Dissertations / Theses on the topic 'Slip boundary condition'

The no-slip boundary condition is the foundation of traditional lubrication theory. It says that fluid adjacent to a solid boundary has zero velocity relative to that solid surface. For most practical applications the no-slip boundary condition is a good model for predicting fluid behavior. However, recent experimental research has found that for special engineered surfaces the no-slip boundary condition is not applicable. Measured velocity profiles suggest that slip is occurring at the interface. In the present study, it is found that judicious application of slip to a bearings surface can lead to improved bearing performance. The focus of this thesis is to analyze the effect an engineered slip/no-slip surface could have on hydrodynamic bearing performance. A heterogeneous pattern is applied to the bearing surface in which slip occurs in certain regions and is absent in others. Analysis is performed numerically for both plane pad slider bearings and journal bearings. The performance parameters evaluated for the bearings are load carrying capacity, side leakage rate and friction force. Fluid slip is assumed to occur according to the Navier relation and the effect of a critical value for slip onset is considered.

This thesis describes the study of the hydrodynamic boundary condition at the solid-liquid interface using the colloidal probe Atomic Force Microscope. Quantitative comparison between measured lubrication forces and theoretical lubrication forces show that the measured forces agree with theory when the no slip boundary condition is employed. We measure an effective slip length of 0 ± 2 nm at shear rates up to 250,000 sec-1. Our results are consistent with the Taylor lubrication equation without the need to invoke a slip length fitting parameter. Our results are also consistent with molecular dynamic simulations that predict no slip at the shear rates that are currently experimentally accessible.

The main contributions of this thesis are an investigation of the effect of slip on dipole-wall collisions and a study of topological changes in two-dimensional bounded flows. The Navier-Stokes equations in the streamfunction-vorticity formulation are solved in both a channel domain, periodic in the streamwise direction and also in a disc domain, with either no-slip or Navier boundary conditions. Navier boundary conditions permit some fluid slip along the wall. The normal velocity component at the wall is always zero, but the tangential component of velocity is proportional to the rate of strain at the boundary, with the constant of proportionality identified as the slip length. Recently, Romain Nguyen van yen, Marie Farge, and Kai Schneider, ``Energy dissipating structures produced by walls in two-dimensional flows at vanishing viscosity'', Physical Review Letters, 106:184502, 2011 investigated the problem of a dipole incident on a rigid boundary using a volume penalisation method. The volume penalisation method approximates a no-slip boundary condition and intrinsically introduces some slip at the boundary, which vanishes as the Reynolds number increases. Their results indicate that energy dissipating structures persist in the vanishing viscosity limit. Here a similar problem was investigated using Fourier collocation techniques in the streamwise direction, and a compact finite difference method in the direction normal to the wall. The boundary conditions were enforced using a linear superposition technique called the influence matrix method, modified to treat Navier boundary conditions. For the no-slip boundary condition no evidence of energy dissipating structures was found in the limit as the viscosity approaches zero, and this result also holds for any fixed slip length. However when the slip length was taken to vary inversely with Reynolds number, the results of Nguyen van yen et al were recovered. To investigate the production of enstrophy at the wall it is useful to track the minima, maxima, and saddle points of the vorticity and streamfunction. As coherent structures of vorticity approach near the wall, vorticity is generated at the wall to satisfy the no-slip or Navier boundary conditions and injected into the domain. The flow throughout the domain may be classified into regions dominated by coherent vortices, and regions dominated by filamentary stretching by a condition called the Okubu-Weiss criterion. The Okubu-Weiss criterion can be derived by considering the linear stability of the stagnation points of the flow, and it can be considered as a measure of curvature of the streamfunction. The Gaussian curvature of both the streamfunction and vorticity can be used to directly classify the flow into hyperbolic and elliptical regions. The curvature of the streamfunction is essentially a modification of the Okubu-Weiss criterion and can be interpreted as a balance between vorticity and strain, but the curvature of the vorticity does not appear to have a simple physical interpretation. The Gauss-Bonnet theorem shows that the total Gaussian curvature of both vorticity and streamfunction vanishes for a doubly-periodic domain and for the channel domain. For a disc domain it is possible to derive similar conservation laws. In the channel domain and a disc domain, the total curvature of the streamfunction vanishes, but for the vorticity there is an additional contribution from the curvature of the boundary. The generation and merger of critical points in bounded flows was monitored in an observational study. In an unbounded domain, the well-known forward energy cascade forces energy to large spatial scales. Large numbers of small scale vortices will therefore merge into domain-filling structures. In bounded domains, the wall acts as a source of enstrophy which constantly injects small scale vortices that disturb the formation of organised, domain-filling circulation cells. Fayeza Al Sulti and Koji Ohkitani, ``Vortex merger and topological changes in two-dimensional turbulence'', Physical Review E, 86.1 016309, 2012 studied vortex merger in unbounded domains by counting the number of elliptic and hyperbolic critical points of vorticity and streamfunction. To identify and classify the critical points Al Sulti and Ohkitani used the zeros of gradient of the field and the eigenvalues of the Hessian matrix at those points. In this thesis a new algorithm is developed, based on estimating the the Poincaré-Hopf index near potential critical points. This algorithm avoids the difficulties of identifying the zeros of gradient of the field, instead only local minima are required. Results showing the motion of critical points in time and the variation in the number of critical points over the simulation for a channel geometry are presented. In addition to the overall theme of the effect of the boundary conditions on the interior flow, the influence matrix method has been improved into a powerful and efficient numerical method for studying viscous fluid flow. The extension to generalised boundary conditions also allows a careful comparison of the volume penalisation method, with its intrinsic approximation to no-slip boundary conditions, against methods that enforce the no-slip boundary conditions exactly.

