Finite element methods

High-order adaptive mesh refinement offers the potential for accurate and efficient resolution of problems in fluid dynamics and other fields where a wide range of length scales is present. A critical requirement for the interface closures used with these methods is stability in the context of hyperbolic systems of partial differential equations. In this study, a class of energy-stable high-order finite-difference interface closures is presented for grids with step resolution changes in one dimension. Asymptotic stability in time for these schemes is achieved by imposing a summation-by-parts condition on the interface closure, which is thus also nondissipative. Interface closures compatible with interior fourth- and sixth-order explicit, and fourth-order implicit centered schemes are presented. Validation tests include linear and nonlinear problems in one and in two dimensions with tensor-product grid refinement. A second class of stable high-order interface closures is presented for two-dimensional cell-centered grids with patch-refinement and step-changes in resolution. For these grids, coarse and fine nodes are not aligned at the mesh interfaces, resulting in hanging nodes. Stability is achieved by again imposing a summation-by-parts condition, resulting in nondissipative closures, at the cost of accuracy at corner interfaces. Interface stencils for an explicit fourth-order finite-difference scheme are presented for each geometry. Validation tests confirm the stability and accuracy of these closures for linear and nonlinear problems. The Richtmyer-Meshkov instability is investigated using a novel first-order perturbation of the two-dimensional Navier-Stokes equations about a shock-resolved base flow. The computational domain is efficiently resolved using the one-dimensional fourth-order interface scheme. Results are compared to analytic models of the instability, showing agreement with predicted asymptotic growth rates in the inviscid range, while significant discrepancies are noted in the transient growth phase. Viscous effects are found to be poorly predicted by existing models.

This research investigates a number of problems, related to Stokes flow and to heat flow. The Stokes flow is inspired by glass flow in the process of making bottles or jars. The heat flow is related to a heat conduction model problem, and a problem about hot-spots formation in the microwave heating. We will discuss the first problem. There are two phases during the industrial process of making glass, viz. the pressing phase and the blowing phase. We consider some mathematical aspects of the pressing phase. The motion of glass at temperatures above 6000C can be described by the Navier-Stokes equations. Since glass is a highly viscous fluid, those equations can be simplified to the Stokes equations. We use two different methods to solve these equations, viz. perturbation and operator methods. The perturbation method is based on the geometry being slowly varying. As a result, we obtain the velocity analytically. This result has a good agreement with numerical results based on finite element modelling. Using the velocity obtained we derive the formula for the force on the plunger. Next, we consider the operator method. Using this method, the Stokes equations can be transformed into an operator equation on the boundary ??? with a tangent vector field a on the boundary ??? as unknown. Solving this operator equation shows, that the solutions of the Stokes equations can be parameterized by aH, the harmonic extension of a to the interior of the domain ??. As an application, we present some full explicit solutions of the Stokes equations for several domains such as the interior and exterior of the unit ball and of the unit disk, an infinite strip, a half space, and a wedge. In the second problem, we consider the heat conduction problem inside two types of geometry, viz. slowly and slightly varying geometry. Using this problem, we show the difference between those geometries. An example that involves the boundary layers at the ends is presented. Finally, we consider a simplified model of the microwave heating of a one-dimensional unit slab. This slab consists of three layers that have different thermal conductivities. We consider only the steady state problem with Dirichlet boundary conditions and continuity of heat flux across the layers. Using a fundamental-mode approximation of eigenfunction expansion, we investigate the effect of thermal conductivity on the formation of hot-spots where the temperature increases catastrophically as a function of d, the amplitude of the applied electric field. First, we consider a unit slab geometry. In this geometry, we find the critical value dcr, for which slight changes in d yields a sudden jump to another stable solution, now with a much higher temperature. Next, we consider a unit slab consisting of three layers of material with different thermal conductivity (µ). We assume the inner layer has the smallest value of µ. We find the temperature in this layer is much higher than that in other layers. Then, we consider only the inner layer. For a given value of d and changing values of µ, we get a temperature jump near some values of µ. This jump shows that there is a critical value of µ and signifies the formation of a hot-spot.

