The work material in metal working operations always showssome kind of anisotropy. In order to simplify the theoreticalanalysis, especially considering bulk deformation processes,anisotropy is usually neglected and the material is assumed tobe isotropic. On the other hand, the analysis that consideredthe influence of anisotropy seldom incorporates the influenceof friction. For predicting the material flow during plasticdeformation and for predicting the final material properties ofthe product, adequate descriptions of both flow stress curvesand coefficients of friction have to be developed. In the present work a number of experimental methods fordetermining the anisotropy have been utilized and compared:Yield loci, strain ratios (R-values) and establishing flowstress-curves in different directions. The results show thatthe yield loci measurements are weak in predicting anisotropywhen the material strain hardening is different in differentdirections. It is concluded that also the strain ration(R-value) measurements are unreliable for describinganisotropy. The most trustable and useful results were foundfrom multi-direction determinations of the flow stresses. Three typical cases of ring upsetting conditions wereanalyzed by theory (3D-FEM) and experiments:     An anisotropic ring, oriented 900 to the axis ofrotational symmetrical anisotropy. The friction coefficientwas the same in all directions     An isotropic ring. The friction coefficient was differentin different directions     An anisotropic ring oriented 00 to the axis of rotationalsymmetrical anisotropy. The friction coefficient was the samein all directions The cases 1) and 2) reveal that the influence of anisotropyon the ring deformation is quite similar to that obtained bychanging the frictional condition. The case 3) exposes that ifthe material flow caused by anisotropy is incorrectly referredto friction, the possible error of the friction coefficient canbe as high as 80% for a pronounced anisotropic material. Amodified two-specimen method (MTSM) has been establishedaccording to an inverse method. Experiments were carried ascylinder upsetting. Here both ordinary cylinders were used aswell as so-called Rastegaev specimen. Also plane straincompression tests were utilized. The results show that MTSM isable to evaluate the validity of a selected mathematical modelwhen both the friction coefficient and the flow stress areunknown for a certain process. MTSM can also be used toestimate the friction coefficient and flow stress provided thatthe selected mathematical model is adequate. <b>Key words:</b>Anisotropy, friction coefficient, flowstress, modified two-specimen method and FE-analysis / NR 20140805

The present study has the objective of evaluating the effect of lubricant features on the friction coefficient by using three different lubricants (two synthetics and one mineral), which are used in automotive transmissions. In order to perform experiments, it was used a ball-on-disc system. The disc was manufactured with the same mechanical properties as transmission gears and the ball was made of 52100 SAE steel. The test parameters were defined based on a dynamic analysis conducted using \"ISOCAD\" which takes into account the geometry of the gear and vehicle dynamics (applied force and speed) during the tribological experiments. In order to set the temperature parameter, the lubricant testing temperature of a standard fuel economy was used. Experiments were conducted in two different lubrication conditions: i (starved) and ii (fully flooded). After the parameters were defined, it was possible to calculate the oil film thickness and, thus define the lubrication regime reached in each condition. Mixed and/or boundary lubrication was obtained in all the tests. Results showed the relationship between the speed and the specific oil film using a Khonsari and Masjedi formulation (2014), which takes into account the effect of surface roughness and, it is based on Downson and Higginson\'s (1981) formulation. The chemical analysis of the lubricant showed a relationship between friction coefficient and the additives used in lubricants. These results also showed the oil performance under different temperatures, speeds and loads. The best results were obtained for a synthetic lubricant. / O presente estudo tem o objetivo de avaliar o efeito de propriedades dos lubrificantes sobre o coeficiente de atrito. Para tanto foram testados três lubrificantes diferentes (dois sintéticos e um mineral), que são utilizados em transmissões automotivas. Os experimentos foram conduzidos utilizando um sistema tribológico denominado \"ball-on-disc\". Os discos foram fabricados com as mesmas propriedades mecânicas das engrenagens de transmissão, já as esferas foram fabricadas com aço SAE 52100. Os parâmetros de teste foram definidos com base na análise dinâmica conduzida utilizando o software \"ISOCAD\", que leva em conta a geometria da engrenagem e a dinâmica do veículo (força aplicada e velocidade) durante os ensaios tribológicos. Para definir os parâmetros de temperatura, foram utilizadas as mesmas temperaturas encontradas durante o ensaio de economia de combustível padronizado. Experimentos foram realizados em duas condições de lubrificação diferentes: i) fornecimento limitado de lubrificante, ou \"Starved\" e ii) condição de lubrificação imersa, ou \"Fully flooded\". Depois dos parâmetros definidos, foi possível calcular a espessura de filme de óleo, de forma a definir o regime de lubrificação alcançado em cada condição. Lubrificação mista e limítrofe foi obtida todas as condições de testes. Os resultados mostraram a relação entre a velocidade e o filme de óleo específico usando uma formulação de Khonsari e Masjedi (2014), que leva em conta o efeito da rugosidade da superfície e baseia-se na formulação de Downson e Higginson (1981). A análise química do lubrificante mostrou que existe uma relação entre o coeficiente de atrito e os aditivos utilizados nos lubrificantes. Estes resultados também mostram o desempenho dos lubrificantes em diferentes temperaturas, velocidades e cargas. Os melhores resultados foram obtidos para um lubrificante sintético.

