Trajectory statistical solutions and Liouville type equations for evolution equations: abstract results and applications

In this article, we first prove, from the viewpoint of infinite dynamical system, sufficient conditions ensuring the existence of trajectory statistical solutions for autonomous evolution equations. Then we establish that the constructed trajectory statistical solutions possess invariant property and satisfy a Liouville type equation. Moreover, we reveal that the equation describing the invariant property of the trajectory statistical solutions is a particular situation of the Liouville type equation. Finally, we study the equations of three-dimensional incompressible magneto-micropolar fluids in detail and illustrate how to apply our abstract results to some concrete autonomous evolution equations

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

Computational Fluid Dynamics 2020

This book presents a collection of works published in a recent Special Issue (SI) entitled “Computational Fluid Dynamics”. These works address the development and validation of existent numerical solvers for fluid flow problems and their related applications. They present complex nonlinear, non-Newtonian fluid flow problems that are (in some cases) coupled with heat transfer, phase change, nanofluidic, and magnetohydrodynamics (MHD) phenomena. The applications are wide and range from aerodynamic drag and pressure waves to geometrical blade modification on aerodynamics characteristics of high-pressure gas turbines, hydromagnetic flow arising in porous regions, optimal design of isothermal sloshing vessels to evaluation of (hybrid) nanofluid properties, their control using MHD, and their effect on different modes of heat transfer. Recent advances in numerical, theoretical, and experimental methodologies, as well as new physics, new methodological developments, and their limitations are presented within the current book. Among others, in the presented works, special attention is paid to validating and improving the accuracy of the presented methodologies. This book brings together a collection of inter/multidisciplinary works on many engineering applications in a coherent manner

Similarity solutions of boundary layer flows in a channel filled by non-newtonian fluids

Similarity solutions of non-Newtonian fluids are getting much attention to researchers because of their practical importance in the field of science and engineering. Currently, most of researchers focus their work on non-Newtonian fluids over a sheet. However, only a few of them pay their attention towards the geometry of channel due to the complexity of governing equations. Therefore, this study attempts to investigate the numerical solutions of new problems of laminar incompressible Nanofluids, Casson fluids and Micropolar fluids under various fluid flow conditions. Each considered fluid involves porous channel walls, stretching or shrinking walls, and expanding or contracting walls with the influence of various physical parameters. Mathematical formulations such as the law of conservation, momentum or angular momentum, heat and mass transfer are performed on the new problems. After the mathematical formulation is developed, the governing fluid flow equations of partial differential equations are then transformed into boundary value problems (BVPs) of nonlinear ordinary differential equations (ODEs) by using the suitable similarity transformations. After converting high order BVPs into the equivalent first order system of BVPs, shootlib function in Maple 18 software is employed to obtain the similarity solutions of nonlinear ODEs. The numerical results in this study are compared with the existing solutions in literature for the purpose of validation. The results are found to be in good agreement with the existing solutions. Multiple solutions of some of the problems particularly in porous channel with expanding or contracting walls also exist for the case of strong suction. This study has successfully find the numerical solutions of the new problems for various fluid flow conditions. The results obtained from this study can serve as a theoretical reference in related fields

Analytical solutions for wall slip effects on magnetohydrodynamic oscillatory rotating plate and channel flows in porous media using a fractional burgers viscoelastic model

A theoretical analysis of magnetohydrodynamic (MHD) incompressible flows of Burger's fluid through a porous medium in a rotating frame of reference is presented. The constitutive model of a Burger's fluid is used based on a fractional calculus formulation. Hydrodynamic slip at the wall (plate) is incorporated and a fractional generalized Darcy model deployed to simulate porous medium drag force effects. Three different cases are considered- namely, flow induced by a general periodic oscillation at a rigid plate, periodic flow in a parallel plate channel and finally Poiseuille flow. In all cases the plate (s) boundary (ies) are electrically-non-conducting and small magnetic Reynolds is assumed, negating magnetic induction effects. The well-posed boundary value problems associated with each case are solved via Fourier transforms. Comparisons are made between the results derived with and without slip conditions. 4 special cases are retrieved from the general fractional Burgers model, viz Newtonian fluid, general Maxwell viscoelastic fluid, generalized Oldroyd-B fluid and the conventional Burger’s viscoelastic model. Extensive interpretation of graphical plots is included. We study explicitly the influence on wall slip on primary and secondary velocity evolution. The model is relevant to MHD rotating energy generators employing rheological working fluids

Stability and well-posedness problems on the partially dissipated Boussinesq equations and the micropolar equations

Fluid Mechanics is a central theme of science concerned with the study of the behavior of fluids when they are in state of motion or rest. When the density of the fluid is constant or its change with the pressure is so small that can be neglected, the fluid is said to be incompressible. Examination of such fluid flow phenomena is carried out with the help of the incompressible Navier-Stokes equations. These fundamental equations provide a mathematical model of the motion of the fluid. In this direction, this thesis is concerned with the study of two closely associated systems, the micropolar equations and the Boussinesq equations. The work being conducted in this thesis includes four main chapters. In the first chapter, we give a small introduction to the concerned equations. The second chapter is devoted to show the existence and uniqueness of the weak solutions to the d-dimensional micropolar equation with general fractional dissipation. Additionally, in the third chapter, we focus first on the stability problem of the 2D Boussinesq equations with vertical dissipation and horizontal thermal diffusion in $\mathbb{R}^2$, then we present some decay properties of the corresponding linearized system. Lastly, the fourth chapter investigates the stability and large-time behavior of the solutions to the 2D Boussinesq equations with horizontal dissipation and vertical thermal diffusion in two different spatial domains

Numerical Simulation

Nowadays mathematical modeling and numerical simulations play an important role in life and natural science. Numerous researchers are working in developing different methods and techniques to help understand the behavior of very complex systems, from the brain activity with real importance in medicine to the turbulent flows with important applications in physics and engineering. This book presents an overview of some models, methods, and numerical computations that are useful for the applied research scientists and mathematicians, fluid tech engineers, and postgraduate students