Existence and Uniqueness of BVPs Defined on Semi-Infinite Intervals: Insight from the Iterative Transformation Method

This work is concerned with the existence and uniqueness of boundary value problems defined on semi-infinite intervals. These kinds of problems seldom admit exactly known solutions and, therefore, the theoretical information on their well-posedness is essential before attempting to derive an approximate solution by analytical or numerical means. Our utmost contribution in this context is the definition of a numerical test for investigating the existence and uniqueness of solutions of boundary problems defined on semi-infinite intervals. The main result is given by a theorem relating the existence and uniqueness question to the number of real zeros of a function implicitly defined within the formulation of the iterative transformation method. As a consequence, we can investigate the existence and uniqueness of solutions by studying the behaviour of that function. Within such a context the numerical test is illustrated by two examples where we find meaningful numerical results.Comment: 28 pages, 7 figures and 4 tables. arXiv admin note: substantial text overlap with arXiv:2003.07971, arXiv:1212.505

A two-step hybrid block method with fourth derivatives for solving third-order boundary value problems

Abstract This manuscript proposes an implicit two-step hybrid block method which incorporates fourth derivatives, for solving linear and non-linear third-order boundary value problems in ODEs. The derivation of the present method is based on collocation and interpolation techniques, and the convergence analysis of the new strategy is proved to be seventh-order convergent. The proposed approach produces discrete approximations at the grid points, obtained after solving an algebraic system of equations. Numerical experiments are studied to show the performance and viability of the proposed approach. The numerical results demonstrated that the new technique gives accurate approximations, which are better than some existing strategies in the available literature and also found to be in good agreement with known analytical solutions

Numerical Spline Algorithm for Solving Falkner–Skan Equation

This paper presents a numerical spline algorithm for the Falkner–Skan equation (FSE) over a semi-infinite interval. This algorithm is based on change of variable from interval  to [0,1], then the FSE is transformed into first initial value problem (IVP) and second IVP for improving convergence.  Spline polynomials with four collocation points are applied directly to the IVPs without their reducing to a system of first-order differential equations. The study shows that purposed algorithm is consistent and convergent with a global truncation error from order eighth. The efficacy of our algorithm is tested by solving the problems of Blasius, Pohlhausen, Homann and Hiemenz flows, and other special cases over various intervals, where the results of comparisons with other methods indicate the efficiency of our algorithm and enable it to provide solutions with high numerical accuracy.

Galerkin-Laguerre Spectral Solution of Self-Similar Boundary Layer Problems

In this work the Laguerre basis for the biharmonic equation introduced by Jie Shen is employed in the spectral solution of self-similar problems of the boundary layer theory. An original Petrov-Galerkin formulation of the Falkner-Skan equation is presented which is based on a judiciously chosen special basis function to capture the asymptotic behaviour of the unknown. A spectral method of remarkable simplicity is obtained for computing Falkner-Skan-Cooke boundary layer flows. The accuracy and efficiency of the Laguerre spectral approximation is illustrated by determining the linear stability of nonseparated and separated flows according to the Orr-Sommerfeld equation. The pentadiagonal matrices representing the derivative operators are explicitly provided in an Appendix to aid an immediate implementation of the spectral solution algorithms

An Examination Of The Effectiveness Of The Adomian Decomposition Method In Fluid Dynamic Applications

Since its introduction in the 1980\u27s, the Adomian Decomposition Method (ADM) has proven to be an efficient and reliable method for solving many types of problems. Originally developed to solve nonlinear functional equations, the ADM has since been used for a wide range of equation types (like boundary value problems, integral equations, equations arising in flow of incompressible and compressible fluids etc...). This work is devoted to an evaluation of the effectiveness of this method when used for fluid dynamic applications. In particular, the ADM has been applied to the Blasius equation, the Falkner-Skan equation, and the Orr-Sommerfeld equation. This study is divided into five Chapters and an Appendix. The first chapter is devoted to an introduction of the Adomian Decomposition method (ADM) with simple illustrations. The Second Chapter is devoted to the application of the ADM to generalized Blasius Equation and our result is compared to other published results when the parameter values are appropriately set. Chapter 3 presents the solution generated for the Falkner-Skan equation. Finally, the Orr-Sommerfeld equation is dealt with in the fourth Chapter. Chapter 5 is devoted to the findings and recommendations based on this study. The Appendix contains details of the solutions considered as well as an alternate solution for the generalized Blasius Equation using Bender\u27s delta-perturbation method

A numerical study of entropy generation, heat and mass transfer in boundary layer flows.

