MATHEMATICAL MODELLING FOR SOME FLOW PROBLEMS IN BIOLOGICAL AND INDUSTRIAL SYSTEMS

Abstract

The thesis embodies the modelling of some biological and industrial problems. It consists of Nine Chapters. The first Chapter is introductory in nature and deals with fundamental concepts of flow and transfer processes related to heat, mass and energy. It constitutes the equations of momentum, energy, mass and bio-heat for one as well as for two phase flow. The heat-balance integral technique, series method, method of separation of variables, gradient transfer technique and finite difference methods have been used in the solution of the problems included. In Chapter II, an analysis has been carried out to discuss the freezing behaviour of tumor, lung and tumor embedded in the lung during cryosurgery. The temperature distribution and position of freezing interface are determined by solving one-dimensional transient bio-heat equation in unfrozen region and heat conduction equation in frozen region. Analytical expressions for the freezing interface of tumor, lung and tumor embedded in lung have derived by employing heat-balance integral technique. The 'results obtained from this method are compared with the theoretical and experimental results available in the literature. Chapter III, is devoted to the study of a group of male industrial workers at Bharat Heavy Electricals Limited, Hardwar (India) in connection with the study of noise-induced raised blood pressure and heart rate. The statistical data for sound pressure level, blood pressure, heart rate and other physiological and psychological conditions are observed for 23 noise exposed persons. Many investigators predicted that noise affects peripheral circulation and heart activation via the sympathetic nervous system, causes constriction of blood vessels, contraction of peripheral arteries, increase in heart rate and raised blood pressure. We have considered the constriction phenomenon to model the noise (iv) induced wall vibrations and heart rate. The flow is taken pulsatile. The effects of increased heart rate on pressure, wall shear stress and velocity profiles are discussed. From practical observations we found that diastolic pressure was high enough (L. 90 mmHg) in more than 60% persons and heart-rate in normal range (--s 80 per minute) was found in very few persons. Chapter IV, deals with the inverse Womersley problem to consider the flow rate as a function of time for pulsatile flow of a dusty fluid in a straight rigid tube. The pressure gradient, wall shear stress and instantaneous velocity profiles have been calculated numerically and shown their graphical representations for the known flow rate waveform as opposed to Womersley solution where the pressure gradient was the known function. The velocity profiles for the fluid particles in the absence of dust particles show a flatness in the area around the artery centreline corresponding to the forward flow, which is typical for pulsatile flow and for high Womersley parameters. The wall shear stress increases as the number density of solid particles increases in the fluid. In Chapter V, a nonlinear pulsatile flow of Newtonian fluid in an elastic tube is studied with the motivation of the problem of blood flow in an artery. The non-linear term of the Navier-Stokes equation as well as the non-linear behaviour and large deformations of the arterial wall are considered in the model. The velocity distribution, wall shear stress and flow rate are calculated numerically by solving the problem using finite difference scheme. The pressure is assumed to be known sinusoidal function. The variation of flow rate and wall shear stress with Womersley parameter a are discussed. In Chapter VI, the effects of presence of a very thin layer slightly viscoelastic fluid very close to the wall have been studied analytically when a slightly viscoelastic fluid was being continuously and uniformly injected through the wall of the pipe in which a purely viscous Newtonian fluid was flowing. The problem has been analysed in two regions; one thin viscoelastic layer near the wall and other, central core region of purely viscous fluid separately, alongwith (v) suitable matching conditions. Finally the wall shear stress and total flux through a cross-section for a particular pressure gradient with and without the presence of the assumed thin layer of the polymer additives has been presented. Chapter VII, deals with an analytical study of the mass transfer in an artificial kidney (tubular dialyser). A model for the two phase concurrent motion with a circular interface has been developed and a first order chemical reaction to account for the production of substances due to metabolism in the red cells has been considered in the blood phase. The attempt has been made to estimate the redistribution of materials across a circular interface and the resistance to the mass transfer in both the phases simultaneously. Component distribution within the blood and the dialysing fluid has been discussed. Results obtained show that the consideration of the chemical reaction in the film gives rise to the additional diffusion. The degree of film saturation has shown to be dependent on the chemical reaction rate. In Chapter VIII, the dispersion of air pollutant around an axisymmetric isolated hill has been studied. For stable stratified and potential flow, the mass concentration equation is solved analytically together with suitable matching conditions. The problem is divided into three regions, Region-I, from source to the stagnation point on the hill, Region-II, from stagnation point to the separation point and Region-III, downstream the separation point i.e. wake. Analytical expressions for predicting the concentration of pollutant for each region, are obtained and numerical results are shown graphically. It is shown that the pollutant concentration is more in the case of hill compared to the concentration obtained by Gaussian model in the absence of hill at the same point and keeping all other conditions same. The dependence of concentration on the source position has been presented.........