The objective of this article is investigation of dynamic response of thick multilayer functionally graded (FG) beam under generalized dynamic forces. The plane stress problem is exploited to describe the constitutive equation of thick FG beam to get realistic and accurate response. Applied dynamic forces are assumed to be sinusoidal harmonic, sinusoidal pulse or triangle in time domain and point load. Equations of motion of deep FG beam are derived based on the Hamilton principle from kinematic relations and constitutive equations of plane stress problem. The numerical finite element procedure is adopted to discretize the space domain of structure and transform partial differential equations of motion to ordinary differential equations in time domain. Numerical time integration method is used to solve the system of equations in time domain and find the time responses. Numerical parametric studies are performed to illustrate effects of force type, graduation parameter, geometrical and stacking sequence of layers on the time response of deep multilayer FG beams.

Scalar hyperbolic balance laws in several space dimensions play a central role in this thesis. First, we deal with a new class of mixed parabolic–hyperbolic systems on all Rn: we obtain the basic well–posedness theorems, devise an ad hoc numerical algorithm, prove its convergence and investigate the qualitative properties of the solutions. The extension of these results to bounded domains requires a deep understanding of the initial boundary value problem (IBVP) for hyperbolic balance laws. The last part of the thesis provides rigorous estimates on the solution to this IBVP, under precise regularity assumptions. In Chapter 1 we introduce a predator–prey model. A non local and non linear balance law is coupled with a parabolic equation: the former describes the evolution of the predator density, the latter that of prey. The two equations are coupled both through the convective part of the balance law and the source terms. The drift term is a non local function of the prey density. This allows the movement of predators to be directed towards the regions where the concentration of prey is higher. We prove the well–posedness of the system, hence the existence and uniqueness of solution, the continuous dependence on the initial data and various stability estimates. In Chapter 2 we devise an algorithm to compute approximate solutions to the mixed system introduced above. The balance law is solved numerically by a Lax–Friedrichs type method via dimensional splitting, while the parabolic equation is approximated through explicit finite–differences. Both source terms are integrated by means of a second order Runge–Kutta scheme. The key result in Chapter 2 is the convergence of this algorithm. The proof relies on a careful tuning between the parabolic and the hyperbolic methods and exploits the non local nature of the convective part in the balance law. This algorithm has been implemented in a series of Python scripts. Using them, we obtain information about the possible order of convergence and we investigate the qualitative properties of the solutions. Moreover, we observe the formation of a striking pattern: while prey diffuse, predators accumulate on the vertices of a regular lattice. The analytic study of the system above is on all Rn. However, both possible biological applications and numerical integrations suggest that the boundary plays a relevant role. With the aim of studying the mixed hyperbolic–parabolic system in a bounded domain, we noticed that for balance laws known results lack some of the estimates necessary to deal with the coupling. In Chapter 3 we then focus on the IBVP for a general balance law in a bounded domain. We prove the well–posedness of this problem, first with homogeneous boundary condition, exploiting the vanishing viscosity technique and the doubling of variables method, then for the non homogeneous case, mainly thanks to elliptic techniques. We pay particular attention to the regularity assumptions and provide rigorous estimates on the solution.