This thesis describes measurements of the gas-solid flow boundary condition at moderate Knudsen number, i.e., where the dimensions of the flow are similar to the mean free path, and thus partial slip is expected. This regime has become more important with increased focus on nano-scale devices, but there is currently no consensus on how the slip length should vary for different solids and gases, or whether it can be controlled. In this thesis, I describe unambiguous measurements showing that partial slip occurs, that the slip length depends both on gas and solid, and that the slip length can be altered in situ. The slip length is determined from analysis of the vibration of a small sphere adjacent to a solid. I also describe applications of these findings both to the separation of gases, and to inhalants. The effect of water films, gas species, organic films, and electric fields on gas flow was studied. Water films had a large, but complex effect. On bare hydrophobilic glass, the tangential momentum accommodation coefficient (TMAC) for nitrogen on hydroxyl-terminated silica changed from 0.25 to 0.88 when the humidity changed from 0 to 98 %. On hydrophobized glass, TMAC changed from 0.20 to 0.56 in the same range. The effect of the gas on TMAC was measured for five different gases (helium, nitrogen, argon, carbon dioxide, hexafluoride sulfur) on octadecyltrichlorosilane-coated glass surfaces. A lower TMAC occurred for greater molar mass, and this trend was explained using a simple model representing both the gas and the monolayer by spheres. The existence of this gas-dependent difference in TMAC suggests that gases can be separated based on their collisions with surfaces. Methods for controlling the flow boundary condition were also developed by adsorbing monolayers on the solid, and altering the monolayers in situ. Both temperature and electric fields altered the boundary condition, and these changes were attributed to changes in the surface roughness. The effect of roughness was modeled with grooved surfaces. Possible applications of this effect of roughness include changing the flow of aerosol droplets for deeper delivery of therapeutic drugs into the lung.<br>Ph. D.

This study is aimed at developing modeling methodologies for simulating the flow of air and aerosol particles through fibrous filter media made up of micro- or nano-fibers. The study also deals with modeling particle deposition (due to Brownian diffusion, interception, and inertial impaction) and particle cake formation, on or inside fibrous filters. By computing the air flow field and the trajectory of airborne particles in 3-D virtual geometries that resemble the internal microstructure of fibrous filter media, pressure drop and collection efficiency of micro- or nano-fiber filters are simulated and compared with the available experimental studies. It was demonstrated that the simulations conducted in 3-D disordered fibrous domains, unlike previously reported 2-D cell-model simulations, do not need any empirical correction factors to closely predict experimental observations. This study also reports on the importance of fibers’ cross-sectional shape for filters operating in slip (nano-fiber filters) and no-slip (micro-fiber filters) flow regimes. In particular, it was found that the more streamlined the fiber geometry, the lower the fiber drag caused by a nanofiber relative to that generated by its micron-sized counterpart. This work also presents a methodology for simulating pressure drop and collection efficiency of a filter medium during instantaneous particle loading using the Fluent CFD code, enhanced by using a series of in-house subroutines. These subroutines are developed to allow one to track particles of different sizes, and simulate the formation of 2-D and 3-D dendrite particle deposits in the presence of aerodynamic slip on the surface of the fibers. The deposition of particles on a fiber and the previously deposited particles is made possible by developing additional subroutines, which mark the cells located at the deposition sites and modify their properties to so that they resemble solid or porous particles. Our unsteady-state simulations, in qualitative agreement with the experimental observations reported in the literature, predict the rate of increase of pressure drop and collection efficiency of a filter medium as a function of the mass of the loaded particles.

We consider the flow of two immiscible fluids in a thin inclined channel subject to gravity and a change in pressure. In particular, we focus on the effects of Navier slip along the channel walls on the long-wave linear stability. Of interest are two different physical scenarios. The first corresponds to two incompressible fluid layers separated by a sharp interface, while the second focuses on a more dense fluid below a compressible gas. From a lubrication analysis, we find in the first scenario that the system is stable in the zero-Reynolds number limit with the slip effects modifying the decay rate of the stable perturbation. In the case of the Rayeligh-Taylor problem, slip along the less dense fluid wall has a destabilizing effect. In the second scenario, fluid inertia is pertinent, and we find neutral stability criteria are not significantly affected with the presence of slip.

Efficiency is one of the key issues in numerical simulation of large-scale problems with complex 3-D geometry. Traditional domain based methods, such as finite element methods, may not be suitable for these problems due to, for example, the complexity of mesh generation. The Boundary Element Method (BEM), based on boundary integral formulations (BIE), offers one possible solution to this issue by discretizing only the surface of the domain. However, to date, successful applications of the BEM are mostly limited to linear and continuum problems. The challenges in the extension of the BEM to nonlinear problems or problems with non-continuum boundary conditions (BC) include, but are not limited to, the lack of appropriate BIE and the difficulties in the treatment of the volume integrals that result from the nonlinear terms. In this thesis work, new approaches and techniques based on the BEM have been developed for 3-D nonlinear problems and Stokes problems with slip BC. For nonlinear problems, a major difficulty in applying the BEM is the treatment of the volume integrals in the BIE. An efficient approach, based on the precorrected-FFT technique, is developed to evaluate the volume integrals. In this approach, the 3-D uniform grid constructed initially to accelerate surface integration is used as the baseline mesh to evaluate volume integrals. The cubes enclosing part of the boundary are partitioned using surface panels. No volume discretization of the interior cubes is necessary. This grid is also used to accelerate volume integration. Based on this approach, accelerated BEM solvers for non-homogeneous and nonlinear problems are developed and tested. Good agreement is achieved between simulation results and analytical results. Qualitative comparison is made with current approaches. Stokes problems with slip BC are of particular importance in micro gas flows such as those encountered in MEMS devices. An efficient approach based on the BEM combined with the precorrected-FFT technique has been proposed and various techniques have been developed to solve these problems. As the applications of the developed method, drag forces on oscillating objects immersed in an unbounded slip flow are calculated and validated with either analytic solutions or experimental results.

The flow properties of polydimethylsiloxane (PDMS) melts at room temperature were studied by measurement of lubrication forces using an Atomic Force Microscopy (AFM) colloidal force probe. A glass probe was driven toward a glass plate at piezo drive rates in the range of 12 â 120 μm/s, which produced shear rates up to ~10⁴ s-1. The forces on the probe and the separation from the plate were measured. Two hypotheses were examined: (1) when a hydrophilic glass is immersed in a flow of polymer melt, does a thin layer of water form at the glass surface to lubricate the flow of polymer and (2) when a polymer melt is subject under a shear stress, do molecules within the melt spatially redistribute to form a lubrication layer of smaller molecules at the solid surface to enhance the flow? To examine the effect of a water lubrication layer, forces were compared in the presence and the absence of a thin water layer. The presence of the water layer was controlled by hydrophobization of the solid. In the second part, the possibility of forming a lubrication layer during shear was examined. Three polymer melts were compared: octamethyltrisiloxane (OMTS, n = 3), PDMS (n _avg = 322), and a mixture of 70 weight% PDMS and 30 weight% OMTS. We examined whether the spatial variation in the composition of the polymer melt would occur to relieve the shear stress. The prediction was that the trimer (OMTS) would become concentrated in the high shear stress region in the thin film, thereby decreasing the viscosity in that region, and mitigating the shear stress.<br>Master of Science

