SOME PROBLEMS OF TRANSPORT PHENOMENON USING MESHFREE METHODS

Abstract

The mathematical formulation of a typical uid ow problem involves a set of ordinary/ partial di erential equations which are based on the conservation laws of uid mechanics along with appropriate boundary conditions. These governing equations of uid motions, known as Navier-Stokes equations, are too complex to have a closed form analytical solution. Therefore, numerical techniques are the only choice for the solution of these problems. In literature, many grid based methods such as weighted residual methods, nite di erence methods, nite element methods, nite volume method etc. have been used for numerical simulation of uid ow problems. It is observed that these classical grid based methods behave poor for irregular domains and complex real life geometries (Aquifer domains, heat exchangers, turbines), problems related with moving boundaries (phase transition problems) or for unde ned geometries ( re propagation) due to problems generated because of discretization. These di culties of meshing and re-meshing associated with grid based methods can be eliminated utilizing meshfree methods, which are very exible numerical tools and for discretization purpose, they require only a set of nodes arbitrarily scattered in the problem domain as well as on the boundaries without any xed connectivity. There are number of meshfree methods available in literature such as Smooth particle hydrodynamics method (SPH), Di use element method (DEM), Element free Galerkin method (EFGM), Meshless local petrov galerkin method (MLPG) etc. It is evident from the latest survey of literature that not much has been done for the solution of uid ow problems using mesh free methods. The work in this thesis is concerned with ow and heat transfer problems of different types of uids such as Newtonian uids, Micropolar uids, Viscoelastic uids and Nano- uids under various simple and complex geometries of practical concern. iii Numerical simulation of the problems is carried out using a meshfree technique known as Element free Galerkin method. The superiority of meshfree methods over conventional grid based methods such as nite element and nite di erence methods, classi es them as the next generation computational methods. Also, in one of the problems, hybrid FEM (Finite element method) and EFGM (Element free Galerkin method) technique is implemented to save the computational time associated with EFGM. Convergence and accuracy of obtained results is found to be quite satisfactory. The whole work of the thesis is divided into eight chapters and chapter-wise summary of the thesis is as follows: Chapter 1 is introductory and it contains a brief outline of di erent uids, (Newtonian, Micropolar, Viscoelastic and Nano- uids) used in this thesis, some basic fundamentals, including governing equations and literature review relating to these uids. Chapter 2 contains a study of ow and heat transfer phenomenon of a viscoelastic

uid over a stretching sheet embedded in a porous medium. In this study, uid viscosity and thermal conductivity are considered as variable and viscous dissipation e ect is also included. Unlike the commonly employed thermal conditions of critical and prescribed surface temperature, the present study uses a convective heating boundary condition also along with the prescribed surface temperature condition. The e ect of various physical parameters e.g. variable uid viscosity, thermal conductivity, heat source/sink parameter, viscoelastic parameter, Biot number etc. on velocity, temperature, local skin friction and local heat transfer is studied. Obtained EFGM results are validated with some previously published results [39] for a special case of the problem. Results are obtained with regular nodal distribution and a grid convergence study is also performed to check their consistency. In Chapter 3, the analysis of a second grade viscoelastic uid ow and heat transfer is done over an oscillatory stretching sheet. In this problem, unsteady MHD (Magneto-hydrodynamic) ow is considered and the e ect of viscous dissipation and joule heating are taken into account. The results illustrating the e ect of various parameters like viscoelastic parameter, magnetic parameter, and relative frequency iv amplitude of the oscillatory sheet to the stretching rate on velocity and temperature eld are reported in terms of graphs and tables. Validation of the results is done with previously published results [39] taking a special case of the problem. Chapter 4 consists of numerical simulation of an unsteady squeezing magneto micropolar ow between two parallel plates. The in uence of micropolar, unsteadiness and magnetic eld parameters on ow characteristics is studied in detail. It has been observed that increasing magnetic eld serves to decelerate the linear velocity and enhances angular velocity of the squeezing ow between plates. Obtained results have shown that with high squeezing of plates, velocities (angular and linear) are depressed considerably while for separating plates, velocities are increased as the distance between plates is increasing. The performance of EFGM results is validated with a grid convergence study and nite element results. Chapter 5 consists of the study of a mixed convection MHD (Magneto-hydrodynamic)