Optimal Tubes: Geodesic Metric, Euler Flow, Moving Domain J.P. Zolesio Numerical Simulation of Pattern Formation in a Rotating Suspension of Non-Brownian Settling Particles Tsorg-Whay Pan and Roland Glowinski On the Homogenization of Optimal Control Problems on Periodic Graphs P.I. Kogut and G. Leugering Lift and Sedimentation of Particles in the Flow of a Viscoelastic Liquid in a Channel G.P. Galdi and V. Heuveline Modeling and Simulation of Liquid-Gas Free Surface Flows A. Caboussat, M. Picasso, and J. Rappaz Transonic Regular Reflection for the Unsteady Transonic Small Disturbance Equation Detail of the Subsonic Solution K. Jegdic, B.L. Keyfitz, and S. Canic Shape Optimization for 3D Electrical Impedance Tomography K. Eppler and H. Harbrecht Analysis of the Shape Gradient in Inverse Scattering P. Dubois and J.P. Zolesio Array Antenna Optimization L. Blanchard and J.P. Zolesio The Stokes Basis for 3D Incompressible Flow Fields G. Auchmuty Nonlinear Aeroelasticity: Continuum Theory-Flutter/Divergence Speed, Plate Wing Model A.V. Balakrishnan Differential Riccati Equations for the Bolza Problem Associated with Point Boundary Control of Singular Estimate Control Systems I. Lasiecka and A. Tuffaha Energy Decay Rates for the Semilinear Wave Equation with Nonlinear Localized Damping and Source Terms-An Intrinsic Approach I. Lasiecka and D. Toundykov Electromagnetic 3D Reconstruction by Level-Set with Zero Capacity Connecting Sets C. Dedeban, P. Dubois, and J.P. Zolesio Shape and Geometric Methods in Image Processing M. Dehaes and M. Delfour Topological Derivatives for Contact Problems J. Sokolowski and A. Zochowski The Computing Zoom J. Henry An Optimization Approach for the Delamination of a Composite Material with Non-Penetration M. Hintermuller, V.A. Kovtunenko, and K. Kunish Adaptive Refinement Techniques in Homogenization Design Method R.H.W. Hoppe and S.I. Petrova Nonlinear Stability of the Flat-Surface State in Faraday Experiment G. Guidoboni A Dynamical Programming Approach in Hilbert Spaces for a Family of Applied Delay Optimal Control Problems Giorgio Fabbri A Posteriori Error Estimates of Recovery Type for Parameter Estimation Problem in Linear Elastic Problem T. Feng, M. Gulliksson, and W. Liu Tube Derivative of Non-Cylindrical Shape Functionals and Variational Formulations R. Dziri and J.P. Zolesio A Stochastic Riccati Equation for a Hyperbolic-Like System with Point and/or Boundary Control C. Hafizoglu

The pulse-tube refrigerator (PTR) is a rather new device for cooling down to extremely low temperatures, i.e. below 4 K. The PTR works by the cyclic compression and expansion of helium that flows through a regenerator made of porous material, a cold heat exchanger, a tube, a hot heat exchanger and an orifice, in series. In a Stirling-type PTR compression and expansion are generated by a piston. The compression increases the temperature of the helium in the tube and makes it flow towards the orifice; the expansion decreases the temperature and makes the helium flow backwards to the regenerator. The net effect of warmer helium flowing in one direction and colder helium in the opposite direction is that of cooling power at the cold heat exchanger. Three PTRs are inter-connected aiming to obtain the desired 4 K lowest temperature. The conservation laws of mass, momentum and energy, and an equation of state, are simplified using asymptotic analysis based on low Mach-numbers. The regenerator is modelled one-dimensionally with Darcy’s law for flow resistance. The tube is modelled either one-dimensionally without resistance or two-dimensionally with axisymmetric laminar viscous flow. The heat transfer in the porous medium of the regenerator and in the solid tube wall is taken into account. The gas can be either ideal or real. All the material properties, including viscosity and conductivity, are taken temperature and pressure dependent. Three single-stage PTRs are connected with the regenerators in series and the tubes in parallel and six flow possibilities at the junctions are considered. Three by-passes (double-inlets) are used to enhance and tune the performance. The governing equations are numerically solved with a finite-difference method of nominally second-order accuracy in space and time. Pressure correction, flux limiter, 1D-2D connections and domain decomposition are the keywords here. Special attention is paid to suitable initial conditions, high resolution in the boundary layers and to the correct calculation of the three-way junctions in multi-stage PTRs. The model describes the fluid dynamics and thermodynamics of the pulse tubes and regenerators. The heat exchangers are assumed to be ideal in the whole analysis. The equation of state for real gas and other real properties of gas and regenerator material, which are temperature and/or pressure dependent, are applied in the three-stage PTR which works with extreme low temperatures, where the ideal gas law does not hold. The numerical methods require special attention. Typically for flow problems we deal with various length scales. Straight forward discretisation will result in unnecessary fine grids and therefore unacceptable computational time. We developed robust and efficient algorithms to deal with boundary layer problems. The employed domain decomposition technique allows for using coarse grids in areas where the solution does not change significantly. It also decouples a larger system into smaller ones, leading to smaller complexities. The objective of computing accurately and efficiently the steady oscillatory flow and heat transfer in a PTR has been achieved. In particular the tiny viscous and thermal boundary layers of a PTR operating at high frequency (20 Hz) could be resolved. The simulated three-stage PTR was able to cool down close to 5 K. The developed software is intended for use in design and optimisation of multi-stage PTRs.