Abstract Fretting is present where the contacting surfaces of mechanical parts are subjected to rubbing and an alternating stress, resulting in fatigue in the material. This is the situation between the hub and shaft in a shrink-fitted assembly. In practice, fretting is found in the same assembly with normal fatigue, and it occurs in the axle in a shrink-fit between an axle and a hub, while normal fatigue cracks are found outside the assembly. Fretting phenomena have been investigated by many authors in laboratories, but the dimensioning criteria for shrink-fit assemblies are insufficient for the construction engineer. Fretting causes a considerable reduction in the fatigue strength of a shrink-fit assembly, and failures caused by fretting are as numerous as failures resulting from normal fatigue. The purpose of this investigation was to examine the effect of hub material, contact pressure, slip amplitude and shaft geometry on fatigue strength. The goal of this investigation was to determine an optimal contact pressure and a favourable fillet radius and axle diameter ratio at which fretting failure can be avoided and maximum normal fatigue strength will be obtained. The torsional fatigue strength of shrink-fitted shaft couplings was estimated using tests that varied the material of the hub, the contact pressure, the geometry of the shaft and the torsional stress amplitude of the shrink-fitted assembly. Based on the information obtained from the test, aluminium as a hub material appears to produce little damage to a steel shaft, whereas bronze appears to cause damage and fatigue. Cast iron and steel caused a medium amount of damage. To increase contact pressure at the shoulder, hub overhang past the shoulder was used. These experiments showed that increasing the contact pressure decreased the slip amplitude, thereby reducing fretting. With low contact pressure, shaft fracturing began from fretting fatigue inside the hub, but with high contact pressure the shaft fractured at the fillet due to normal fatigue. Selecting the fillet radius according to the contact pressure makes it possible to dimension the shrink-fit shaft assembly to prevent fretting. The shrink-fitted shaft should be designed according to the normal fatigue limit, because fretting fatigue may occur when the number of load cycles exceeds 2×107. Fretting can be prevented by using a sufficiently high contact pressure and by choosing optimal ratios for the fillet radius and the diameter. To avoid fretting, the slip amplitude should be under 3 μm. This is achieved by using a contact pressure of over 100 N/mm2, calculated according to Lamé's theory. A method for dimensioning a shrink-fitted shaft with respect to fretting fatigue was presented based on a specific geometry (Ø 50 mm) and the materials used in the tests. The method was applied in verifying the fretting fatigue of a shrink-fitted shaft with a diameter of 300 mm. / Tiivistelmä Kitkaväsyminen esiintyy yleisesti, kun koneenosien kontaktipintoihin kohdistuu edestakainen jännitysamplitudi hankaavan liikkeen lisäksi johtaen materiaalin väsymiseen. Tällainen tilanne vallitsee navan ja akselin välissä kutistusliitoksessa. Kitkaväsyminen esiintyy käytännössä samassa kutistusliitoksessa kuin normaali väsyminenkin ja se ilmenee navan ja akselin välissä yleensä akselissa, kun taas tavanomainen väsyminen tapahtuu akselissa liitoksen ulkopuolella. Kitkaväsymistä on tutkittu paljonkin, mutta kutistusliitoksen mitoituskriteerit kitkaväsymisen osalta ovat puutteelliset. Kitkaväsyminen aiheuttaa merkittävän väsymislujuuden heikkenemisen sekä täten väsymisiän alenemisen kutistusliitoksessa ja sen aiheuttamat vauriot ovat määrältään samaa luokkaa tavanomaisen väsymisen kanssa. Tämän tutkimuksen tarkoituksena oli selvittää