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.This study lies at the interface between mathematical modelling of fluid flows and numerical methods for differential equations. It is an investigation, through modelling techniques, of entropy generation in Newtonian and non-Newtonian fluid flows with special focus on nanofluids. We seek to enhance our current understanding of entropy generation mechanisms in fluid flows by investigating the impact of a range of physical and chemical parameters on entropy generation in fluid flows under different geometrical settings and various boundary conditions. We therefore seek to analyse and quantify the contribution of each source of irreversibilities on the total entropy generation. Nanofluids have gained increasing academic and practical importance with uses in many industrial and engineering applications. Entropy generation is also a key factor responsible for energy losses in thermal and engineering systems. Thus minimizing entropy generation is important in optimizing the thermodynamic performance of engineering systems. The entropy generation is analysed through modelling the flow of the fluids of interest using systems of differential equations with high nonlinearity. These equations provide an accurate mathematical description of the fluid flows with various boundary conditions and in different geometries. Due to the complexity of the systems, closed form solutions are not available, and so recent spectral schemes are used to solve the equations. The methods of interest are the spectral relaxation method, spectral quasilinearization method, spectral local linearization method and the bivariate spectral quasilinearization method. In using these methods, we also check and confirm various aspects such as the accuracy, convergence, computational burden and the ease of deployment of the method. The numerical solutions provide useful insights about the physical and chemical characteristics of nanofluids. Additionally, the numerical solutions give insights into the sources of irreversibilities that increases entropy generation and the disorder of the systems leading to energy loss and thermodynamic imperfection. In Chapters 2 and 3 we investigate entropy generation in unsteady fluid flows described by partial differential equations. The partial differential equations are reduced to ordinary differential equations and solved numerically using the spectral quasilinearization method and the bivariate spectral quasilinearization method. In the subsequent chapters we study entropy generation in steady fluid flows that are described using ordinary differential equations. The differential equations are solved numerically using the spectral quasilinearization and the spectral local linearization methods

A numerical study of heat transfer and entropy generation in Powell-Eyring nanofluid flows.

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.The heat transfer in non-Newtonian nanofluid flow through different geometries is an important research area due to the wide application of these fluids in biomedical, chemical and thermal engineering processes. The continuous generation of entropy leads to exergy loss which reduces the performance and efficiency of any physical system, therefore, the minimization of entropy generation becomes necessary. In this thesis, we present a numerical study of heat transfer and entropy generation in non-Newtonian nanofluid flows. We study the flow of a Powell-Eyring nanofluid, using models developed from experimental data. The equations that model the flow are, in each case, reduced to systems of nonlinear differential equations using Lie group theory scaling transformations. Accurate, efficient and rapidly converging spectral numerical techniques including the spectral quasilinearizzation, spectral local linearization and bivariate spectral quasilinearization methods are used to find the numerical solutions. The results show, among other findings, that increasing either the nanoparticle volume fraction or thermal radiation parameter enhances the nanofluid temperature, entropy generation and the Bejan number. In addition, we find that the Nusselt number increases with the temperature ratio parameter and thermal radiation. The results from this study may find use in the design of cooling devices to enhance and optimize the performance of thermal systems

Low order modelling of flow-control techniques for turbulent skin-friction reduction

In the present thesis a linearized formulation of the Navier-Stokes equations is used to study two main subjects: the generation of near-wall streaks in turbulent boundary layers and the response of turbulent wall-bounded flows to streamwise-travelling waves of spanwise oscillations of the bounding walls. For the purposes of the present work, these oscillations of the wall have been considered as a flow control mechanism. A mathematical model, based on a velocity-vorticity formulation linearized around a turbulent mean base flow, is adopted to simulate the fluid flow equations. A hybrid spectral-finite differences solver has been employed to numerically implement the linearized system. A review on turbulent streak generation is presented and the concepts of exponential growth, algebraic growth and viscous dissipation of small-scale perturbations are linked to the concept of transient growth. Mechanisms of generation of near-wall streaks are explored using a large set of sources of excitation for the linearized Navier- Stokes equations (LNSE). Two types of sources, here labelled as Excitation Mechanisms (EM), are employed: a body-force source and an initial condition. The selection of parameters for the excitation mechanisms is performed based on the definition of a multi-step optimization problem. The different EMs studied consist of a restricted number of parameters, and therefore can be considered as a Low Order Model (LOM) for the generation of streaks. It is shown that both types of EM produce satisfactory results for the streak generation process evaluated in terms of experimentally expected optimal spanwise scales. Finally, a large set of numerical experiments are conducted to evaluate the response of the LNSE with optimised EM, to the flow control by a spanwise oscillating wall. By comparing the results between the response of the LNSE considered here to this type of flow control against a drag-reduction map obtained by DNS in other studies, it is possible to assess the correlation between streak evolution and disruption of the skin-friction drag. A good agreement between these two responses is found for the parameter space of the streamwise-travelling spanwise oscillation waves