We consider the solution of elliptic partial differential equations with random input data. In particular, we are interested in the mean and the correlation of the solution of these partial differential equations. Once the correlation is available, the variance can be computed from its diagonal. If the dependence of the solution on the random input data is linear and the mean of the input data is available, the mean of the solution can be computed whenever the corresponding partial differential equation can be solved. When the correlation of the input data is available, the correlation of the solution is given as the solution of a higher dimensional partial differential equation in the product domain. A Galerkin discretization yields a matrix equation with a typically densely populated right-hand side. This and the higher dimension of the problem makes the equation prohibitively expensive to solve. Since existing approaches to these correlation equations are known to struggle for roughly correlated input data, we use hierarchical matrices and their corresponding arithmetic to represent and solve the matrix equation. The regularity assumptions required for hierarchical matrices can be justified on sufficiently smooth domains. The feasibility of the approach is illustrated for finite element and boundary element discretizations and several specialities of hierarchical matrices for finite element discretizations are discussed. An almost linear scaling with respect to the dimension of the used finite element spaces is verified. The nonlinear dependence of the solution on the random input data is exemplarily discussed for the case of random domains. Extending previous perturbation approaches, we can linearize the problem and can compute the mean and the correlation up to third order accuracy in almost linear time. We discuss that these spaces can become even fourth order accurate under certain circumstances. A full convergence analysis is presented, which shows that higher order boundary elements are required for the solution of the problem. Therefore, we develop a fast multipole method for higher order boundary element methods on parametric surfaces.

Accurate prediction of loads on mechanical couplings is crucial in assessment of loads on coupled structures, and in reliable prediction of the motions of coupled bodies. This thesis presents a numerical articulation method, which is capable of modeling various mechanical couplings. The proposed numerical method relies on contact elements. The first objective of the work is to prove the eligibility of contact elements in representation of mechanical couplings, and to highlight the shortcomings and positive aspects of contact elements for this representation. The second objective of the thesis is to study the influence of different hydrodynamic, kinematic, and coupling models on the motion behavior of articulated bodies and coupling forces. To accomplish these objectives extensive and systematic studies on a twofold pushing convoy are conducted. The convoy bodies are interconnected with hinge joints. The hinges are modeled by means of kinematic constraints, and with contact elements. Frequency and time domain boundary element methods with linearized motions of equation, as well as a Reynolds-averaged Navier-Stokes (RANS) code coupled with a nonlinear kinematic solver are used to compute the body motion responses and coupling forces. The simulation results are compared to model basin measurements. It is shown that the contact element model can simulate mechanical couplings efficiently and it provides a suitable method to idealize free and suppressed modes at articulation locations. Furthermore, it is shown, that RANS in association with the nonlinear equation of motion and kinematically constrained joints, as well as boundary element methods in adherence to the contact element model provide more accurate prediction of articulation forces, and motion responses of coupled bodies. However, frequency and time domain boundary element methods with kinematically constrained joints still deliver agreeable and reliable results. Moreover, a case study is presented, in which a river-sea pushing convoy is applied in time domain numerical simulations. The convoy bodies are interconnected with prismatic joints and are free to heave and pitch relative to each other. Prismatic joints are modeled by means of contact elements. The case study demonstrates a successful application of contact elements in simulation of complexly articulated multibody systems. Furthermore, the findings obtained in this work contribute sustainably to the numerical approaches in assessment of hydrodynamic and articulation loads on mechanically coupled floating bodies.

