Orientadores: Philippe Remy Bernard Devloo, Sonia Maria Gomes / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Civil, Arquitetura e Urbanismo / Made available in DSpace on 2018-08-15T11:57:47Z (GMT). No. of bitstreams: 1 Forti_TiagoLuisDuarte_D.pdf: 8887357 bytes, checksum: dc0e7acaa76c7778f2ffa9e3f9d24f9b (MD5) Previous issue date: 2010 / Resumo: Este trabalho dedica-se ao estudo dos métodos de Elementos Finitos e de Galerkin Descontínuo combinados. Nele, o Método de Galerkin Descontínuo é tratado como uma variante do Método de Elementos Finitos tradicional em que as funções do espaço de interpolação são descontínuas entre elementos. Procura-se a melhor combinação dos métodos, identicando em que condições cada método se sobressai. São abordados problemas elípticos de segunda ordem com singularidade e problemas de convecção. Em problemas elípticos, propõe-se utilizar funções de enriquecimento em elementos de Galerkin descontínuo. Os elementos enriquecidos são posicionados na vizinhança de singularidades, enquanto que nas regiões distantes, empregam-se elementos contínuos. Em problemas de convecção, propõe-se utilizar elementos descontínuos na vizinhança de choques e elementos contínuos em regiões em que a solução é suave. Uma estratégia de adaptação entre elementos contínuos e de Galerkin descontínuo é apresentada. Os resultados são mostrados em termos de erro de aproximação e, para problemas convectivos, em amplitude de oscilações / Abstract: The present work is dedicated to study the continuous Finite Element Method (FEM) and the Discontinuous Galerkin Method (DGM) combined in the same simulation. In this work the DGM is dealt with as a variant of the Finite Element Method where the interpolation space is formed by discontinuous functions between elements. In this work, we propose a formulation which combines FEM and DGM in the same simulation identifying when each method has better performance. The proposed formulation is applied to second-order elliptic problems with singular solution and to convection problems. For elliptic problems, we propose the use of local enrichment function in the approximation space of discontinuous elements. Elements with enrichment functions are employed in the vicinity of singularities. In other regions, continuous elements are employed. For convection problems, we propose to use discontinuous elements in regions where the solution presents shocks and continuous elements where the solution is smooth. A strategy to automatically decide which type of element is to be adopted is proposed. The results are compared in terms of approximation errors and for convective problems also in terms of amplitude of oscillations / Doutorado / Estruturas / Doutor em Engenharia Civil

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-12-21T13:46:29Z No. of bitstreams: 1 miguelcutipacoaquira.pdf: 878116 bytes, checksum: 1d11a64ceebc2f6da2e3f07d46ef0468 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-12-22T12:51:57Z (GMT) No. of bitstreams: 1 miguelcutipacoaquira.pdf: 878116 bytes, checksum: 1d11a64ceebc2f6da2e3f07d46ef0468 (MD5) / Made available in DSpace on 2016-12-22T12:51:57Z (GMT). No. of bitstreams: 1 miguelcutipacoaquira.pdf: 878116 bytes, checksum: 1d11a64ceebc2f6da2e3f07d46ef0468 (MD5) Previous issue date: 2016-05-12 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O uso de espuma para o controle da mobilidade é uma técnica potencial que melhora a eficiência na recuperação avançada de óleo. As propriedades da espuma são controladas pela dinâmica de criação e destruição seguindo os modelos mais usados de balanço de populaçãoemodelosdeequilíbriolocal,considerandoalgumashipótesescomodeslocamento unidimensional, método do fluxo fracionário. O surfactante como componente da fase liquida é responsável da criação de espuma. Em muitos artigos por simplicidade a concentração do surfactante é considerada constante. Neste trabalho não é considerado esta simplificação. O objetivo deste trabalho é desenvolver um modelo onde a concentração do surfactante é descrita por uma equação de balanço. O modelo é completado por equações de balanço de massa de água, gás e a concentração de bolhas de espuma. A geração e destruição de bolhas é descrita pela dinâmica do modelo cinético de primeira ordem. Para estudar matematicamenteomodelousamosferramentasdeequaçõesdiferenciaisordináriaseondas viajantes. Para estados de equilíbrio adequados mostramos a existência de ondas viajantes. Para o caso particular, desprezando a pressão capilar, a existência foi rigorosamente provada. Para o caso geral, uma investigação numérica foi realizada. / Theuseoffoamtocontrolthemobilityisapotentialtechniquethatimprovestheefficiency of the enhanced oil recovery. The properties of the foam are controlled by the dynamics of creation and destruction following the most used population balance models and models in local equilibrium. Under some assumptions, one-dimensional displacements, the fractional flow method. The surfactant as a component of the water phase is responsible for the foam generation e destruction. Some papers neglect this component for simplicity. In the present work the surfactant concentration is considered. Inthepresentworkthesurfactantphaseisconsideredinthemodelasseparatebalancelaw. The model is complete with mass balance equations of water, gas and the concentration of bubbles foam. The bubble generation and destruction is described by dynamic of the first order kinetic model. The mathematically study was based on ordinary differential equation tools and traveling waves analysis. For reasonable equilibrium conditions we study the existence of the traveling wave solution. For the particular case neglecting the capillary pressure, the existence was proved rigorously. For the general case numerical investigation was performed.