napamateriaalin, liitospaineen, liukuma-amplitudin ja geometrian vaikutus väsymislujuuteen. Tavoitteena oli määrittää optimaalinen liitospaine sekä sopivat olakkeen pyöristyssäde ja akselisuhde, jotta kitkaväsyminen voidaan välttää ja saavuttaa maksimaalinen normaali väsymislujuus. Kitkaväsymislujuutta väännön suhteen testattiin varioimalla napamateriaalia, liitospainetta, geometriaa ja jännitysamplitudia. Testien perusteella alumiininen napamateriaali sopi hyvin teräsakselin kanssa, kun taas pronssi aiheutti akseliin pintavaurion ja sitä kautta väsymisilmiön. Teräs- ja valurautanapa olivat näiden kahden materiaalin välissä. Liitospaineen nostamiseksi olakkeen reunalla käytettiin navan ylitystä olakkeen yli. Kokeet osoittivat kitkaväsymisen vähenevän korkeammilla liitospaineilla liukuman aletessa. Alhaisella liitospaineella akselin vaurioituminen alkoi kitkaväsymisenä navan sisältä, mutta hyvin korkealla liitospaineella murtuminen tapahtui olakkeesta tavanomaisena väsymisenä. Valitsemalla pyöristyssäde liitospaineen perusteella kutistusliitos on mitoitettavissa kitkaväsymistä vastaan. Kutistusliitos tulisi mitoittaa tavanomaisen väsymisen mukaan, koska kitkaväsymismurtuma voi tapahtua kuormanvaihtoluvulla yli 2×07. Kitkaväsyminen on ehkäistävissä käyttämällä riittävän korkeata liitospainetta sekä sopivaa olakkeen pyöristyssädettä yhdistyneenä oikeaan akselisuhteeseen. Kitkaväsyminen estyy kun liukuma-amplitudi on alle 3 μm. Tämä on saavutettavissa liitospaineella yli 100 N/mm2 laskettuna Lamén teorian mukaan. Tutkimuksessa on esitetty mitoitusmenetelmä kitkaväsymisen suhteen perustuen Ø 50 mm akselilla sekä käytetyillä materiaaleilla tehtyihin testeihin. Menetelmää on sovellettu kitkaväsymisen tarkasteluun kutistusliitokseen, jonka akselin halkaisija on 300 mm.

Aim of the diploma thesis is set to present a new method for evaluation of the friction coefficient of the wastegate valve system. A test fixture was designed and manufactured for this purpose. Simulations in ANSYS and MSC Adams support test results. The developed method provides an accurate and affordable solution for investigating of friction coefficient for various valves of turbochargers wastegate.

Axial piston pumps and motors are widely used in heavy-duty applications and play a fundamental role in hydrostatic and power split drives. The mechanical power losses in hydraulic piston pumps come from the friction between parts in relative motion. The improvement, albeit marginal, in overall efficiency of these components may significantly impact the global efficiency of the machine. The friction between slipper and swash plate is a functional key in an axial piston pump, especially when the pump (at low rotational speed or at partial displacement) works in the critical areas where the efficiency is low. The application of special surface treatments have been exploited in pioneering works in the past, trying different surface finishing or adding ceramic or heterogeneous metallic layers. The potential of structured coatings at nanoscale, with superhydrophobic and oleophobic characteristics, has never been exploited. Due to the difficulty to reproduce the real working conditions of axial piston pump slippers, it has been made a hydraulic test bench properly designed in order to compare the performance of nano-coated slippers with respect to standard ones. The nano-coated and standard slippers have been subjected to the following working conditions: a test at variable pressure and constant rotational speed, a test at constant pressure and variable rotational speed. The comparison between standard and nanocoated slippers, for both working conditions, shows clearly that more than 20% of friction reduction can be achieved using the proposed nano-coating methodology.