This paper presents a surface contact static stress of a spur gear system combined with dynamic characteristic using transient Finite Element Method (FEM). Traditionally, the static stress analysis is done separately with dynamic properties due to limitation of complex equation and avoiding of error occurred. However, in this paper, static stress information is combined with the dynamic mechanism due to the time consuming during the design and analysis stage. A transient FEM analysis is carried out to formulate and solve large systems of algebraic equations in order to obtain a relationship between the contact parameter and the kinematics function. The methodology of the research is started with static stress analysis on tooth surface contact of a pair of the spur gear. Finite element modeling is run by choosing a certain static condition. The loading conditions are applied suitable with the gear mechanism. Degree of freedom controlled is based on the transmission system. The process is repeated until diagnosing work is satisfied. The result of the surface contact stress is visualized at each condition. Modeling of spur gear system is continued by combining stress analysis with dynamic characteristic via transient finite element method. Analysis of gear mechanism is obtained by investigate the stress distribution on real time application. Time range is set at the beginning of the analysis. Duration of the analysis is depended on a time frame chosen. By the transient FEM analysis, the stress occur at each step of the work cycle is performed. Results of the kinematics functions are derived and qualitative kinematics variations due to contact changes in time-step domain is identified. The simulation results from static and transient FEM are compared due to the validating procedure. The finite element results are in good agreement compared with the theory calculation.

This paper presents a numerical investigation for laminar forced convection flow of a radiating gas in a rectangular duct with a solid element that makes a backward facing step. The fluid is treated as a gray, absorbing, emitting and scattering medium. The governing differential equations consisting the continuity, momentum and energy are solved numerically by the computational fluid dynamics techniques. Since the present problem is a conjugate one and both gas and solid elements are considered in the computational domain, simultaneously, the numerical solution of Laplace equation is obtained in the solid element for temperature calculation in this area. Discretized forms of these equations are obtained using the finite volume method and solved by the SIMPLE algorithm. The radiative transfer equation (RTE) is also solved numerically by the discrete ordinate method (DOM) for computation of the radiative term in the gas energy equation.The streamline and isotherm plots in the gas flow and the distributions of convective, radiative and total Nusselt numbers along the solid-gas interface are presented. Besides, the effects of radiation conduction parameter and also solid to gas conduction ratio as two important parameters on thermo hydrodynamic characteristics of such thermal system are explored. It is revealed that the radiative Nusselt number on the interface surface is much affected by RC parameter but the radiation conduction parameter has not considerable effect on the convective Nusselt number. Comparison between the present numerical results with those obtained by other investigators for the case of non-conjugate problems shows good consistency.