Made available in DSpace on 2014-07-29T16:02:23Z (GMT). No. of bitstreams: 1 versaofinal_lidiane.pdf: 708631 bytes, checksum: c97f21b6c4fd093f9c0ec8f92aa22110 (MD5) Previous issue date: 2010-04-09 / In this dissertation we describe the solutions to the Riemann problem for a system of two conservation laws written in the normal from according to classification of Schaeffer-Shearer in [9]. Through changes of variables Schaeffer-Shearer determined the normal form for a nonlinear system of two conservation laws with an isolated umbilical point in state space. The normal form consists of a system of two equations, with homogeneous and quadratic functions of flow that depend only on two parameters. Also in [9] were established four distinct regions in terms of parameters, denoted by I, II, III and IV, in which varying pair of parameters in each region, the curves of waves that make up the solution of the Riemann problem have the same configuration. In this dissertation we consider the case in which the pair of parameters belongs to region IV, and in the particular case in which one of the parameters is null. In this case, the classic Lax criterion for admissibility of shocks (discontinuity solutions) generally is sufficient to obtain uniqueness of solution. Although, for some initial states, it is necessary to admit in solution also the called compressive shocks, which do not satisfy the Lax criterion. / Nesta dissertação determinamos as soluções do problema de Riemann para um sistema de duas leis de conservação escrito na forma normal segundo a classificação de Schaeffer-Shearer, em [9]. Através de mudanças de variáveis, Schaeffer-Shearer determinaram a forma normal para um sistema não linear de duas leis de conservação com um ponto umbílico isolado no espaço de estados. A forma normal consiste de um sistema de duas equações, com funções de fluxo quadráticas homogêneas que dependem apenas de dois parâmetros. Também em [9] foram determinadas quatro regiões distintas no plano dos parâmetros, denotadas por I, II, III e IV, onde, variando o par de parâmetros em cada região, as curvas de onda que compõem a solução do problema de Riemann tem a mesma configuração. Nesta dissertação consideramos o caso em que o par de parâmetros pertence a região IV, e ainda no caso particular em que um dos parâmetros é nulo. Neste caso, o clássico critério de Lax para admissibilidade dos choques (soluções descontínuas), em geral, é suficiente para se obter unicidade de solução. Embora, para alguns estados iniciais, é necessário admitir na solução também os chamados choques compressivos, que não satisfazem o critério de Lax.

