A numerical method for the incompressible Navier-Stokes equations with application to two-phase flow

A finite difference numerical method is developed for the simulation of time-dependent incompressible Navier-Stokes equations As in the original projection method developed by Chorin, we first solve diffusion-convection equations to predict a intermediate velocity field which are then projected onto the space of divergence-free field. We integrate the diffusion-convection by θ implicit scheme and solve the pressure Poisson equation by successive over-relaxation method. The second order centered difference is employed to discretize both the convective and viscous terms We develop two-dimensional and three-dimensional computer programs in Fortran language Numerical results are presented We couple our inpressible Navier-Stokes solver with an interacting continuum model for two-phase flows with heat, mass transfer and phase change The Volume of Fluid method is employed in the present study Two example simulations are presented, convective melting of solid particles in a fluid under micro-gravity, and a three dimensional driven cavity problem

Modelling and simulation of the sea-landing of aerial vehicles using the Particle Finite Element Method

In this paper the Particle Finite Element Method (PFEM) is applied to the simulation of the sea-landing of an unmanned aerial vehicle (UAV). The problem of interest consists in modelling the impact of the vehicle against the water surface, analyzing the main kinematic and dynamic quantities (such as loads exerted upon the capsule at the moment of the impact). The PFEM, a methodology well-suited for free-surface flow simulation is used for modelling the water while a rigid body model is chosen for the vehicle. The vehicle under consideration is characterized by low weight. This leads to difficulties in modelling the fluid–structure interaction using standard Dirichlet–Neumann coupling. We apply a modified partitioned strategy introducing the interface Laplacian into the pressure Poisson's equation for obtaining a convergent FSI solution. The paper concludes with an industrial example of a vehicle sea-landing modelled using PFEM

A finite volume approach for the numerical analysis and solution of the Buckley-Leverett equation including capillary pressure

The study of petroleum recovery is significant for reservoir engineers. Mathematical models of the immiscible displacement process contain various assumptions and parameters, resulting in nonlinear governing equations which are tough to solve. The Buckley-Leverett equation is one such model, where controlling forces like gravity and capillary forces directly act on saturation profiles. These saturation profiles have important features during oil recovery. In this thesis, the Buckley-Leverett equation is solved through a finite volume scheme, and capillary forces are considered during this calculation. The detailed derivation and calculation are also illustrated here. First, the method of characteristics is used to calculate the shock speed and characteristics curve behaviour of the Buckley-Leverett equation without capillary forces. After that, the local Lax-Friedrichs finite-volume scheme is applied to the governing equation (assuming there are no capillary and gravity forces). This mathematical formulation is used for the next calculation, where the cell-centred finite volume scheme is applied to the Buckley- Leverett equation including capillary forces. All calculations are performed in MATLAB. The fidelity is also checked when the finite-volume scheme is computed in the case where an analytical solution is known. Without capillary pressure, all numerical solutions are calculated using explicit methods and smaller time steps are used for stability. Later, the fixed-point iteration method is followed to enable the stability of the local Lax-Friedrichs and Cell-centred finite volume schemes using an implicit formulation. Here, we capture the number of iterations per time-steps (including maximum and average iterations per time-step) to get the solution of water saturation for a new time-step and obtain the saturation profile. The cumulative oil production is calculated for this study and illustrates capillary effects. The influence of viscosity ratio and permeability in capillary effects is also tested in this study. Finally, we run a case study with valid field data and check every calculation to highlight that our proposed numerical schemes can capture capillary pressure effects by generating shock waves and providing single-valued saturation at each position. These saturation profiles help find the amount of water needed in an injection well to displace oil through a production well and obtains good recovery using the water flooding technique

Glosarium Matematika

273 p.; 24 cm

Complex numbers from 1600 to 1840

This thesis uses primary and secondary sources to study advances in complex number theory during the 17th and 18th Centuries. Some space is also given to the early 19th Century. Six questions concerning their rules of operation, usage, symbolism, nature, representation and attitudes to them are posed in the Introduction. The main part of the thesis quotes from the works of Descartes, Newton, Wallis, Saunderson, Maclaurin, d'Alembert, Euler, Waring, Frend, Hutton, Arbogast, de Missery, Argand, Cauchy, Hamilton, de Morgan, Sylvester and others, mainly in chronological order, with comment and discussion. More attention has been given tp algebraists, the originators of most advances in complex numbers, than to writers in trigonometry, calculus and analysis, who tended to be users of them. The last chapter summarises the most important points and considers the extent to which the six questions have been resolved. The most important developments during the period are identified as follows: (i) the advance in status of complex numbers from 'useless' to 'useful'. (ii) their interpretation by Wallis, Argand and Gauss in arithmetic, geometric and algebraic ways. (iii) the discovery that they are essential for understanding polynomials and logarithmic, exponential and trigonometric functions. (iv) the extension of trigonometry, calculus and analysis into the complex number field. (v) the discovery that complex numbers are closed under exponentiation, and so under all algebraic operations. (vi) partial reform of nomenclature and symbolism. (vii) the eventual extension of complex number theory to n dimensions

Numerical Approximate Methods for Solving Linear and Nonlinear Integral Equations