Cette thèse est consacrée à l'analyse mathématique et numérique d'un problème d'interaction fluide-structure stationnaire, couplant un fluide newtonien, visqueux et incompressible, modélisé par les équations de Stokes 2D et une structure déformable, décrite par les équations d'une poutre 1D. Le fluide et la structure sont couplés via une condition aux limites de type frottement à l'interface.Dans l'étude théorique, nous montrons un résultat d'existence et unicité de solutions faibles, dans le cadre de petits déplacements, du problème de couplage fluide structure avec une condition de glissement de type Tresca en utilisant le théorème de point fixe de Schauder.Dans l'analyse numérique, nous étudions d'abord, l'approximation du problème de Stokes avec la condition de Tresca par une méthode d'éléments finis mixtes à quatre champs. Nous montrons ensuite une estimation d'erreur a priori optimale pour des données régulières et nous réalisons des tests numériques. Enfin, nous présentons un algorithme de point fixe pour la simulation numérique du problème couplé avec des conditions aux limites non linéaires<br>This PHD thesis is devoted to the theoretical and numerical analysis of a stationary fluid-structure interaction problem between an incompressible viscous Newtonian fluid, modeled by the 2D Stokes equations, and a deformable structure modeled by the 1D beam equations.The fluid and structure are coupled via a friction boundary condition at the fluid-structure interface.In the theoretical study, we prove the existence of a unique weak solution, under small displacements, of the fluid-structure interaction problem under a slip boundary condition of friction type (SBCF) by using Schauder fixed point theorem.In the numerical analysis, we first study a mixed finite element approximation of the Stokes equations under SBCF.We also prove an optimal a priori error estimate for regular data and we provide numerical examples.Finally, we present a fixed point algorithm for numerical simulation of the coupled problem under nonlinear boundary conditions

In most applications concerning a fluid flowing over a solid surface, the no-slip velocity condition was widely used because it is simple and produces the results in agreement with experiments. However, this dynamical boundary condition is not appropriate when the flow under consideration is at a micro or nano length scale.In order to model this effect at the macroscopic scale, the Navier boundary conditions have been introduced, with the slip length as a parameter. When the fluid is a gas, this length is related to the tangential momentum accommodation coefficient (TMAC) and the mean free path, according to the Maxwell model. The aim of this work is to systematically address this model using a multi-scale approach and to extend it by incorporating both the morphology and the anisotropy of a surface. The thesis consists of five chapters. In Chapter 1, the basics of the kinetic theory of gases, the Boltzmann equation and related solutions (Navier-Stokes-Fourier, Burnett, Grad, Direct Simulation Monte Carlo ...) are briefly presented. The models of gas-wall interaction and slip models introduced in the fluid mechanics are also recalled. The chapter ends with a description of the computational method used for the molecular dynamics simulations performed in this work. Chapter 2 is dedicated to the development of a simple technique to simulate the pressure driven flows. The principle is to rely on the atomistic formulas of the stress tensor (Irving Kirkwood, Method of Plane, Virial Stress) and to modify the periodic conditions by maintaining the difference between the kinetic energy of the ingoing and outgoing particles of the simulation domain. Several types of channels are studied with this technique. The results (temperature, velocity ...) are discussed and compared. Chapter 3 deals with the study of the gas-wall interaction potential by the ab-initio method. The code CRYSTAL 09 is used to obtain the potential between an atom of argon (Ar) and a surface of platinum (Pt) <111> as a function of distance. Then the gas-wall potential is decomposed into binary potential and approached by an analytic function. This function is then implemented in a MD code to simulate the gas-wall collisions and determine the TMAC coefficient. In Chapter 4, the effect of morphology is studied. The multi-body Quantum Sutton Chen (QSC) potential is used for Pt <100> solid and the binary potential proposed in the previous chapter for the Ar-Pt couple is employed. The QSC potential is needed to reproduce the surface effects that affect the final results. Different surfaces are treated : smooth, nanostructured surface and, random surface obtained by Chemical vapor deposition (CVD). The TMAC is determined using a generalized approach, i.e. depending on the angle of incident flux of gas atoms on the surface. The surface anisotropy and the scattering kernel are also examined. In Chapter 5, we propose a model of anisotropic slip for fluids based on accommodation tensor. The model is obtained by the analytical approximate calculations developed in the framework of the kinetic theory. We thus generalize Maxwell's equation by showing that the slip length tensor is directly related to the accommodation tensor. The model is in good agreement with the MD results. Thanks to our MD simulations, we develop a suitable technique for reproducing the anisotropy of the accommodation tensor. The thesis ends with a conclusion section in which we suggest some perspectives for a continuation of this work

A model for stress and strain accumulation in strike slip earthquake faults is presented in which a finite width cuboidal fault region is embedded between two cuboidal tectonic plates. Elasto-plastic continuum constitutive equations model the gouge in the fault and the tectonic plates are linear elastic solids obeying the generalised Hooke's law. The model predicts a velocity field which is comparable to surface deformations. The plastic behaviour of the fault material allows the velocities in the tectonic plate to increase to values which are independent of the distance from the fault. Both of the non-trivial stress and strain components accumulate most significantly in the vicinity of the fault. The release of these strains during a dynamic earthquake event would produce the most severe deformations at the fault which is consistent with observations and the notion of an epicenter. The accumulations in the model, however, are at depths larger than would be expected. Plastic strains build up most significantly at the base of the fault which is in yield for the longest length of time but additionally is subject to larger temperatures which makes the material more ductile. The speed of propagation of the elasto-plastic boundary is calculated and its acceleration towards the surface of the fault may be indicative of a dynamic earthquake type event.