Provamos que o. modelo CPn-1 não permit e formação de par até terceira ordem em teoria de perturbação. A matriz-S dos modelos CPn-1 e Thirring SU(n) foi calculada em perturbação até 2 loops. O cálculo mostra que a matriz-S tem algumas diferenças em relação à esperada. Além disso calculamos a carga não local quantizada do modelo cpn-1 em teoria de perturbação renormalizada 1/n e provamos que ela não é conservada, no entanto quando fermionss são acoplados de uma maneira mínima ou supersimétrica a anomalia se cancela. / We prove that the CPn-1 model does not accomodite pair formation up to third order in perturbation theory. The S-matrix of the Cpn-1 and SU(n) Thirring models was calculated perturbatively up to 2 loops. The calculation shows that the S-matrix has some deviations from the expected exact one. Furthermore, we calculate the quantized nonlocal charge of the CPn-1 model in the framework of renormalized l/n perturbation theory and prove that it is not conserved, nevertheless when fermions are coupled in a minimal or supersymmetric way the anomaly vanishes.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-04-06T17:55:59Z No. of bitstreams: 1 angelenriqueramirezgutierrez.pdf: 1737893 bytes, checksum: 1069699a1f9f64ab614c7bc8c26ddae0 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-04-24T04:12:46Z (GMT) No. of bitstreams: 1 angelenriqueramirezgutierrez.pdf: 1737893 bytes, checksum: 1069699a1f9f64ab614c7bc8c26ddae0 (MD5) / Made available in DSpace on 2016-04-24T04:12:46Z (GMT). No. of bitstreams: 1 angelenriqueramirezgutierrez.pdf: 1737893 bytes, checksum: 1069699a1f9f64ab614c7bc8c26ddae0 (MD5) Previous issue date: 2013-07-18 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Alguns problemas parabólicos podem ser reescritos na forma de problema de complementaridade e aparecem em muitas aplicações como em fluxos de líquidos no interior num domínio, difusão, fluxo de calor envolvendo mudança de fase e reações químicas. Estes tipos de problemas apresentam muitas dificuldades analíticas e numéricas, normalmente devido à evolução no tempo ou fronteira móvel. Como a solução analítica é muito difícil de obter, é importante o estudo de métodos numéricos que permitam a busca de uma solução aproximada da solução exata para tais tipos de problemas. Estuda-se leis de conservação e os tipos de soluções associadas ao Problema de Riemann, essencialmente leis de balanço que expressam o fato de que alguma substância é conservada. O estudo desta teoría é importante porque frequentemente as leis de conservação aparecem quando nos problemas parabólicos são desprezados os termos difusivos de segunda ordem. Apresenta-se também um método numérico que é uma variação do método de Newton para resolver sistemas não lineares. O método é baseado num esquema de diferenças finitas implícito e um algoritmo de complementaridade não linear, FDA–NCP. O método dado tem a vantagem de fornecer uma convergência global em relação ao método de diferenças finitas com o método de Newton que só tem convergência local. A teoria é aplicada ao modelo de difusão de oxigênio num tecido e ao modelo de combustão de oxigênio in situ, os dois modelos são problemas parabólicos linear e não linear respectivamente e que podem ser reescritos na forma de problema de complementaridade. O primeiro modelo que pode ser aplicado no tratamento de células cancerígenas conduz a um problema de fronteira livre enquanto no segundo modelo, consideramos um processo unidimensional de injeção de ar dentro de um meio poroso que contém inicialmente combustível sólido e onde ocorre combustão gas–sólido, assim o modelo envolve a lei de balanço do calor, lei molar do combustível e a lei de gases ideais. Além disso, estuda-se a onda térmica e a onda de combustível associadas. / Some parabolic problems can be rewritten in complementarity form and appear in many applications such as liquid flows within a domain, diffusion, heat flow involving phase change and chemical reactions. These types of problems have many analytical and numerical difficulties, usually due to the evolution in time or moving boundary. Since the analytical solution is very difficult to obtain, so it is important to study numerical methods that allow the search for an approximate solution of the exact solution for these types of problems. We study the conservation laws and the types of solutions associated with the Riemann Problem, these types of laws are essentially balance laws that express the fact that some substance is balanced. The study of this theory is important because the conservation laws often appear when the parabolic problems are neglected the diffusive terms of second order. It also presents a numerical method which is a variation of the Newton’s method for solving nonlinear systems, the method is based on an implicit finite difference scheme and an algorithm complementary non-linear FDA–NCP. The given method has the advantage of providing a global convergence with respect to the finite difference method with Newton’s method which has only local convergence. The theory is applied to the model difussion in tissue of oxygen and oxygen combustion model in situ, this two models are linear and nonlinear parabolics problems respectively and which can be rewritten in the form of complementarity problem. The first model that can be applied in the treatment of cancer cells leads to a free boundary problem, while the second model, consider a one-dimensional process of air injection inside a porous medium initially containing solid fuel and where combustion occurs gas - solid thus the model involves the heat balance law, law and the fuel molar ideal gas law, in addition, studies the thermal wave and the wave associated fuel.