Integral equation has been one of the essential tools for various area of applied mathematics. In this work, we employed different numerical methods for solving both linear and nonlinear Fredholm integral equations. A goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study on numerical methods for solving integral equations. Integral equations can be viewed as equations which are results of transformation of points in a given vector spaces of integrable functions by the use of certain specific integral operators to points in the same space. If, in particular, one is concerned with function spaces spanned by polynomials for which the kernel of the corresponding transforming integral operator is separable being comprised of polynomial functions only, then several approximate methods of solution of integral equations can be developed. This work, specially, deals with the development of different wavelet methods for solving integral and intgro-differential equations. Wavelets theory is a relatively new and emerging area in mathematical research. It has been applied in a wide range of engineering disciplines; particularly, wavelets are very successfully used in signal analysis for waveform representations and segmentations, time frequency analysis, and fast algorithms for easy implementation. Wavelets permit the accurate representation of a variety of functions and operators. Moreover, wavelets establish a connection with fast numerical algorithms. Wavelets can be separated into two distinct types, orthogonal and semi-orthogonal. The preliminary concept of integral equations and wavelets are first presented in Chapter 1. Classification of integral equations, construction of wavelets and multi-resolution analysis (MRA) have been briefly discussed and provided in this chapter. In Chapter 2, different wavelet methods are constructed and function approximation by these methods with convergence analysis have been presented. In Chapter 3, linear semi-orthogonal compactly supported B-spline wavelets together with their dual wavelets have been applied to approximate the solutions of Fredholm integral equations (both linear and nonlinear) of the second kind and their systems. Properties of these wavelets are first presented; these properties are then utilized to reduce the computation of integral equations to some algebraic equations. Convergence analysis of B-spline method has been discussed in this chapter. Again, in Chapter 4, system of nonlinear Fredholm integral equations have been solved by using hybrid Legendre Block-Pulse functions and xiii Bernstein collocation method. In Chapter 5, two practical problems arising from chemical phenomenon, have been modeled as Fredholm- Hammerstein integral equations and solved numerically by different numerical techniques. First, COSMO-RS model has been solved by Bernstein collocation method, Haar wavelet method and Sinc collocation method. Second, Hammerstein integral equation arising from chemical reactor theory has been solved by B-spline wavelet method. Comparison of results have been demonstrated through illustrative examples. In Chapter 6, Legendre wavelet method and Bernoulli wavelet method have been developed to solve system of integro-differential equations. Legendre wavelets along with their operational matrices are developed to approximate the solutions of system of nonlinear Volterra integro-differential equations. Also, nonlinear Volterra weakly singular integro-differential equations system has been solved by Bernoulli wavelet method. The properties of these wavelets are used to reduce the system of integral equations to a system of algebraic equations which can be solved numerically by Newton's method. Rigorous convergence analysis has been done for these wavelet methods. Illustrative examples have been included to demonstrate the validity and applicability of the proposed techniques. In Chapter 7, we have solved the second order Lane-Emden type singular differential equation. First, the second order differential equation is transformed into integro-differential equation and then solved by Legendre multi-wavelet method and Chebyshev wavelet method. Convergence of these wavelet methods have been discussed in this chapter. In Chapter 8, we have developed a efficient collocation technique called Legendre spectral collocation method to solve the Fredholm integro-differential-difference equations with variable coefficients and system of two nonlinear integro-differential equations which arise in biological model. The proposed method is based on the Gauss-Legendre points with the basis functions of Lagrange polynomials. The present method reduces this model to a system of nonlinear algebraic equations and again this algebraic system has been solved numerically by Newton's method. The study of fuzzy integral equations and fuzzy differential equations is an emerging area of research for many authors. In Chapter 9, we have proposed some numerical techniques for solving fuzzy integral equations and fuzzy integro-differential equations. Fundamentals of fuzzy calculus have been discussed in this chapter. Nonlinear fuzzy Hammerstein integral equation has been solved by Bernstein polynomials and Legendre wavelets, and then compared with homotopy analysis method. We have solved nonlinear fuzzy Hammerstein Volterra integral equations with constant delay by Bernoulli wavelet method and then compared with B-spline wavelet method. Finally, fuzzy integro-differential equation has been solved by Legendre wavelet method and compared with homotopy analysis method. In fuzzy case, we have applied two-dimensional numerical methods which are discussed in chapter 2. Convergence analysis and error estimate have been also provided for Bernoulli wavelet method. xiv The study of fractional calculus, fractional differential equations and fractional integral equations has a great importance in the field of science and engineering. Most of the physical phenomenon can be best modeled by using fractional calculus. Applications of fractional differential equations and fractional integral equations create a wide area of research for many researchers. This motivates to work on fractional integral equations, which results in the form of Chapter 10. First, the preliminary definitions and theorems of fractional calculus have been presented in this chapter. The nonlinear fractional mixed Volterra-Fredholm integro-differential equations along with mixed boundary conditions have been solved by Legendre wavelet method. A numerical scheme has been developed by using Petrov-Galerkin method where the trial and test functions are Legendre wavelets basis functions. Also, this method has been applied to solve fractional Volterra integro-differential equations. Uniqueness and existence of the problem have been discussed and the error estimate of the proposed method has been presented in this work. Sinc Galerkin method is developed to approximate the solution of fractional Volterra-Fredholm integro-differential equations with weakly singular kernels. The proposed method is based on the Sinc function approximation. Uniqueness and existence of the problem have been discussed and the error analysis of the proposed method have been presented in this chapte