Microfluidics has been the focal point of research in various disciplines due to its advantages of portability and cost effectiveness, and the ability to perform complex tasks with precision. In the past two decades microfluidic technology has been used to cool integrated circuits, for exoplanetary chemical analysis, for mimicking cellular environments, and in the design of specialized organ-on-a-chip devices. While there have been considerable advances in the complexity and miniaturization of microfluidic devices, particularly with the advent of microfluidic large-scale integration (mLSI) and microfluidic very-large-scale-integration (mVLSI), in which there are hundreds of thousands of flow channels per square centimeter on a microfluidic chip, there remains an actuation overhead problem: these small, complex microfluidic devices are tethered to extensive off-chip actuation machinery that limit their portability and efficiency. Insects, in contrast, actively and efficiently handle their respiratory air flows in complex networks consisting of thousands of microscale tracheal pathways. This work analytically and experimentally investigates the viability of incorporating some of the essential kinematics and actuation strategies of insect respiratory systems in microfluidic devices. Mathematical models of simplified individual tracheal pathways were derived and analyzed, and insect-mimetic PDMS-based valveless microfluidic devices were fabricated and tested. It was found that not only are these devices are capable of pumping fluids very efficiently using insect-mimetic actuation techniques, but also that the fluid flow direction and magnitude could be controlled via the actuation frequency alone, a feature never before realized in microfluidic devices. These results suggest that insect-mimicry may be a promising direction for designing more efficient microfluidic devices.<br>Ph. D.<br>Microfluidics or the study of fluids at the microscale has gained a lot of interest in the recent past due to its various applications starting from electronic chip cooling to biomedical diagnostic devices and exoplanetary chemical analysis. Though there has been a lot of advancements in the functionality and portability of microfluidic devices, little has been achieved in the improvement of the peripheral machinery needed to operate these devices. On the other hand insects can expertly manipulate fluids, in their body, at the microscale with the help of their efficient respiratory capabilities. In the present study we mimic some essential features of the insect respiratory system by incorporating them in microfluidic devices. The feasibility of practical application of these techniques have been tested, at first, analytically by mathematically modeling the fluid flow in insect respiratory tract mimetic microchannels and tubes and then by fabricating, testing and analyzing the functionality of microfluidic devices. The mathematical models, using slip boundary conditions, showed that the volumetric fluid flow through a trachea mimetic tube decreased with the increase in the amount of slip. Apart from that it also revealed a fundamental difference between shear and pressure driven flow at the microscale. The microfluidic devices exhibited some unique characteristic features never seen before in valveless microfluidic devices and have the potential in reducing the actuation overhead. These devices can be used to simplify the operating procedure and subsequently decrease the production cost of microfluidic devices for various applications.

On s'intéresse aux équations stationnaires de Stokes posées dans un domaine extérieur tridimensionnel décrivant l'écoulement d'un fluide visqueux et incompressible autour d'obstacle supposé borné. La particularité ici réside dans les conditions au bord de l'obstacle que nous avons imposées. En effet, nous supposons que l'obstacle a une certaine rugosité et par conséquent, le fluide n'adhère pas au bord de l'obstacle mais, au contraire, il existe une friction dont on suppose décrite par les conditions aux limites de type Navier. Ces dernières modélisent d'une part l'imperméabilité de l'obstacle et d'autre part le fait que la composante tangentielle de la vitesse du fluide sur l'obstacle est proportionnelle au tenseur des déformations. Ce problème a été bien étudié lorsqu'il est posé dans un domaine borné. Les espaces de Sobolev classiques fournissent, dans ce cas, un cadre fonctionnel adéquat pour une étude complète. Cependant lorsque le domaine n'est pas borné, ces espaces ne sont plus adaptés car il est nécessaire de décrire le comportement à l'infini des solutions. On choisit alors de poser le problème dans des espaces de Sobolev avec des poids polynomiaux qui précisent la croissance ou la décroissance des fonctions à l'infini. Dans ce travail, nous commençons par effectuer une analyse hilbertienne du problème. Le point-clé ici est d'établir des inégalités de type Korn avec poids afin d'obtenir la coercivité de la forme bilinéaire associée à la formulation variationnelle. Nous continuons par démontrer des résultats d'existence, d'unicité et de régularité de solutions fortes et très faibles. Enfin, nous étudions l'extension de certains résultats en théorie L^p<br>In this manuscript, we study the three-dimensional stationary Stokes equations set in a exterior domain. The problem describes the flow of a viscous and incompressible fluid past a bounded obstacle. The distinctif feature here relies on the fact that the obstacle is assumed to a rough boundary. As a result, the fluid may slip on the boundary of the obstacle and, to take into account this property, we use the Navier boundary conditions. On the one hand, They model the impermeability of the obstacle, and on the other hand, the fact that the tangential component of the fluid velocity on the obstacle is proportional to the stress tensor. This problem has been well studied when set in a bounded domain. The standard Sobolev spaces provides, in this case, an adequate functional framework for a complete study. Since in our case, the domain is unbounded, these spaces are not adapted since it is necessary to describe the behaviour of the solutions to infinity. Therefore, we choose to set the problem in weighted Sobolev spaces where the weights describe the behaviour at infinity of the function (growth or decay).In this work, we first start by performing the mathematical analysis in the Hilbert setting. The key point here is to establish variant weighted Korn’s inequalities in order to get the coercivity of the bilinear form associated to the variational formulation. Next, we proved the existence, uniqueness of strong and very weak solutions. Finally, we study the extension of some of thses results to a weightedL^p-theory

