Hydrodynamics of Viscous Fluid

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.

Collection of Solved Nonlinear Problems for Remote Shaping and Patterning of Liquid Structures on Flat and Curved Substrates by Electric and Thermal Fields

Different numerical methods have been proposed for the solution of partial differential equations (PDE). Most of them are based on a variational principle which recasts the PDE into an equivalent integral equation. One of the most common principles is the Galerkin method, which has some specific disadvantages for some types of PDE. In this work an alternative variational principle, the least squares finite element method, will be tested with respect to its application for transient fluid-structure interaction problems. The accurracy of different formulations which were proposed for the Navier-Stokes equations in literature will be tested. In a next step these formulations will be coupled with a standard Galerkin approach for the structure. After that a new formulation for the linear equations of elastodynamics is developed and analysed with respect to its stability and accuracy. With this formulation it is possible do develop a pure least squares formulation for the strongly coupled fluid-structure problem. Finally the different formulations are tested with respect to their accuracy and efficiency.

Edge noise is generated if turbulence interacts with solid edges. Reduction of trailing edge noise of airfoils can be achieved by replacing the solid material at the trailing edge by inlays of porous permeable material. The acoustic benefit of approximately 6 dB of such treatment is known from experiments. Enroute to numerically optimized porous properties, this paper presents a first principle based Computational Aeroacoustics (CAA) method for predicting the acoustic effect of a porous NACA0012 trailing edge. In a hybrid two-step CFD/CAA procedure the turbulence statistics from a solution of the Volume Averaged Navier-Stokes (VANS) equations is used as a basis for the prediction of turbulent-boundary-layer trailing-edge noise (TBL-TEN). For the acoustic part of the calculation, the Acoustic Perturbation Equations (APE) are solved in the flow field. Inside the porous regions, a different set of governing equations, referred to as Linear Perturbation Equations (LPE) will be solved. The LPE represent a modified form of the Linearized Euler Equations (LEE) with the APE vorticity source term shifted to the right-hand side. The new set of equations is derived by volume averaging the Navier-Stokes equations and decomposing the flow variables into a time-averaged mean part and a fluctuating part and isolating the vorticity source term to the right-hand side of the momentum equation. The LPE are verified by an analytical solution. The simulation results of a NACA0012 airfoil geometry with and without porous trailing edge treatment are compared to wind tunnel measurements. The noise reduction effect of such a trailing edge treatment is successfully demonstrated.

Thermodynamic and Gasdynamic Aspects of a Boiling Liquid Expanding Vapour Explosion

Aim of this paper is to investigate the chemical reaction effects on transient MHD free convective flow of a viscous incompressible electrically conducting fluid along an inclined isothermal non-conducting plate in the presence of transverse magnetic field and viscous dissipation. The dimensionless governing equations are unsteady coupled and non-linear partial differential equations. An analytic method fails to give a solution. Hence an explicit finite difference scheme of DuFort - Frankel method, which is unconditionally stable is employed. The effect of the chemical parameter (Kr), magnetic parameter (M), buoyancy ratio parameter (N), Schmidt number (Sc), Eckert number (Ec) on the velocity, temperature and concentration fields are extensively discussed with the help of graphs.

Generalized multi-symplectic integrators for a class of Hamiltonian nonlinear wave PDEs

I. SOLITON ON A SLOPING BEACH AND RELATED PROBLEMS The problem of the behaviour of a soliton on a slowly varying beach is considered. It is shown that for a correct description, the full Boussinesq equations rather than a Korteweg-de Vries type approximation must be used. Using both energy conservation and two-timing expansions, the behaviour of the soliton is analysed. The slowly varying soliton is found not to conserve mass and momentum and it has been suggested that to conserve these quantities, both forward and reflected waves must be added behind the soliton, these waves being solutions of the linear shallow water equations. It is shown that to the order of approximation of the Boussinesq equations, only a forward wave (or tail) behind the soliton is necessary to fulfill mass and momentum conservation. A perturbed Korteweg-de Vries equation for which the perturbation adds energy to the soliton is considered. It is found that a tail is formed behind the soliton. The development of this tail into new solitons is analysed. II. MODULATED CAPILLARY WAVES An exact hodograph solution for symmetric and antisymmetric capillary waves on a fluid sheet (of possibly infinite thickness) has been previously found. Using this solution, an exact averaged Lagrangian for slowly varying capillary waves is calculated. Modulation equations can be found from this averaged Lagrangian, but due to the algebraic complexity of the equations, the limit of waves on a thin fluid sheet is considered. From the modulation equations, the stability of symmetric and antisymmetric capillary waves on a thin fluid sheet is found. The modulation equations for antisymmetric waves form a hyperbolic system and the simple wave solutions for this system are calculated. These simple wave solutions are interpreted physically.