In order to ensure safe travel, railway vehicles must be stable under every condition along the track. Thus, a vehicle can be certified for operation only when it can fulfil certain criteria related to the ride stability. The stability of the vehicle is highly dependent on the wheel-rail friction coefficient: higher friction results in worse ride. So, to ensure a good evaluation of the stability, the friction should be high enough during tests. The same applies to the risk of wheel flange climbing. At the present time, the wheel-rail friction can not be measured directly but there are different procedures utilized to ensure that the conditions are suitable for testing the stability of the vehicle. In this study an algorithm is proposed to estimate the wheel-rail friction coefficient by using quantities which can be measured in reality. The algorithm is tried out in computer simulations. The algorithm has two parts: in Part 1 the friction coefficient is proposed to be equal to the ratio of the total creep force divided by the normal force; in Part 2 the total creep and spin creep are estimated to observe their correlation to the estimated friction. The contact angle in Part 1 is estimated by a contact point function. In the simulations, different conditions are tried. There are four horizontal radii: tangent track, R1300m, R1000m, and R400m. Three friction coefficients are used: 0.5, 0.4 and 0.3. In addition to this, track irregularities are included. A single vehicle is simulated in two modes: capable and incapable of passive radial steering. The track irregularities caused high values of the proposed estimated friction coefficient. The values in some instances were close or equal to the input friction coefficient of the simulation. Thus, if the highest values of the estimated friction were taken over a certain distance or time, the friction of the simulation could be approximated. In most cases, the total creep was following the trend of the estimated friction. The total creep and spin creep were used as a quality factor to determine how close the estimated friction was to the simulation’s friction. In this study when the total creep was greater than 0.006 and the spin creep was less than 1.0 m-1, the estimated friction was close to the input friction. The closeness was dependent on the simulation’s friction. Higher input friction resulted in larger deviation compared to lower friction. A sensitivity analysis has been performed by deliberately introducing errors in the position of the contact point and the angle of attack. The analysis shows that the errors are not critical when the contact point is close to the tread circle. When the contact point is close to the flange, a good measurement of the wheel profile and the contact point position required to obtain accurate results. On the other hand, the errors affect the friction estimate for high spin and low total creepage. These results are discarded by the algorithm, the influence of the errors is minimized.

Lack of drivers knowledge about the abrupt changes in pavement friction and poor performance of the vehicle stability, traction and ABS controllers on the low friction surfaces are the most important factors affecting car crashes. Due to its direct relation to vehicle stability, accurate estimation of tire-road characteristics is of interest to all vehicle and tire companies. Many studies have been conducted in this field and researchers have used different tools and have proposed different algorithms. One such concept is the Intelligent Tire. The application of intelligent tire in tire-road characterization is investigated in this study. Three different test setups were used in this research to study the application of the intelligent tires to improve mobility; first, a wheeled ground robot was designed and built. A Fuzzy Logic algorithm was developed and validated using the robot for classifying different road surfaces such as asphalt, concrete, grass, and soil. The second test setup is a portable tire testing trailer, which is a quarter car test rig installed in a trailer and towed by a truck. The trailer was equipped with different sensors including an accelerometer attached to the center of the tire inner-liner. Using the trailer, acceleration data was collected under varying conditions and a Neural Network (NN) algorithm was developed and trained to estimate the contact patch length, effective tire rolling radius and tire normal load. The third test setup developed for this study was an instrumented Volkswagen Jetta. Different sensors were installed to measure vehicle dynamic response. Additionally, one front and one rear tire was instrumented with an accelerometer attached to their inner-liner. Two intelligent tire based algorithms, a tire pressure estimation algorithm and a road condition monitoring algorithm, were developed and trained using the experimental data from the instrumented VW Jetta. The two-step pressure monitoring algorithm uses the acceleration signal from the intelligent tire and the wheel angular velocity to monitor the tire pressure. Also, wet and dry surfaces are distinguished using the acceleration signal from the intelligent tire and the wheel angular velocity through the surface monitoring algorithm. Some of the model based tire-road friction estimation algorithms, which are widely used for tire-road friction estimation, were also introduced in this study and the performance of each algorithm was evaluated in high slip and low slip maneuvers. Finally a new friction estimation algorithm was developed, which is a combination of experiment based and vehicle dynamic based approaches and its performance was also investigated. / PHD

The paper overviews the modes of vibration of the principal component parts of a brake and their contribution to system instability during noise generation. It is shown that both in-plane and out-of-plane vibration is present and that both can be related to the vibration of the pad. It is further shown that the pad and its region often provide a solution or 'fix' towards noise prevention and it is this area that forms the focus of this investigation. The collective evidence, proposals and associated theory are applied to real brake case studies when it is demonstrated that disc/pad interface 'spragging' may be the source of brake noise. Measurements of the position of the dynamic centre of pressure (CoP) support the theoretical predictions that a leading CoP induces brake noise. Design proposals are suggested that may be applied early in the design phase as a means to reduce the propensity of a brake to generate noise.