The class of hyperbolic conservation laws model the phenomena of non-linear wave propagation, including the presence and propagation of discontinuities and expansion waves. Such nonlinear systems can generate discontinuities in the so-lution even for smooth initial conditions. Presence of discontinuities results in break down of a solution in the classical sense and to show existence, weak for-mulation of a problem is required. Moreover, closed form solutions are di cult to obtain and in some cases such solutions are even unavailable. Thus, numerical algorithms play an important role in solving such systems. There are several dis-cretization techniques to solve hyperbolic systems numerically and Finite Volume Method (FVM) is one of such important frameworks. Numerical algorithms based on FVM are broadly classi ed into two categories, central discretization methods and upwind discretization methods. Various upwind and central discretization methods developed so far di er widely in terms of robustness, accuracy and ef-ciency and an ideal scheme with all these characteristics is yet to emerge. In this thesis, novel upwind and central schemes are formulated for various hyper-bolic systems, with the aim of maintaining right balance between accuracy and robustness. This thesis is divided into two parts. First part consists of the formulation of upwind methods to simulate genuine weakly hyperbolic (GWH) systems. Such systems do not possess full set of linearly independent (LI) eigenvectors and some of the examples include pressureless gas dynamics system, modi ed Burgers' sys-tem and further modi ed Burgers' system. The main challenge while formulating an upwind solver for GWH systems, using the concept of Flux Di erence Splitting (FDS), is to recover full set of LI eigenvectors, which is done through addition of generalized eigenvectors using the theory of Jordan Canonical Forms. Once the defective set of LI eigenvectors are completed, a novel (FDS-J) solver is for-mulated in such a manner that it is independent of generalized eigenvectors, as they are not unique. FDS-J solver is capable of capturing various shocks such as -shocks, 0-shocks and 00-shocks accurately. In this thesis, the FDS-J schemes are proposed for those GWH systems each of which have one particular repeated eigenvalue with arithmetic multiplicity (AM) greater than one. Moreover, each ux Jacobian matrix corresponding to such systems is similar to a unique Jordan matrix. After the successful treatment of genuine weakly hyperbolic systems, this strategy is further applied to those weakly hyperbolic subsystems which result on employ-ing various convection-pressure splittings to the Euler ux function. For example, Toro-Vazquez (TV) splitting and Zha-Bilgen (ZB) type splitting approaches to split the Euler ux function yield genuine weakly hyperbolic convective parts and strict hyperbolic pressure parts. Moreover, the ux Jacobian of each convective part is similar to a Jordan matrix with at least two lower order Jordan blocks. Based on the lines of FDS-J scheme, we develop two numerical schemes for Eu-ler equations using TV splitting and ZB type splitting. Both the new ZBS-FDS and TVS-FDS schemes are tested on various 1-D shock tube problems and out of two, contact capturing ZBS-FDS scheme is extended to 2-dimensional Euler system where it is tested successfully on various test cases including many shock instability problems. Second part of the thesis is associated with the development of simple, robust and accurate central solvers for systems of hyperbolic conservation laws. The idea of splitting schemes together with the notion of FDS is not easily extendable to systems such as shallow water equations. Thus, a novel central solver Convection Isolated Discontinuity Recognizing Algorithm (CIDRA) is formulated for shallow water equations. As the name suggests, the convective ux is isolated from the total ux in such a way that other ux, in present case other ux represents celerity part, must possess non-zero eigenvalue contribution. FVM framework is applied to each part separately and ux equivalence principle is used to x the coe cient of numerical di usion. CIDRA for SWE is computed on various 1-D and 2-D benchmark problems and extended to Euler systems e ortlessly. As a further improvement, a scalar di usion based algorithm CIDRA-1 is designed for v Euler systems. The scalar di usion coe cient depends on that particular part of the Rankine-Hugoniot (R-H) condition which involves total energy of the system as a direct contribution. This algorithm is applied to a variety of shock tube test cases including a class of low density ow problems and also to various 2-D test problems successfully. vi