This paper presents the development of a mathematical model of non-isothermal, unsteady flow of a real gas in deviation of Darcy's law and an algorithm for its numerical implementation. A formula for the flow velocity for non-linear binomial law of filtration was obtained. The formula includes a radial coordinate pressure derivative, thermo physical gas properties, and geophysical rock properties. There is also a partial differential equation and boundary condition at the well bottomhole for non-linear two-dimensional law of filtration. A method for simultaneous numerical integration for finite differences of differential equations shows an unsteady non-isothermic gas flow in the reservoir and energy conservation equation with initial and boundary conditions on the well bottom and the external drainage radius; functions of pressure and temperature at different time variances and their distribution along radial coordinates were calculated using a developed mathematical model. Graphs for these functions for linear and non-linear flow in reservoirs with different reservoir features for different well rates were built. Analysis of these graphs shows that taking into account the deviation of Darcy’s law of gas filtration in high permeable reservoirs for various flow rates does not make substantial corrections to the calculated values of pressure and temperature in comparison with the corresponding characteristics for flow. Calculations of bottomhole pressure - assuming that gas flow is steady - have calculation errors that are less than 0.2%, considered in the numerical implementation of task solution. With a low-permeable reservoir - when well production rates are low (for example, 169 thousand sm 3/day) - the bottomhole pressure decrease is small; therefore, a decrease of temperature at the bottomhole zone will also be small. The difference between the calculated values of these characteristics for linear and nonlinear filtering laws will also be negligible. Error of the value of bottomhole pressure calculated in assumptions that flow is steady does not exceed 0.7%. There is a deviation of Darcy's law in gas flow through the low-permeable reservoir at high well production flow rates (for example, 845 thousand sm 3/day), which leads to a substantial reduction of the reservoir pressure and temperature. Pressure stabilization for gas flow in the low-permeable reservoir does not depend on flow rate value. Error of value of bottomhole pressure, calculated in assumption that flow is steady exceeds 300%.

Coupling Mortar Finite Element and Boundary Element Methods for 2D Navier-Stokes Equations Courant Element: Before and After Iterative Methods for Solving Stiff Elliptic Problems Straight and Curved Finite Elements of Class C1 and Some Applications to Thin Shell Problems Exact Controllability to Solve the Helmholtz Equation with Absorbing Boundary Conditions Cubic Version of FEM in Elliptic Problems with Interfaces and Singularities Least Squares Mixed Finite Elements Turbulence Modelling in Finite Element Industrial Applications Necessary and Sufficient Conditions for the Numerical Approximation of a Partial Differential Equation Depending on a Small Parameter Efficient Solution Methods for Compressible Flow Computations Parallel Finite Volume Algorithms for Solving the Time-Domain Maxwell Equations on Nonstructured Meshes Coupling Between Nonlinear Maxwell and Heat Equations for an Induction Heating Problem: Modelling and Numerical Methods Solving the 3D Harmonic Maxwell Equations with Finite Elements, Lagrange Multipliers, and Iterative Methods Some Applications of the Hierarchic High Order MITC Finite Elements for Reissner-Mindlin Plates Domain Decomposition for Immiscible Displacement in Single Porosity Systems An Error Estimator for Nonconforming Approximations of a Nonlinear Problem Some Observations on Raviart-Thomas Mixed Finite Elements in p Extension for Parabolic Problems Mixed Finite Element Methods in Fluid Structure Systems A Black-Box Solver for the Solution of General Nonlinear Functional Equations by Mixed FEM A Remark on the Asymptotic Behaviour of Parabolic Variational Inequalities and Their Finite Element Approximation by the Courant Element Domain Decomposition vs. Adaptivity Material Optimization of Composites. Part Contents.

Multiple-soliton Solutions for Nonlinear Partial Differential Equations

The objective of this paper is to apply the generalized Riccati equation mapping method for constructing the exact traveling wave solutions of a nonlinear partial differential equation describing the dynamics of DNA modeling. This model consists of two long elastic homogeneous strands connected with each other by an elastic membrane. Hyperbolic and trigonometric function solutions of this model are obtained. Comparison between our new results and the well-known results are given. Key words: Nonlinear dynamics of DNA model, generalized Riccati equation mapping method, exact traveling wave solutions, generalized Riccati equation.