Cette thèse traite de l'utilisation de méthodes de projection pour la simulation de fluides à seuil, incompressibles et à densité variable. Elle se découpe en trois parties. La première partie est consacrée à la présentation du modèle mathématique, à la construction du schéma de semi-discrétisation en temps du modèle, ainsi qu'à l'analyse du schéma ainsi obtenu. Pour construire le schéma, nous avons d'une part adapté un schéma de type pas de temps fractionné existant pour les écoulements newtoniens au cas des écoulements à seuil et d'autre part, nous avons utilisé une formulation en terme de projection pour le tenseur des contraintes qui nous permet de résoudre le couplage entre la vitesse et la contrainte dans l'équation de conservation de la quantité de mouvement avec une méthode de type Uzawa. Nous démontrons mathématiquement la stabilité et la convergence du schéma, et on obtient une estimation d’erreur en temps d’ordre un pour la vitesse, la densité, la viscosité ainsi que le seuil de plasticité. La deuxième partie est consacrée à la discrétisation en espace et à l’implémentation du schéma. La discrétisation en espace est effectuée à l’aide d’une méthode de type volumes finis sur grilles décalées. L’implémentation de l’algorithme a été réalisée à partir d’un code Fortran 90 parallélisé et utilisant les bibliothèques PetsC et MPI. La dernière partie de cette thèse est consacrée à la réalisation de simulations numériques à l'aide du code de calcul que l’on a ainsi construit. Dans la configuration de l’instabilité de Rayleigh-Taylor, on réalise des simulations en faisant varier le seuil de plasticité, en décrivant l’évolution de la position des deux phases et la localisation des zones rigides. Dans la configuration de rupture de barrage, nous mettons en oeuvre les conditions aux limites de type glissement à seuil, plus réalistes physiquement que les conditions de Dirichlet, et nous comparons les résultats obtenus à l’aide de notre nouveau schéma à la littérature existante<br>This thesis deal with the use of projection methods for incompressible viscoplastic flows with a variable density. This manuscript is organized following three main lines. The first part is devoted to the mathematical model, the time-discretization of the model and the analysis of the numerical scheme. To construct the numerical scheme, on one side we adapt a time-stepping method already used for newtonian flows to viscoplastic flows and on the other side, we use a projection formulation of the stress tensor to solve the coupling between the velocity and the plastic part of the stress tensor in the momentum equation with an Uzawa-like method. Stability and error analysis of the numerical scheme are provided and a first-order estimate of the time error is derived for the velocity field, the density, the viscosity and the yield stress. The second part is devoted to the space discretization and the implementation of the scheme. A second-order cell-centred finite volume scheme on staggered grids is applied for the spatial discretization. The implementation of the numerical scheme has been performed using a Fortran 90 code and using the PetsC and MPI library The last part of the manuscript is devoted to numerical simulations. In the Rayleigh-Taylor instability configuration, we perform simulations by varying the yield stress, and describing the evolution of the interface and the localization of the rigid zones. In the dambreak configuration, we use Stick-Slip boundary conditions and we compare our results to the existing literature

abstract: This dissertation studies two outstanding microscale fluid mechanics problems: 1) mechanisms of gas production from the nanopores of shale; 2) enhanced mass flow rate in steady compressible gas flow through a micro-conduit. The dissertation starts with a study of a volumetric expansion driven drainage flow of a viscous compressible fluid from a small capillary and channel in the low Mach number limit. An analysis based on the linearized compressible Navier-Stokes equations with no-slip condition shows that fluid drainage is controlled by the slow decay of the acoustic wave inside the capillary and the no-slip flow exhibits a slip-like mass flow rate. Numerical simulations are also carried out for drainage from a small capillary to a reservoir or a contraction of finite size. By allowing the density wave to escape the capillary, two wave leakage mechanisms are identified, which are dependent on the capillary length to radius ratio, reservoir size and acoustic Reynolds number. Empirical functions are generated for an effective diffusive coefficient which allows simple calculations of the drainage rate using a diffusion model without the presence of the reservoir or contraction. In the second part of the dissertation, steady viscous compressible flow through a micro-conduit is studied using compressible Navier-Stokes equations with no-slip condition. The mathematical theory of Klainerman and Majda for low Mach number flow is employed to derive asymptotic equations in the limit of small Mach number. The overall flow, a combination of the Hagen-Poiseuille flow and a diffusive velocity shows a slip-like mass flow rate even through the overall velocity satisfies the no-slip condition. The result indicates that the classical formulation includes self-diffusion effect and it embeds the Extended Navier-Stokes equation theory (ENSE) without the need of introducing additional constitutive hypothesis or assuming slip on the boundary. Contrary to most ENSE publications, the predicted mass flow rate is still significantly below the measured data based on an extensive comparison with thirty-five experiments.<br>Dissertation/Thesis<br>Doctoral Dissertation Mechanical Engineering 2019

碩士<br>國立臺灣大學<br>機械工程學研究所<br>95<br>Based on the core-spreading vortex method developed by Leonard and the blobs-splitting-and-merging scheme developed by Huang, this thesis develops a new numerical method for two-dimensional viscous incompressible flows with solid boundaries. The no-penetration boundary condition is satisfied by placing a vortex sheet along the boundary, which strength must be adapted to cancel the slip velocity on the boundary induced by all the other flow components. The strength of the vortex sheet is computed in the present work by the constant panel method. To simulate the diffusion of the vortex sheet into the flow field as time goes on, Koumoutsakos’ analytical solution is employed, in which an effective vorticity flux is derived and used for solving the vorticity diffusion equation. The solution is then discretized into blobs (called “ -blobs”) in the vicinity of the boundary. Moreover, to prevent the vorticity from entering into the body, the concept of “residual vorticity” is introduced in the sense that partial circulation of the vortex sheet is remained at the boundary without being diffused into the flow field. Blobs very close to the wall are thus unnecessary. Moreover, blobs may move too close to the boundary because of advection errors or other numerical errors. It may cause serious fluctuations in evaluating the strength of the vortex sheet. In order to reduce the fluctuations, these near-wall blobs (NWB) are also manipulated in use of the concept of “residual vorticity”. Finally, we apply the so-developed solver to a simulation of the flow past an impulsively started circular cylinder at different Reynolds numbers. The simulation results are compared with previous experimental as well as numerical data. The validity and the accuracy of this newly developed Navier-Stokes solver are confirmed.

博士<br>國立中山大學<br>海洋環境及工程學系研究所<br>101<br>In the present study, a new and simple method for determining the slip velocity (also called the bed velocity) on the solid-fluid interface in the wave boundary layer is proposed. Based on experimental and numerical results, when waves travel over a rigid permeable seabed, a nonzero slip velocity exists at the solid-fluid interface. The defect of a small slip velocity has been found to occur in previous studies and is usually encountered in fluid-porous layer problems. In the wave-rigid permeable seabed problem, the slip effect depends on the properties of the seabed. The slip velocity boundary condition (SVBC) is one specification of the slip conditions and is usually applied to explain the slip phenomenon in a fluid-porous layer problem. However, the traditional SVBC or the slip velocity is only considered in a single flow, and the application of SVBC in harmonic motion is still an open problem that necessitates a simple formula for determining the slip velocity in realistic cases. The Stokes’ second problem and the slip length model (SLM) are applied to derive a new slip velocity and a slip factor in this paper. Both the permeability and the roughness of the seabed are chosen as the characteristic length of slip motion. The analytical solution shows that the new slip velocity depends on the wave period and the pressure gradient, and the slip factor is related to the wave Reynolds number, the permeability, and the roughness of the seabed. The resultant slip velocity shows good agreement with the experimental results. Using the slip factor to determine the bed shear stress in the laminar/turbulent-rough regimes is the second part in this study. A simple relationship is developed to theoretically estimate the wave friction factor in various flow regimes in the porous media based on the slip factor formula. The theoretical formula shows that the wave friction factor varies inversely with the relative bed roughness,A/ks , over a rough bed and that it can be conveniently determined if wave conditions and sediment parameters are known without using a specific regression formula deduced from experiments. A laboratory experiment that directly measures the wave-driven bed shear stress dominant in the turbulent regime with a permeable bed is used to examine the newly-derived relationship. In the laminar regime, the comparison demonstrates that the theoretical results determined by the proposed formula are in good agreement with existing measurements. In the turbulent-rough regime, the influence of eddy viscosity is considered in the present model and the zero-equation model is used to estimate an average eddy viscosity. The theoretical wave friction factor is reasonably close to the experimental measurement, and considerably better than that obtained by other existing regressions. It is also found that the wave friction factor in the small zone can be described by the present model, with comparisons showing that the slip factor theory can be extended to estimate the wave friction factor in the turbulent-rough regime. Additionally, the proposed relationship is demonstrated to be effectively used in an alternate rough bed. Experimental results further indicate that the wave friction factor in porous medium is affected by the permeability of the sediment. Based on many comparisons with previous results, it is concluded that the method provided by the present study can be applied for determining the slip velocity and bed shear stress and setting up the parameter in the real case and numerical model.

