of the liquid sheet. For low viscosity regime, the elastic forces damp the capillary waves that arise during the retraction of Newtonian sheets.