The entry of digital signal processors in modern control systems not only allows for extended diagnostics compared to analog systems but also for sophisticated and tricky extensions of the control algorithms. With modern DSP- and FPGA-technology, the processing speed of digital systems is no longer inferior to analog systems in many applications. A higher degree of digitalization leads to an increased complexity of the systems and hence to higher requirements on their operators. The focus of research and development in the field of high frequency control has changed in the last few years and moved towards the direction of software development and complexity management. In the presented thesis, a frame for an automation concept of modern high frequency control systems is developed. The developed automation is based on the concept of finite state machines (FSM), which is established in industry for years. A flexible framework was developed, in which procedures communicate using standardized interfaces and can be exchanged easily. With that, the developer of high frequency control components as well as the operator on shift shall be empowered to improve and adapt the automation to changed conditions without special programming skills required. Along the automation concept a number of algorithms addressing various problems were developed which satisfy the needs of modern high frequency control systems. Among the developed and successfully tested algorithms are the calibration of incident and reflected wave of resonators without antennas, the fast adaptive compensation of repetitive errors, the robust estimation of the phase advance in the control loop and the latency adjustment for the rejection of instabilities caused by passband modes. During the development of the resonator theory, high value was set on the usability of the equation in algorithms for high frequency control. The usage of the common nomenclature of control theory emphasizes the underlying mathematical structures of the equations. Several physical limitations and requirements, for example the limits of the vector sum calibration, were newly and adequately calculated based on the developed theory. The linear accelerator of the Free-Electron Laser in Hamburg (FLASH) served as the main platform for testing of the algorithms and concepts. The developed automation, in particular the flexible and transparent framework and methods for the reduction of the complexity of the various communication channels (quantization) is not only suited for high fequency control but also for other aspects of an accelerator and beyond.

Machine Learning is a modern and actively developing field of computer science, devoted to extracting and estimating dependencies from empirical data. It combines such fields as statistics, optimization theory and artificial intelligence. In practical tasks, the general aim of Machine Learning is to construct algorithms able to generalize and predict in previously unseen situations based on some set of examples. Given some finite information, Machine Learning provides ways to exract knowledge, describe, explain and predict from data. Kernel Methods are one of the most successful branches of Machine Learning. They allow applying linear algorithms with well-founded properties such as generalization ability, to non-linear real-life problems. Support Vector Machine is a well-known example of a kernel method, which has found a wide range of applications in data analysis nowadays. In many practical applications, some additional prior knowledge is often available. This can be the knowledge about the data domain, invariant transformations, inner geometrical structures in data, some properties of the underlying process, etc. If used smartly, this information can provide significant improvement to any data processing algorithm. Thus, it is important to develop methods for incorporating prior knowledge into data-dependent models. The main objective of this thesis is to investigate approaches towards learning with kernel methods using prior knowledge. Invariant learning with kernel methods is considered in more details. In the first part of the thesis, kernels are developed which incorporate prior knowledge on invariant transformations. They apply when the desired transformation produce an object around every example, assuming that all points in the given object share the same class. Different types of objects, including hard geometrical objects and distributions are considered. These kernels were then applied for images classification with Support Vector Machines. Next, algorithms which specifically include prior knowledge are considered. An algorithm which linearly classifies distributions by their domain was developed. It is constructed such that it allows to apply kernels to solve non-linear tasks. Thus, it combines the discriminative power of support vector machines and the well-developed framework of generative models. It can be applied to a number of real-life tasks which include data represented as distributions. In the last part of the thesis, the use of unlabelled data as a source of prior knowledge is considered. The technique of modelling the unlabelled data with a graph is taken as a baseline from semi-supervised manifold learning. For classification problems, we use this apporach for building graph models of invariant manifolds. For regression problems, we use unlabelled data to take into account the inner geometry of the input space. To conclude, in this thesis we developed a number of approaches for incorporating some prior knowledge into kernel methods. We proposed invariant kernels for existing algorithms, developed new algorithms and adapted a technique taken from semi-supervised learning for invariant learning. In all these cases, links with related state-of-the-art approaches were investigated. Several illustrative experiments were carried out on real data on optical character recognition, face image classification, brain-computer interfaces, and a number of benchmark and synthetic datasets.