碩士<br>國立虎尾科技大學<br>機械設計工程系碩士班<br>106<br>Nowadays, Microfluidic channels have applications in a wide range of fields, therefore the momentum and thermal transport characteristics of microflows. This work mainly studies the influence of the slip boundary conditions on the electro-osmotic flow of power-law fluids in a microtube. The numerical results for velocity and temperature distributions, Nusselt number and friction coefficient are obtained by solving the potential field, momentum equation and energy equation. The governing parameters include the flow behavior index（n）,dimensionless slip length（）,dimensionless Joule heating parameter（S）and ratio of tube radius to Debye length（Z）. The results show that when the flow behavior index（n）decreases, the dimensionless velocity profile increases. For shear thinning fluids, the dimensionless velocity profile increases with increasing the ratio of tube radius to Debye length（Z）when is Z small, while the opposite trend is observed when Z is sufficiently large. When the dimensionless slip length（）decreases, the dimensionless velocity profile decreases. An increase in the value of Z may increase or decrease the dimensionless temperature, depending on the thermal boundary conditions, i.e. surface heating or cooling. When Z is sufficiently large, the effect of Z on the dimensionless temperature becomes insignificant. Also, singularities occur in the Nusselt number variations for surface cooling as Z value is sufficiently large. The friction coefficient increases with the increase of Z for small and moderate values of Z, but the friction coefficient decreases with increasing Z when it is large enough. Moreover, increasing the dimensionless slip length（）can effectively reduce the friction coefficient.

碩士<br>國立中央大學<br>數學系<br>102<br>In general, we usually impose the no-slip boundary condition when simulating the problem of fluid dynamics. But recently, some experimental evidences this condition is not applicable in small-scale system or other situations. Many researchers propose to use the slip boundary condition instead. Then the result would be consistent with real appearance. Thus, we speculate the typical appearance would change when we apply the slip boundary condition. Therefore, we assume there exist slip behavior. We simulate with slip boundary condition to observe the difference between no-slip. In this thesis, we first introduce the background of slip boundary condition and the model we used. Then we derive the variational formulation of the Navier-Stokes equation with the slip boundary condition and the resulting large, sparse nonlinear system of equations is solved by the parallel Newton-Krylov-Schwarz algorithm. We validate our parallel fluid code by considering a test case with an available analytical solution. We apply parallel Galerkin/least squares finite element flow code with the slip boundary condition to two benchmark problems -- lid-driven cavity flows and sudden expansion flows. We investigate numerically how the slip condition effects the physical behavior of the fluid flows, including the critical Reynolds number for the pitchfork bifurcation and the performance of the nonlinear and linear iterative methods for solving resulting linear sparse nonlinear system of equations.

Zhao, Xiaozheng = 滑移邊界條件下湍流熱對流的實驗研究 / 趙晓争.<br>"September 2010."<br>Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.<br>Includes bibliographical references (leaves 52-57).<br>Abstracts in English and Chinese.<br>Zhao, Xiaozheng = Hua yi bian jie tiao jian xia tuan liu re dui liu de shi yan yan jiu / Zhao Xiaozheng.<br>Abstract --- p.i<br>摘要 --- p.ii<br>Acknowledgement --- p.iv<br>Contains --- p.iv<br>List of Figures --- p.vii<br>List of Tables --- p.xi<br>Chapter 1 --- Introduction --- p.1<br>Chapter 1.1 --- Turbulence --- p.1<br>Chapter 1.2 --- Turbulent Rayleigh-Benard Convection --- p.2<br>Chapter 1.2.1 --- Physical Picture --- p.2<br>Chapter 1.2.2 --- Governing Equations and Characteristic Parameters --- p.5<br>Chapter 1.2.3 --- Nu Scaling --- p.7<br>Chapter 1.2.4 --- Boundary Layer --- p.8<br>Chapter 1.3 --- Motivations of the Present Work --- p.10<br>Chapter 2 --- Experimental Setup --- p.13<br>Chapter 2.1 --- The Convection Cell --- p.13<br>Chapter 2.2 --- Temperature Probe and Translation Stage --- p.15<br>Chapter 2.3 --- Calibration of the Thermistors --- p.17<br>Chapter 2.4 --- Data Acquisition Units --- p.18<br>Chapter 2.5 --- The Working Fluids --- p.19<br>Chapter 2.6 --- Heat Leakage Prevention --- p.21<br>Chapter 3 --- Heat Transfer and Thermal Boundary Layer Measurement --- p.23<br>Chapter 3.1 --- The Setup and Experimental Procedure --- p.23<br>Chapter 3.2 --- The Mean Temperature and Temperature Fluctuation Profiles across the Interfaces --- p.24<br>Chapter 3.2.1 --- Profiles across the Water-FC77 Interface --- p.24<br>Chapter 3.2.2 --- Profiles across the FC77-Mercury Interface --- p.27<br>Chapter 3.3 --- Nu Results --- p.29<br>Chapter 3.3.1 --- Results Obtained with Assumption of Pure Conduction --- p.30<br>Chapter 3.3.2 --- Results from Mean Temperature Profile --- p.32<br>Chapter 3.3.3 --- Comparison of the Two Methods --- p.33<br>Chapter 3.4 --- Boundary Layer Thickness --- p.37<br>Chapter 3.5 --- Summary --- p.39<br>Chapter 4 --- Influence of Flow in the Water (Mercury) Layer on the FC77 Layer --- p.41<br>Chapter 4.1 --- Experimental Setup --- p.41<br>Chapter 4.2 --- Main Results --- p.42<br>Chapter 4.3 --- Probability Density Function and Temperature Oscillation --- p.44<br>Chapter 4.4 --- Summary --- p.50<br>Chapter 5 --- Conclusions and Perspective --- p.51<br>Chapter 5.1 --- Conclusions --- p.51<br>Chapter 5.2 --- Perspective for Future Work --- p.52