In this thesis, we have explored the commonalities and connections between different classes of quantum systems that do not thermalize. Specifically, we have (1) shown that localized systems possess conservation laws like integrable systems, which can be constructed in a systematic way and used to detect localization-delocalization transitions , (2) studied the phenomenon of many-body localization in a model with a single particle mobility edge, (3) shown that interesting finite-size scaling emerges, with universal exponents, when athermal quantum systems are forced to thermalize through the application of perturbations and (4) shown that these scaling laws also arise when a perturbation causes a crossover between quantum systems described by different random matrix ensembles. We conclude with a brief summary of each chapter. In Chapter 2, we have investigated the effects of finite size on the crossover between quantum integrable systems and non-integrable systems. Using exact diagonalization of finite-sized systems, we have studied this crossover by obtaining the energy level statistics and Drude weight associated with transport. Our results reinforce the idea that for system size L → ∞, non-integrability sets in for an arbitrarily small integrabilitybreaking perturbation. The crossover value of the perturbation scales as a power law ∼ L−3 when the integrable system is gapless and the scaling appears to be robust to microscopic details and the precise form of the perturbation. In Chapter 3, we have studied the crossover among different random matrix ensembles CHAPTER 6. CONCLUSION 127 [Poissonian, Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE) and Gaussian Symplectic Ensemble (GSE)] realized in different microscopic models. We have found that the perturbation causing the crossover among the different ensembles scales to zero with system size as a power law with an exponent that depends on the ensembles between which the crossover takes place. This exponent is independent of microscopic details of the perturbation. We have also found that the crossover from the Poissonian ensemble to the other three is dominated by the Poissonian to GOE crossover which introduces level repulsion while the crossover from GOE to GUE or GOE to GSE associated with symmetry breaking introduces a subdominant contribution. Finally,we have conjectured that the exponent is dependent on whether the system contains interactions among the elementary degrees of freedom or not and is independent of the dimensionality of the system. In Chapter 4, we have outlined a procedure to construct conservation laws for Anderson localized systems. These conservation laws are found as power series in the hopping parameters. We have also obtained the conservation laws for the disorder free Aubry-Andre model, where the states are either localized or extended depending on the strength of a coupling constant. We have formulated a specific procedure for averaging over disorder, in order to examine the convergence of the power series. Using this procedure for the Aubry-Andre model, we show that integrals of motion given by our construction are well-defined in the localized phase but not so in the extended phase. Finally, we also obtain the integrals of motion for a model with interactions to lowest order in the interaction. In Chapter 5, we have studied many body localization and investigated its nature in the presence of a single particle mobility edge. Employing the technique of exact diagonalization for finite-sized systems, we have calculated the level spacing distribution, time evolution of entanglement entropy, optical conductivity and return probability to characterize the nature of localization. The localization that develops in the presence of interactions in these systems appears to be different from regular Many-Body Localization (MBL) in that the growth of entanglement entropy with time is linear (like in CHAPTER 6. CONCLUSION 128 a thermal phase) instead of logarithmic but saturates to a value much smaller than the thermal value (like for MBL). All other diagnostics seem consistent with regular MBL

The main purpose of this work is to prove short-time structural stability of compressible vortex sheets with surface tension. The main result can be summarised as follows. Assume we start with an initial vortex-sheet configuration which consists of two inviscid fluids with density bounded below flowing smoothly past each other, where a strictly positive fixed coefficient of surface tension produces a surface tension force across the common interface, balanced by the pressure jump. We assume the fluids are modelled by the compressible Euler equations in three space dimensions with a very general equation of state relating the pressure, entropy and density in each fluid such that the sound speed is positive. Then, for a short time, which may depend on the initial configuration, there exists a unique solution of the equations with the same structure, that is, two fluids with density bounded below flowing smoothly past each other, where the surface tension force across the common interface balances the pressure jump. The mathematical approach consists of introducing a carefully chosen artificial viscosity-type regularisation which allows one to linearise the system so as to obtain a collection of transport equations for the entropy, pressure and curl together with a parabolic-type equation for the velocity. We prove a high order energy estimate for the non-linear equations that is independent of the artificial viscosity parameter which allows us to send it to zero. This approach loosely follows that introduced by Shkoller et al in the setting of a compressible liquid-vacuum interface. Although already considered by Shkoller et al, we also make some brief comments on the case of a compressible liquid-vacuum interface, which is obtained from the vortex sheets problem by replacing one of the fluids by vacuum, where it is possible to obtain a structural stability result even without surface tension.

