MODELLING OF NON-NEWTONIAN FLUID PROBLEMS AND THEIR SOLUTIONS

Abstract

The thesis presents the theoretical analysis of wire coating extrusion process inside pressure type die. Efforts at obtaining better insight into the process must be mainly theoretical rather than experimental. But the hope, of course, is that the better insight than experimental so gained will provide practical benefits such as better control of the process and of product quality, higher rates and more accurate and less costly die design. In this thesis, two types of problems have been studied, (i) problems within the die and (ii) problems outside the die. The studies are performed with several elastic fluid models such as Phan-Thien and Tanner, second grade, third grade, elastico-viscous and Oldroyd- 8-constant fluid models and for the inelastic power law fluid model. There are ten chapters in this thesis. Chapter 1 is introductory and discusses mainly mathematical modeling, wire coating operation and the phenomena inbuilt to it, in detail. In addition, it discusses the physical properties of the non-Newtonian fluids that have been considered here. Finally, it deals with the literature related to the analysis of coating process. Chapter 2 is concerned with the study of non-isothermal PTT fluid in wire coating analysis in a finite length pressure type die. The analysis is carried out by neglecting the exit and entrance effects. The expressions for axial velocity, average velocity, volume flow rate, shear and normal stresses, thickness of coated wire, force on the total surface of wire and the temperature distribution are obtained. The effects of the Deborah number, Brinkman number, elongation parameter and the ratio of the pressure drop to that of velocity of fluid are discussed. A domain for  Dec2 is found such that outside this domain, the shear and normal stresses show insignificant effects. Chapter 3 is devoted to the study of wire coating for heat transfer flow of a viscoelastic PTT fluid with slip boundary conditions. The investigations are carried out by considering nonzero pressure gradient in the axial direction. The wall shear stress, flow analysis and the role of slip parameter are the areas of investigation. The effect of  Dec2 and the slip parameter on the velocity of melt polymer, volume flow rate, thickness of coated wire, shear and normal stresses and on temperature distributions are studied. It is observed that the shear stress across the gap must follows a linear variation irrespective of the constitutive equation but its magnitude depends on the model parameters. In case of normal stress, this reduction is in the form of parabolic and the profiles overshoot at the centre of the annulus. Chapter 4 is to explore the wire coating analysis in a pressure type die by considering third grade fluid for constant and variable viscosity depends on temperature. For temperature dependent viscosity, two models are under discussion (i) Reynolds model and (2) Vogel’s model. The coupled momentum and energy equations are solved with the help of regular perturbation method. The non-Newtonian behavior of the fluid is discussed with the influence of perturbation parameter. Also, the solution of the problem is discussed for different Reynolds and Vogel’s model parameters. Chapter 5 is targeted to study the wire coating with a bath of Oldroyd 8-constant fluid taking into account the effect of pressure variation in the axial direction. The influence of pseudoplastic and dilatant parameters is investigated on the flow behavior such as velocity, average velocity, volume flow rate and shear stress of the fluid and on the temperature distributions. Also the influence of pressure gradient and the drag flow are examined. Furthermore, the effect of viscosity parameter  0 is discussed on shear stress. The aim of chapter 6 is to investigate an unsteady flow of a second grade fluid in a cylindrical die of finite length. In this problem, wire is dragged in a pool of melt polymer in the axial direction inside the die. The pressure gradient along the flow direction is assumed to be zero. The flow phenomena satisfying the continuity equation are modeled mathematically with the help of Navier Stokes equations and solutions for velocity distribution is derived in two different cases (i) when the wire is dragged in the molten polymer and (ii) when the wire is dragged with cosine oscillation in the melt polymer in a die. An exact solution is obtained in case (i) and an Optimal Homotopy Asymptotic Method (OHAM) is applied for handling solution of the problem in case (ii). The velocity field has been examined with passage of time and the effect of oscillation is investigated in the region of fluid flow. The stability analysis of this technique is discussed on some examples related to the problem under discussion. Chapter 7 gives an analytical investigation of post-treatment of wire coating with heat transfer analysis. The fluid is considered as third grade fluid. The investigation is performed by considering the slippage exists at the contact surfaces of wire, polymer and the gas. The mathematical model is derived for the fluid flow in a die. The governing equations are solved for the velocity field and temperature distribution using the regular Perturbation Method (PM) and OHAM. The explicit expressions for the flow rate, average velocity, force on total surface of wire and thickness of coated wire are derived. The solutions are examined under the effect of various parameters. The analysis of post-treatment of wire coating with heat transfer analysis is studied in chapter 8. The fluid is assumed to be satisfies the power law model. For temperature distribution, three different cases have been discussed (i) temperature of the wire is constant while it is varying linearly on the surface of the coated wire (ii) temperature of the wire varying linearly while it is constant on the surface of the coated wire (iii) temperature of the wire and the surface of coated wire are varying linearly at the same temperature gradient. The analysis for velocity field, volume flow rate, average velocity, shear rate, force on total surface wire and thickness of coated wire are carried out for the power law index parameter n is or is not equal to 1. Temperature distribution is studied separately in each of the three cases. The maximum temperature rise is investigated which depends upon the non-dimensional parameter S 0 . Chapter 9 deals with the post-treatment of wire coating analysis with heat transfer analysis. The fluid is assumed to be satisfies the elastico-viscous fluid model. The pressure gradient is considered to be constant in the direction of drag of wire. The analytical expressions for axial velocity, average velocity, volume flow rate, shear stress, normal stress, thickness of coated wire, the force on the total wire and the temperature distribution are derived by means of regular PM and Modified Homotopy Perturbation Method (MHPM). The influences of elastic number Re , cross-viscous number  c , velocity ratio U and the non-dimensional parameter S are studied on the solutions of the problem. It is concluded that an increase in the elastic number decreases, the flow rate whereas thickness of coated wire and force on the total wire increases. Chapter 10 is devoted to briefly review our main conclusions and future work directions.