碩士<br>國立虎尾科技大學<br>機械設計工程系碩士班<br>106<br>This paper mainly discusses the effect of electro-osmotic flow on the heat transfer between flat plates of non-Newtonian fluid under different boundary conditions. From the present formulation, the key parameters governing this problem include the flow behavior index, dimensionless slip length, dimensionless Joule heating parameters, the ratio of viscous dissipation to Joule heating, and the ratio of half channel width to Debye-length. For purely electro-osmotic flow, the flow and heat transfer characteristics are analyzed for a reasonable range of the aforesaid key parameters. For comparison purpose, results for Newtonian fluid are also presented in the analysis. It was found that the effect of dimensionless slip length on temperature trend is not noticeable, but a little increase or reduction in temperature profile was observed. When the ratio of half channel width to Debye-length increases, the temperature distribution range is reduced and the overall temperature rises for the case of surface heating. When the ratio of half channel width to Debye-length decreases, the temperature distribution range is enhanced and the overall temperature reduces for surface cooling. In the surface heating case, the dimensionless temperature decreases and the temperature trend increases as the fluid behavior index increases. For surface cooling, the dimensionless temperature increases and the temperature trend decreases with increasing the fluid behavior index. The reduction in the ratio of viscous heating to Joule heating tends to increase the Nusselt number for surface heating, while a reversed trend was found for surface cooling.

This thesis is divided in three main chapters devoted to the study of finite element approximations of fluid flows with special nonlinearities coming from boundary con- ditions. In Chapter 1, we consider the finite element approximations of steady Navier-Stokes and Stokes equations driven by threshold slip boundary conditions. After re-writing the problems in the form of variational inequalities, a fixed point strategy is used to show existence of solutions. Next we prove that the finite element approximations for the Stokes and Navier Stokes equations converge respectively to the solutions of each continuous problem. Finally, Uzawa’s algorithm is formulated and convergence of the procedure is shown, and numerical validation tests are achieved. Chapter 2 is concerned with the finite element approximation for the stationary power law Stokes equations driven by slip boundary conditions of “friction type”. It is shown that by applying a variant of Babuska-Brezzi’s theory for mixed problems, convergence of the finite element approximation formulated is achieved with classi- cal assumptions on the regularity of the weak solution. Solution algorithm for the mixed variational problem is presented and analyzed in details. Finally, numerical simulations that validate the theoretical findings are exhibited. In Chapter 3, we are dealing with the study of the stability for all positive time of Crank-Nicolson scheme for the two-dimensional Navier-Stokes equation driven by slip boundary conditions of “friction type”. We discretize these equations in time using the Crank-Nicolson scheme and in space using finite element approximation. We prove that the numerical scheme is stable in L2 and H1-norms with the aid of different versions of discrete Grownwall lemmas, under a CFL-type condition.<br>Thesis (PhD)--University of Pretoria, 2014.<br>lk2014<br>Mathematics and Applied Mathematics<br>PhD<br>Unrestricted

The fast-paced growth in microelectromechanical systems (MEMS), microfluidic fabrication, porous media applications, biomedical assemblies, space propulsion, and vacuum technology demands accurate and practical transport equations for rarefied gas flows. It is well-known that in rarefied situations, due to strong deviations from the continuum regime, traditional fluid models such as Navier-Stokes-Fourier (NSF) fail. The shortcoming of continuum models is rooted in nonequilibrium behavior of gas particles in miniaturized and/or low-pressure devices, where the Knudsen number (Kn) is sufficiently large. Since kinetic solutions are computationally very expensive, there has been a great desire to develop macroscopic transport equations for dilute gas flows, and as a result, several sets of extended equations are proposed for gas flow in nonequilibrium states. However, applications of many of these extended equations are limited due to their instabilities and/or the absence of suitable boundary conditions. In this work, we concentrate on regularized 13-moment (R13) equations, which are a set of macroscopic transport equations for flows in the transition regime, i.e., Kn≤1. The R13 system provides a stable set of equations in Super-Burnett order, with a great potential to be a powerful CFD tool for rarefied flow simulations at moderate Knudsen numbers. The goal of this research is to implement the R13 equations for problems of practical interest in arbitrary geometries. This is done by transformation of the R13 equations and boundary conditions into general curvilinear coordinate systems. Next steps include adaptation of the transformed equations in order to solve some of the popular test cases, i.e., shear-driven, force-driven, and temperature-driven flows in both planar and curved flow passages. It is shown that inexpensive analytical solutions of the R13 equations for the considered problems are comparable to expensive numerical solutions of the Boltzmann equation. The new results present a wide range of linear and nonlinear rarefaction effects which alter the classical flow patterns both in the bulk and near boundary regions. Among these, multiple Knudsen boundary layers (mechanocaloric heat flows) and their influence on mass and energy transfer must be highlighted. Furthermore, the phenomenon of temperature dip and Knudsen paradox in Poiseuille flow; Onsager's reciprocity relation, two-way flow pattern, and thermomolecular pressure difference in simultaneous Poiseuille and transpiration flows are described theoretically. Through comparisons it is shown that for Knudsen numbers up to 0.5 the compact R13 solutions exhibit a good agreement with expensive solutions of the Boltzmann equation.