In previous work, a complete dynamic inflow model for flow above the rotor disk has been developed; and numerical results have been presented in both the frequency domain and the time domain. In this paper, we extend the inflow model to all three components of inflow below the disk. The inflow model, which is expressed in terms of a finite number of states, is derived in a mathematically rigorous way. The essence of the extension is that if one computes the co-states of the inflow equations (along with the normal states), then one can find the velocity in the hemisphere below the plane of the rotor disk (including the velocity within the wake) with accuracy equal to that of the flow above the rotor plane. The derivation is for the case of general skew angle. Numerical comparisons with exact solutions for the z-component of flow in axial flow––given for some special cases––illustrate the effectiveness of the new model. The simulations also illustrate that the model is valid for either the frequency domain or the time domain. Nomenclature n m a : real part for cosine induced inflow expansion coefficients n m b : imaginary part for cosine induced inflow expansion coefficients [ ] D :

Introduction Generalized Transport Phenomena Approach to Problem Analysis General Content Conservation Laws and Field Equations Concentrations, Velocities, and Fluxes Thermodynamics of Purely Viscous Fluid Mixtures Conservation of Mass for a One-Component System Conservation of Mass for a Mixture Modification of Field Equations for Mass Transfer Conservation of Linear Momentum for One-Component Systems Conservation of Linear Momentum for a Mixture Conservation of Moment of Momentum for One-Component Systems Conservation of Moment of Momentum for a Mixture Strategies for the Solution of Mass Transfer Problems Boundary Conditions Definitions Jump Balances for Mass Conservation Jump Balances for Linear Momentum Conservation Postulated Boundary Conditions at Phase Interfaces Boundary Conditions in the Absence of Mass Transfer Utilization of Jump Balances Additional Comments on Boundary Conditions Boundary Conditions and Uniqueness of Solutions Constitutive Equations Constitutive Principles First-Order Theory for Binary Systems Combined Field and Constitutive Equations for First-Order Binary Theory First-Order Theory for Ternary Systems Special Second-Order Theory for Binary Systems Viscoelastic Effects in Flow and Diffusion Validity of Constitutive Equations Parameters in Constitutive Equations General Approach in Parameter Determination Diffusion in Polymer-Solvent Mixtures Diffusion in Infinitely Dilute Polymer Solutions Diffusion in Dilute Polymer Solutions Diffusion in Concentrated Polymer Solutions - Free-Volume Theory for Self-Diffusion Diffusion in Concentrated Polymer Solutions - Mutual Diffusion Process Diffusion in Crosslinked Polymers Additional Properties of Diffusion Coefficients Special Behaviors of Polymer-Penetrant Systems Volumetric Behavior of Polymer-Penetrant Systems Sorption Behavior of Polymer-Penetrant Systems Antiplasticization Nonequilibrium at Polymer-Penetrant Interfaces Mathematical Apparatus Basic Definitions Classification of Second-Order Partial Differential Equations Specification of Boundary Conditions Sturm-Liouville Theory Series and Integral Representations of Functions Solution Methods for Partial Differential Equations Separation of Variables Method Separation of Variables Solutions Integral Transforms Similarity Transformations Green's Functions for Ordinary Differential Equations Green's Functions for Elliptic Equations Green's Functions for Parabolic Equations Perturbation Solutions Weighted Residual Method Solution Strategy for Mass Transfer Problems Proposed Solution Methods Induced Convection Solutions of a General Set of Mass Transfer Problems Mixing of Two Ideal Gases Steady Evaporation of a Liquid in a Tube Unsteady-State Evaporation Analysis of Free Diffusion Experiments Dissolution of a Rubbery Polymer Bubble Growth from Zero Initial Size Stability Behavior and Negative Concentrations in Ternary Systems Analysis of Impurity Migration in Plastic Containers Efficiency of Green's Function Solution Method Mass Transfer in Tube Flow Time-Dependent Interfacial Resistance Laminar Liquid Jet Diffusion Analysis Analysis of the Diaphragm