ow of a Newtonian uid over a vertical power-law stretching sheet. The impact of buoyancy, exponent and magnetic parameter on velocity, temperature, skin friction and heat transfer rate is discussed. A signi cant increment in heat transfer rate with increase of exponent parameter (power-law stretching parameter) is observed which demonstrates the wide impact of stretching process on heat transfer rate in some of the engineering and manufacturing processes such as cooling of metallic sheets and drawing of plastic sheets. Numerical simulation is performed with a regular nodal distribution and obtained results are validated with FEM results and previously published results [13] for a special case of the problem. Chapter 6 consists of a problem of natural convection within a wavy enclosure with corner heating e ects. Such types of problems of complicated geometries are di cult to handle with regular grid based techniques because of tedious mesh generation procedure while utilizing meshfree techniques, discretization of problem domain using arbitrarily distributed nodes, becomes quite easy. Here, partial heating or corner heating is considered and its impact on temperature pro les and heat transfer rate is investigated. Also, the side wall of the enclosure is considered to be wavy. It has been observed that the shape of the surface in uences the rate of heat transfer. Wavy surfaces have higher heat transfer rate as compared to at surfaces. v The impact of other parameters such as Rayleigh number and Prandtl number on velocity and temperature distributions is shown in terms of isotherms and streamlines. EFGM results are validated with some benchmark results [24] available in literature. In Chapter 7, a study of phase transition between solid and liquid which takes place within a square enclosure with the help of natural convection, is done. For numerical simulation of this problem, hybrid FEM/EFGM methodology is utilized. For mesh re nement, extra nodes are inserted in the vicinity of phase change front at each time step. It results in denser and irregular nodal distribution near the phase transition region. Typically, computational time of element free galerkin method is higher than that of nite element method. Therefore, by using EFGM only in that portion of physical problem where phase transition occurs, the hybrid FEM/EFGM strategy could reduce the computational time of EFGM while still maintaining its accuracy. The impact of Rayleigh number (Ra), Prandtl number (Pr), Stefan number (ste) on thermal and ow eld is investigated. Also, the consistent performance of the results obtained with this hybrid approach is validated with those already available in literature [165] for some special cases. Chapter 8, consists of study of a moving boundary problem, in which phase transition occurs during the cryosurgery process of killing undesired tumor tissues. This study consists of the freezing mechanism for a liver tumor tissue. The target tumor tissue is loaded with nano-particles in order to improve the freezing capacity of probe and to regulate the orientation and size of ice-ball formed during cryosurgery. The latest nano- uid model which includes the e ects of particles size, their distribution, concentration and the interfacial layer at the particle/liquid interface, is utilized and their impact on freezing process is investigated in detail. Real advantage of element free Galerkin method can be explored out in such type of phase transition problem. The size of ice-ball generated during freezing, is increasing with time and for accurate computation of temperature elds in this region, denser meshing is required. Hence, it leads to re-meshing of the whole computational domain at each time step. In element free Galerkin method, depending upon the size of the vi ice-ball generated, it is quite easier to insert nodes in its surrounding region, without re-meshing the whole domain. In our study, we observed that among all the nano-particles used (Al2O3; Fe3O4; Au), gold (Au) nano-particles have maximum freezing e ciency and also the size of ice-ball generated during cooling is found to be maximum with gold particles under the same cooling condition. For validation of the study, temperature elds at the tumor centre obtained by Li et al.[90] using FEM and 􀀀 FEM are compared with present EFGM results. Finally, the thesis ends with the future scope of the research work, appendices and bibliography.