Tese de doutoramento em Ciência e Engenharia de Polímeros e Compósitos<br>This thesis presents an intensive study on slip flows of Newtonian and Non-Newtonian fluids carried out both analytically and numerically. Industrial applications of these flows are found in classical industries such as in polymer processing (for instance in extrusion) and in more modern applications as in lab-on-chip devices. Analytical solutions for flows under slip, are presented for both Newtonian, inelastic Newtonian and Non-Newtonian flows. These analytical solutions were missing in the literature and are helpful for both theoretical analysis and computer code validation. Slip boundary conditions were implemented in a computational fluid dynamics code, based on the finite volume method framework. New techniques for their implementation were devised, allowing to obtain convergence for reasonable Weissenberg numbers, using an appropriate iterative procedure to couple velocity, pressure and stress. The numerical code was then used to simulate benchmark problems, such as the 4:1 contraction, 1:4 expansion and slip-stick flows under slip, usually found in polymer processing.. The theoretical study of development length for Newtonian and viscoelastic fluids was performed for a channel flow under slip. New correlations for predicting the development length in micro and macro channels were devised. Other part of these thesis was dedicated to the theoretical study of electro-osmotic flows of complex fluids, for which analytical solutions for simple flows under the influence of the linear and nonlinear Navier slip boundary conditions were devised.<br>Nesta tese é apresentado um estudo intensivo de escoamentos com escorregamento, envolvendo fluidos newtonianos e não-newtonianos. Este tipo de escoamentos pode ser encontrado na indústria, por exemplo no processo de extrusão e ainda em aplicações mais modernas, tais como aparelhos baseados no conceito “lab-on-chip”. São apresentadas soluções analíticas com escorregamento para fluidos newtonianos, e diversos fluidos não-newtonianos, tanto inelásticos como viscoelásticos. Estas soluções analíticas não existiam na literatura e são úteis tanto em termos teóricos assim como para a validação de códigos numéricos. As condições de fronteira de escorregamento foram implementadas num código numérico de mecânica dos fluidos computacional baseado no métodos dos volumes finitos. Foram desenvolvidas novas técnicas para a implementação destas condições de fronteira, permitindo obter convergência para valores razoáveis do número de Weissenberg, usando um processo iterativo para acoplar os campos de velocidades, pressões e tensões. Este código numérico foi depois usado para simular problemas de referência, tais como os escoamentos numa contracção súbita 4:1, numa expansão 1:4 e o escoamento “slip-stick”, sob a influência do escorregamento na parede. Estas geometrias são frequentemente utilizadas em processamento de polímeros, o que substancia a utilidade prática destas simulações. Foi feito um estudo numérico e teórico para descobrir o comprimento de desenvolvimento do escoamento num canal simples com escorregamento na parede, de fluidos newtonianos e viscoelásticos. Foram ainda propostas novas correlações para prever esse comprimento. Outra parte do trabalho foi dedicada ao estudo de electro-osmose, onde novas soluções analíticas para escoamentos simples sob a influência do modelo de escorregamento linear de Navier, são apresentadas.<br>Fundação para a Ciência e a Tecnologia (FCT) - SFRH/BD/37586/2007<br>Fundo Social Europeu Europeu (FSE) - Human Potential Operational Programme

Гудзь І. В. Наближене розв’язання крайових нелінійних інтегральних рівнянь фрикційного контакту з частковим проковзуванням циліндричного штампа та пружного півпростору: кваліфікаційна робота магістра спеціальності 111 "Математика" / наук. керівник Ю. М. Стрєляєв. Запоріжжя : ЗНУ, 2020. 60 с.<br>UA : Робота викладена на 60 сторінках друкованого тексту, містить 8 рисунків, 1 таблицю, 36 джерел, 1 додатків. Об’єкт дослідження: інтегральні рівняння, що описують напружено-деформований стан пружних тіл, які перебувають в умовах контактної взаємодії. Предмет дослідження: контакт циліндричного штампа та пружного півпростору. Мета роботи: застосувати метод крайових нелінійних інтегральних рівнянь до розв’язання задачі про фрикційний контакт кругового штампа з плоскою підошвою і пружного півпростору при дії на штамп нормального і дотичного навантаження.. Методи дослідження: метод крайових нелінійних інтегральних рівнянь В роботі викладено методику розв’язання квазістатичної контактної задачі теорії пружності за допомогою зведення цієї задачі до системи крайових нелінійних інтегральних рівнянь. Розглянуто алгоритм чисельного розв’язання отриманих інтегральних рівнянь. За допомогою представленого алгоритму отримано новий чисельний розв’язок задачі про контакт зі зчепленням і проковзуванням жорсткого циліндричного штампа з плоскою підошвою і пружного півпростору при нормальному і дотичному навантаженні штампа. Проаналізовано контактні напруження та конфігурацію зон зчеплення при поступовому збільшенні дотичного навантаження, а також умови початку повного проковзування штампа. Отримані результати можуть бути використані в подальших дослідженнях просторових задач контактної механіки.<br>EN : The work is presented on 60 pages of printed text, 8 figures, 1 table, 36 references, 1 supplements. The object of the study is contact of a cylindrical stamp and elastic half-space. The aim of the study is to apply the method of boundary nonlinear integral equations to solve the problem of frictional contact of a circular stamp with a flat sole and an elastic half-space under the action of a normal and tangential load on the stamp.. The methods of research are the method of boundary nonlinear integral equations. The method of solving the quasi-static contact problem of the theory of elasticity by reducing this problem to a system of boundary nonlinear integral equations is presented in the paper. The algorithm of numerical solution of the obtained integral equations is considered. Using the presented algorithm, a new numerical solution of the problem of contact with the adhesion and slippage of a rigid cylindrical stamp with a flat sole and an elastic half-space at normal and tangential loading of the stamp is obtained. The contact stresses and the configuration of the adhesion zones with a gradual increase in the tangential load, as well as the conditions of the beginning of complete slippage of the stamp are analyzed. The obtained results can be used in further studies of spatial problems of contact mechanics.

Qualitative properties of solutions to equations of fluid mechanics Mgr. Jakub Tichý Supervisor: doc. Mgr. Petr Kaplický, Ph.D. Department: Department of Mathematical Analysis Abstract This thesis is devoted to the boundary regularity of weak solutions to the system of nonlinear partial differential equations describing incompressible flows of a certain class of generalized Newtonian fluids in bounded domains. Equations of motion and continuity equation are complemented with perfect slip boundary conditions. For stationary generalized Stokes system in Rn with growth condi- tion described by N-function Φ the existence of the second derivatives of velocity and their regularity up to the boundary are shown. For the same system of equa- tions integrability of velocity gradients is proven. Lq estimates are obtained also for classical evolutionary Stokes system via interpolation-extrapolation scales. Hölder continuity of velocity gradients and pressure is shown for evolutionary generalized Navier-Stokes equations in R2 . Keywords Generalized Stokes and Navier - Stokes equations, incompressible fluids, perfect slip boundary conditions, regularity up to the boundary