Summary A conjugate forced convection with viscous dissipation heat transfer problem of a second-grade visco-elastic fluid past a flat plate fin has been studied. Governing equations include heat conduction equation of the fin, and continuity equation, momentum equationand energy equationofthe fluid,were analyzed by a combination of a series expansion method, the similarity transformation and a second-order accurate finite-difference method. Solutions of a stagnation flow (β = 1.0) at the fin tip and a flat plate shape (wedge flow β = 0.0) on the fin surface were obtained by a generalized Falkner-Skan flow derivation. These solutionswere used to iterate with the heat conduction equation of the fin to obtain distributionsof the local convective heat transfer coefficient and the fin temperature. Ranges of dimensionless parameters, the Prandtl number (Pr), the elastic number (E), the viscous dissipation parameter (Ec) and the conduction-convection coefficient (Ncc) are from 0.1 to 100, 0.001 to 0.01, 0 to 0.1 and 0.05 to 2.0, respectively. Results indicated that Results indicated that elastic effect in the flow can increase the local heat transfer coefficient and enhance the heat transfer of a flat plate fin. Also, same as results from Newtonian fluid flow and conduction analysis of a flat plate fin, a better heat transfer is obtained with a larger Ncc, E, Ec and Pr.

Coal fired power plants contribute significantly to greenhouse gas emission, notably CO2. A novel method to effectively reduce the amount of CO2 emission is to cofire a high fraction of biomass and coal at oxygen enriched environments. However, this technique has not been demonstrated on a large-scale yet. The objective of this PhD research is to perform a detailed modeling study of combustion of a single biomass particle and to establish reduced models which can be used for describing the main characteristics of pyrolyzing and combusting single biomass particles to be employed in design codes of industrial furnaces. To accomplish the goals of this PhD thesis, the research has been carried out in two main stages. First, the sub-processes involved in the biomass combustion are identified and described with a one-dimensional mathematical model based on conservation of mass, energy and momentum. The model encompasses the kinetics of biomass pyrolysis, homogeneous reactions and heterogeneous char oxidation and gasification reactions, coupled with transient transport equations. Subsequently, this model is implemented in an in-house code and a comprehensive numerical study on pyrolysis and combustion of single biomass particles is conducted. The accuracy of the model is examined by comparing its predictions with several experimental data obtained from the literature on pyrolysis and combustion of various types of single biomass particles. The computer code based on the detailed model allows one to observe time and space evolution of several parameters including biomass and char densities, gaseous species mass fractions, porosity, internal pressure, mass flux of volatiles within the pores of the solid matrix, and temperature. The model is used for simulation of combustion of particles of three common shapes; i.e. slab, cylinder and sphere. The results of the detailed modeling study reveal that the combustion of a single biomass particle at the conditions of industrial furnaces (small particles and high heating conditions) consists of three main sub-processes: preheating, pyrolysis, and char oxidation. Therefore, any simplified/reduced particle model should account for these three processes. In the next stage of the project, simplified models are developed to predict the main characteristics of pyrolyzing and combusting single biomass particles. Initially, the preheating stage is modeled using a time and space integral method, which allows one to convert the partial differential form of the heat transfer equation into an algebraic equation. This treatment is then applied to model the pyrolysis process. Two possible regimes are identified: thermally thin and thermally thick particles. A model is established for both regimes, which consists of a set of algebraic equations. This treatment highly simplifies the pyrolysis model so that it can be used in practical applications, which may involve thousands or even millions of particles. The validation of the simplified preheating and pyrolysis models is carried out using various experiments and the results of the detailed model based on partial differential equations (PDEs). The char particle oxidation and gasification processes as the last stages of the particle combustion process, are modeled using the shrinking core approximation. The accuracy of this model is assessed using the experiments reported in past studies. The model is used to study the dynamics of biomass char combustion at oxy-fuel conditions. The effects of the main process parameters on the maximum particle temperature and burnout time are examined. It is found that oxy-fuel combustion with an oxygen mass fraction of 0.3 and higher may lead to a considerable reduction in particle temperature and burnout time compared to the conventional operating case with air as the gasifying agent. In the last stage of this PhD research, the simplified models of biomass particle pyrolysis and char combustion are combined to establish a reduced model for combustion of a single biomass particle. The accuracy of this simplified combustion model is examined using the measured data available in the literature as well as the results of the detailed model. As a conclusion, the simplified models developed in this thesis for pyrolysis and combustion of single biomass particles are efficient enough to capture the main process parameters, and computationally cheaper than the PDE -based models so that they can be used in the design codes of biomass furnaces.

This study provides an approximate analytical solution to the fractional vibration problem of thin plate governing anomalous motion of plate subjected to in-plane loads. The method of variable separable is employed to transform the fractional partial differential equations under consideration into a fractional ordinary differential equation in temporal variable and a bi-harmonic plate equation in spatial variable. The technique of conformable fractional derivative is utilized to solve the resulting fractional differential equation and the approach of finite sine integral transform method is used to solve the accompanying bi-harmonic plate equation. The deflection field which measures the transverse displacement of the plate is expressed in terms of product of Bessel and trigonometric functions via the temporal and spatial variables respectively. The obtained solution reduces to the solution of the free vibration problem of thin plate in literature. This work shows that conformable fractional derivative is an efficient mathematical tool for tracking analytical solution of fractional partial differential equation governing anomalous vibration of thin plates.