The topographical variations of the seas around Taiwan are great, which make the tides complicated. Taiwan is located in the juncture of the tropical and subtropical area. Geographically, it is located within the region of northwestern Pacific typhoon path. These seasonal and geographical situations causing Taiwan frequently threaten by typhoons during summer and autumn. In addition to natural disasters, the coastal area is over developed for the last few decades, which destroys the balance between nature and man. Storms and floods constantly threaten the lowland areas along the coast. An accurate and efficient storm surge model can be used to predict tides and storm surges. The model can be calibrated and verified with the field observations. Data measured by instruments at the tidal station constituting daily tidal variations and storm surge influences during typhoons. The model can offer both predictions to the management institutions and to the general public as pre-warning system and thus taking disaster-prevention measures. This study implements the numerical model, developed by Yu (1993) and Yu et al. (1994) to calculate the hydrodynamic in the seas around Taiwan. The main purpose of this study is to make a calibration and sensitivity analysis of the model parameters. Tidal gauge data around Taiwan coastal stations collected from June to October 2005 are used for the analysis and the comparison between the modeled data and the observations. Two steps have been taken for the model calibration and sensitivity analysis. First step is to calibrate the model for accurate prediction of the astronomical tide, and then the compound tide with meteorological influences. For the calibration of the astronomical tides, sensitivity analysis has been carried out by adjusting the horizontal diffusion coefficient and the bottom friction coefficients used in the model. The sensitivity of the time-step size used in the model and model grids fitted to coastlines are also checked. A depth dependent Chézy numbers are used in the model to describe bottom friction. The model has a better result when the Chézy value varied within 65 to 85. Modifying grids fitted to the coastline has improved the model results significantly. By improving the dynamic phenomenon brought about by the land features, the model calculation fits the real tidal phenomenon better. The analysis has shown that the model is less sensitive to the horizontal diffusion coefficient. Data from 22 tidal stations around Taiwan have been used for the comparisons. The maximum RMSE (root-mean-square error) is about 10 cm at WAi-Pu, whereas the minimum RMSE is about 1 cm for the stations along eastern coast. The calibration of the compound tide is divided into three cases. The first case is to calibrate the forecasted wind field. This has been done by comparing the forecasted wind field from the Central Weather Bureau with the satellite data obtained from QuikSCAT¡XLevel 3. The satellite wind speed has been applied to adjust the forecasted wind speed. The adjusted forecast wind field has shown improvement to the model predictions in the tidal stations south of Taichung, slightly improved in the eastern coast. The second case is tuning the drag coefficient on sea surface used by the hydrodynamic model. Several empirical formulas to describe the sea surface drag have been tested. The model result has shown little influence using various drag formulations. The third case is to single the influences by the meteo-inputs, i.e. the wind field and the atmospheric pressure. The tidal level is more sensitive to the variation of the atmospheric pressure through out the tests carried out during typhoon periods. The model simulation for 2006 using the best selected parameters has shown that the model is consisted with good stability and accuracy for both stormy and calm weather conditions.

Le titre de ma thèse de doctorat est "Equations de Stokes et de Navier-Stokes avec la con- dition de Navier", où j’ai considéré l’écoulement d’un fluide newtonien visqueux, incompressible dans un domaine borné de R3. L’écoulement du fluide est décrit par les équations bien connues de Navier-Stokes, données par le système suivant ∂t − ∆u + (u • ∇)u + ∇π = 0, div u = 0 dans Ω × (0, T )u • n = 0, 2[(Du)n]τ + αuτ = 0 sur Γ × (0, T )u(0) = u0 dans Ω (0.1) dans un domaine borné Ω ⊂ R3 de frontière Γ, éventuellement non simplement connexe, de classe C1,1. La vitesse initiale u0 et le coefficient de friction α, scalaire, sont des fonctions don- nées. Les vecteurs unitaires normal extérieur et tangents à Γ sont notés n et τ respectivement et Du = 1 (∇u + ∇uT ) est le tenseur des déformations. Les fonctions u et π décrivent respective- ment les champs de vitesses et de pression du fluide dans Ω satisfaisant la condition aux limites (0.1.2).Cette condition aux limites, proposée par H. Navier en 1823, a été abondamment étudiée ces dernières années, qui pour de nombreuses raisons