Cell Dissolved Organic Carbon Removal from Marine Aquariums Unsteady Diffusion in a Block Copolymer Drying of Solvent-Coated Polymer Films Flow and Diffusion Past a Flat Plate with Solid Dissolution Gas Absorption in Vertical Laminar Liquid Jet Utilization of Polymers in Drug Delivery Gas Absorption and Diffusion into a Falling Liquid Film Perturbation Solutions of Mass Transfer Moving Boundary Problems Dissolution of a Plane Surface of a Pure Gas Phase Bubble Dissolution Singular Perturbations in Moving Boundary Problems Dropping Mercury Electrode Sorption in Thin Films Numerical Analysis of Mass Transfer Moving Boundary Problems Diffusion and Reaction Design of a Tubular Polymerization Reactor Transport Effects in Low-Pressure CVD Reactors Solution of Reaction Problems with First-Order Reactions Plug Flow Reactors with Variable Mass Density Bubble Dissolution and Chemical Reaction Danckwerts Boundary Conditions for Chemical Reactors Transport in Nonporous Membranes Assumptions Used in the Theory for Membrane Transport Steady Mass Transport in Binary Membranes Steady Mass Transport in Ternary Membranes Unsteady Mass Transport in Binary Membranes Phase Inversion Process for Forming Asymmetric Membranes Pressure Effects in Membranes Analysis of Sorption and Desorption Derivation of a Short-Time Solution Form for Sorption in Thin Films Sorption to a Film from a Pure Fluid of Finite Volume A General Analysis of Sorption in Thin Films Analysis of Step-Change Sorption Experiments Integral Sorption in Glassy Polymers Integral Sorption in Rubbery Polymers Oscillatory Diffusion and Diffusion Waves Dispersion and Chromatography Formulation of Taylor Dispersion Problem Dispersion in Laminar Tube Flow for Low Peclet Numbers Dispersion in Laminar Tube Flow for Long Times Dispersion in Laminar Tube Flow for Short Times Analysis of an Inverse Gas Chromatography Experiment Effects of Pressure Gradients on Diffusion: Wave Behavior and Sedimentation Wave Propagation in Binary Fluid Mixtures Hyperbolic Waves Dispersive Waves Time Effects for Parabolic and Hyperbolic Equations Sedimentation Equilibrium Viscoelastic Diffusion Experimental Results for Sorption Experiments Viscoelastic Effects in Step-Change Sorption Experiments Slow Bubble Dissolution in a Viscoelastic Fluid Transport with Moving Reference Frames Relationships Between Fixed and Moving Reference Frames Field Equations in Moving Reference Frames Steady Diffusion in an Ultracentrifuge Material Time Derivative Operators Frame Indifference of Material Time Derivatives Frame Indifference of Velocity Gradient Tensor Rheological Implications Appendix: Vector and Tensor Notation General Notation Conventions Vectors Tensors Results for Curvilinear Coordinates Material and Spatial Representations Reynolds' Transport Theorem

The entropy generation analysis of non-Newtonian fluid in rotational flow between two concentric cylinders is examined when the outer cylinder is fixed and the inner cylinder is revolved with a constant angular speed. The viscosity of non-Newtonian fluid is considered at the same time interdependent on temperature and shear rate. The Nahme law and Carreau equation are used to modeling dependence of viscosity on temperature and shear rate, respectively. The viscous dissipation term is adding elaboration to the formerly highly associate set of governing motion and energy equations. The perturbation method has been applied for the highly nonlinear governing equations of base flow and found an approximate solution for narrowed gap limit. The effect of characteristic parameter such as Brinkman number and Deborah number on the entropy generation analysis is investigated. The overall entropy generation number decays in the radial direction from rotating inner cylinder to stationary outer cylinder. The results show that overall rate of entropy generation enhances within flow domain as increasing in Brinkman number. It, however, declines with enhancing Deborah number. The reason for this is very clear, the pseudo plastic fluid between concentric cylinders is heated as Brinkman number increases due to frictional dissipation and it is cooled as Deborah number increases which is due to the elasticity behavior of the fluid. Therefore, to minimize entropy need to be controlled Brinkman number and Deborah number.