Ce travail est motivé par une demande croissante d'informations quantitatives sur l'évolution du littoral. Nous avons étudié deux approches pour l'analyse de la dynamique sédimentaire. Les deux techniques aboutissent à la résolution de modèles non-locaux pour le fond. L'étude mathématique a porté sur l'analyse de l'existence et l'unicité de perturbations autour des ondes progressives solutions du modèle de Fowler. Nous avons montré que les solutions constantes de l'équation de Fowler sont instables. Pour la simulation numérique de ce modèle, nous avons dans un premier temps considéré des schémas aux différences finies explicites pour lesquels nous avons obtenu des critères de stabilité numérique. Dans un second temps, nous avons utilisé une approche par splitting de sorte à pouvoir résoudre la convection, puis la diffusion et l'anti-diffusion fractionnaire de façon exacte. Ensuite, il est apparu que nous pouvions utiliser les principes de minimisation pour décrire l'évolution d'un lit érodable sous l'action de l'eau où le fond est considéré comme une structure déformable de faible rigidité s'adaptant en minimisant une certaine fonctionnelle d'énergie. Il est intéressant de constater que cette seconde approche peut être liée à la première car elle débouche aussi sur une équation de type Exner avec un terme non-local. En nous inspirant du modèle morphodynamique non-local de Fowler, nous concluons cette thèse par une application exotique au traitement de signal où nous proposons une nouvelle méthode de filtrage. / This work is motivated by a growing demand for quantitative information on the evolution of the coastline.We have studied two approaches for the analysis of sand morphodynamics.Both techniques lead to the resolution of nonlocal models for the seabottom.The mathematical study focused on the analysis of the existence and uniqueness of perturbations around the travelling-waves solutions of the Fowler model. We have shown that constant solutions of Fowler's equation are unstable.For the numerical simulation of this model, we have first considered explicit finite difference schemes for which we got numerical stability criteria. We have next used an approach by splitting method in order to solve first the convection, then the diffusion/fractional anti-diffusion exactly. We have also used minimization principles to describe the evolution of an erodible bed sheared by a fluid flow where the seabed is considered as a deformable structure with low stiffness whichadapts itself by minimizing a certain energy functional. It is interesting to note that this secondapproach can be linked to the first one because it also leads to a new Exner equation with a nonlocal term for the flux. Inspired by Fowler's morphodynamical model, we conclude this dissertation with an unexpected application to signal processing.

Les premières étapes d’activation des lymphocytes T sont cruciales pour déterminer leur comportement, ainsi que leur prolifération. Ces étapes dépendent fortement des conditions initiales, particulièrement de l’avidité du récepteur du lymphocyte (TCR) pour le ligand spécifique provenant de l’antigène. La reconnaissance du virus entraine une séquence des réactions biochimiques mettant en œuvre de protéines membranaires et cellulaires. Le processus peut être mesuré par cytométrie en flux. On propose ici plusieurs modèles de différents niveaux de complexité. Ces modèles décrivent une relation entre la population de lymphocytes T et leurs composants intracellulaires et extracellulaires. Ils conduisent à des systèmes d’EDO et d’EDP dont la résolution permet d’étudier la dynamique de la densité de population des lymphocytes au cours du processus d'activation. En outre, différentes hypothèses sont proposées pour le processus d'activation des cellules filles après prolifération. Les équations de bilan de population (EBPs) sont résolues par une nouvelle méthode validée par une solution analytique quand elle existe, ou par comparaison à différentes méthodes numériques disponibles dans la littérature. L’avantage de cette nouvelle méthode est d’être utilisable dans certains cas où les méthodes classiques ne le sont pas. / The early steps of activation are crucial in deciding the fate of T-cells leading to the proliferation. These steps strongly depend on the initial conditions, especially the avidity of the T-cell receptor for the specific ligand and the concentration of this ligand. The recognition induces a rapid decrease of membrane TCR-CD3 complexes inside the T-cell, then the up-regulation of CD25 and then CD25–IL2 binding which down-regulates into the T-cell. This process can be monitored by flow cytometry technique. We propose several models based on the level of complexity by using population balance modeling technique to study the dynamics of T-cells population density during the activation process. These models provide us a relation between the population of T-cells with their intracellular and extracellular components. Moreover, the hypotheses are proposed for the activation process of daughter T-cells after proliferation. The corresponding population balance equations (PBEs) include reaction term (i.e. assimilated as growth term) and activation term (i.e. assimilated as nucleation term). Further the PBEs are solved by newly developed method that is validated against analytical method wherever possible and various approximate techniques available in the literature.