convient parfois mieux que la condition aux limites de Dirichlet sans glissement : elle offre plus de liberté et est susceptible de fournir une solution physiquement acceptable au moins pour certains des phénomènes paradoxaux résultant de la condition de non-glissement, comme par exemple le paradoxe de D’Alembert ou le paradoxe de non-collision.Ma thèse comporte trois parties. Dans la première, je cherche à savoir si le problème (0.1) est bien posé en théorie Lp, en particulier l’existence, l’unicité de solutions faibles, fortes dans W 1,p(Ω) et W 2,p(Ω) pour tout p ∈ (1, ∞), en considérant la régularité minimale du coefficient de friction α. Ici α est une fonction, pas simplement une constante qui reflète les diverses propriétés du fluide et/ou de la frontière, ce qui nous permet d’analyser le comportement de la solution par rapport au coefficient de frottement.Utilisant le fait que les solutions sont bornées indépendamment de α, on montre que la solution des équations de Navier-Stokes avec la condition de Navier converge fortement vers une solution des équations de Navier-Stokes avec la condition de Dirichlet, correspondant à la même donnée initiale dans l’espace d’énergie lorsque α → ∞. Des résultats similaires ont été obtenus pour le cas stationnaire.Le dernier chapitre concerne les estimations pour le problème de Robin pour le laplacien : l’opérateur elliptique de second ordre suivant, sous forme divergentielle dans un domaine bornéΩ ⊂ Rn de classe C1, avec la condition aux limites de Robin a été considéré div(A∇)u = divf + F dans Ω, ∂u+ αu = f n + g sur Γ.∂n (0.2) Les coefficients de la matrice symétrique A sont supposés appartenir à l’espace V MO(R3). Aussi α est une fonction appartenant à un certain espace Lq . En plus de prouver l’existence, l’unicité de solutions faibles et fortes, nous obtenons une borne sur u, uniforme par rapport à α pour α suffisamment large, en norme Lp. Pour plus de clarté, nous avons étudié séparément les deux cas: l’estimation intérieure et l’estimation au bord. / My PhD thesis title is "Navier-Stokes equations with Navier boundary condition" where I have considered the motion of an incompressible, viscous, Newtonian fluid in a bounded do- main in R3. The fluid flow is described by the well-known Navier-Stokes equations, given by thefollowing system 1 )t − L1u + (u ⋅ ∇)u + ∇n = 0, div u = 01u ⋅ n = 0, 2[(IDu)n]r + aur = 0 in Q × (0, T )on Γ × (0, T ) (0.1) 11lu(0) = u0 in Qin a bounded domain Q ⊂ R3 with boundary Γ, possibly not connected, of class C1,1. The initialvelocity u0 and the (scalar) friction coefficient a are given functions. The unit outward normal and tangent vectors on Γ are denoted by n and r respectively and IDu = 1 (∇u + ∇uT ) is the rate of strain tensor. The functions u and n describe respectively the velocity2 and the pressure of a fluid in Q satisfying the boundary condition (0.1.2).This boundary condition, first proposed by H. Navier in 1823, has been studied extensively in recent years, among many reasons due to its contrast with the no-slip Dirichlet boundary condition: it offers more freedom and are likely to provide a physically acceptable solution at least to some of the paradoxical phenomenons, resulting from the no-slip condition, for example, D’Alembert’s paradox or no-collision paradox.My PhD work consists of three parts. primarily I have discussed the Lp -theory of well-posedness of the problem (0.1), in particular existence, uniqueness of weak and strong solutions in W 1,p (Q) and W 2,p (Q) for all p ∈ (1, ∞) considering minimal regularity on the friction coefficienta. Here a is a function, not merely a constant which reflects various properties of the fluid and/or of the boundary. Moreover, I have deduced estimates showing explicitly the dependence of u on a which enables us to analyze the behavior of the solution with respect to the friction coefficient.Using this fact that the solutions are bounded with respect to a, we have shown the solution of the Navier-Stokes equations with Navier boundary condition converges strongly to a solution of the Navier-Stokes equations with Dirichlet boundary condition corresponding to the sameinitial data in the energy space as a → ∞. The similar results have also been deduced for thestationary case.The last chapter is concerned with estimates for a Laplace-Robin problem: the following second order elliptic operator in divergence form in a bounded domain Q ⊂ Rn of class C1, withthe Robin boundary condition has been considered1div(A∇)u = divf + F in Q, 11 )u + u = f ⋅ n + g on Γ. (0.2) 2The coefficient matrix A is symmetric and belongs to V MO(R3). Also a is a function belonging to some Lq -space. Apart from proving existence, uniqueness of weak and strong solutions, we obtain the bound on u, uniform in a for a sufficiently large, in the Lp -norm. We have separately studied the two cases: the interior estimate and the boundary estimate to make the main idea clear in the simple set up.