| dc.description.abstract | This thesis presents the analysis of the numerical investigations of some unsteady laminar incompressible boundary?layer flow problems. It consists of five chapters.
The first chapter deals with a general introduction to boundary?layer theory and a brief account of the development of unsteady flow problems. It also presents a survey of the various methods of solving boundary?layer equations. The remaining four chapters are concerned with the investigation of specific problems. Each of these chapters begins with an introduction and a brief survey of the literature relevant to the problem considered. This is followed by the mathematical formulation of the problem, description of the method of solution and discussion about the results obtained. Each chapter ends with a conclusion.
The second chapter presents the analysis of the unsteady flow of an electrically conducting incompressible fluid over a two?dimensional or an axi?symmetric body in the presence of a transverse magnetic field. Non?similar solution has been obtained by assuming the free?stream velocity to depend on streamwise location as well as on time, while the wall temperature is assumed to be time?dependent only. Computations are carried out for accelerating and oscillating nature of the free?stream velocity for the case of a cylinder and a sphere which are typical representatives of two?dimensional and axi?symmetric bodies. The effects of viscous dissipation and Joule heating are included in the analysis. The effects of magnetic field, the mass transfer, and wall temperature on the velocity and thermal field have been studied. The skin?friction coefficient and heat transfer are observed to be increasing as magnetic parameter increases. The point of zero skin friction is observed to be moved downstream due to the application of magnetic field. Though the skin?friction coefficient is zero at the stagnation point (x = 0.0), the skin?friction parameter is non?zero and finite. The skin?friction coefficient is observed to increase up to some streamwise location and then decrease, while the heat transfer continuously decreases. Both the quantities are observed to increase due to suction and decrease due to injection. The effect of injection is to move the point of zero skin friction upstream. The variation in wall temperature is observed to influence only the heat transfer and the temperature field. When the wall temperature continuously increases with time, heat is transferred from the wall to the fluid. When the wall temperature decreases with time, heat is transferred from the wall to the fluid up to some time and then the direction of heat transfer is reversed.
The heat transfer and temperature field are observed to be strongly influenced by the viscous dissipation and Prandtl number. The effect of unsteadiness in the free?stream velocity is more significant on skin?friction coefficient than on the heat transfer. Qualitatively similar results are observed both for the case of a cylinder and a sphere.
The third chapter presents the results of the investigations on unsteady flow of a viscous incompressible fluid over a moving wall in the presence of a transverse magnetic field. This is presented in two parts. Part A deals with the unsteady motion of an incompressible viscous fluid over a continuous moving wall, in the presence of a transverse magnetic field, where the unsteadiness in the flow is due to the time?dependent motion of the wall. The free stream is assumed to have zero velocity. Viscous dissipation and Joule heating effects are included in the analysis. The computations are carried out for exponentially decaying, accelerating and decelerating nature of the motion of the wall. The effect of magnetic parameter is to increase the skin?friction coefficient and to decrease the heat transfer. Both are observed to increase due to suction and decrease due to injection. The viscous dissipation and Prandtl number influence only the heat transfer and the temperature field. Heat transfer is observed to be increasing with Prandtl number.
Part B deals with the analysis of the unsteady flow of a viscous incompressible fluid over a moving wall in the presence of a transverse magnetic field, where the unsteadiness in the flow is due to the sudden impulse imparted to the wall. The wall is at rest initially and is set into sudden impulsive motion. The free stream is assumed to be at rest. Due to the impulsive motion of the wall, the velocity of the fluid shoots up from the initial zero value and decreases with the passage of time to reach an ultimate steady value. The transition of the fluid motion from unsteady to steady takes place in a small interval of time and is influenced by different flow parameters. The skin?friction coefficient and heat transfer start with high values in the beginning and reach steady?state values with the passage of time, with a steep fall in their values in a small interval of time. The effect of magnetic parameter and mass transfer are not felt initially, but are found to be influencing the ultimate steady?state values and the period of transition. The steady?state value of skin?friction coefficient is observed to be increasing with the increase of the magnetic parameter and due to suction. The period of transition is reduced. However, the effect of magnetic field on heat transfer is not very much appreciable. The viscous dissipation prolongs the transient behaviour of the flow.
Chapter four presents the analysis of the problem of unsteady three?dimensional stagnation?point flow of a viscous incompressible fluid over a stretching surface. The surface is stretched with time?dependent velocity components and the free?stream velocity components are also assumed to be time?dependent. Semi?similar solution has been obtained for accelerating and exponentially decaying nature of the flows. Locally self?similar solution is obtained by assuming the velocity components and square of the surface mass transfer to vary inversely as a linear function of time. The influence of the stretching ratios, the mass transfer and the Prandtl number on the velocity and thermal field has been considered. When the stretching ratios are positive and are increasing, the skin?friction coefficients in the two directions are observed to be decreasing, whereas the heat transfer continues to increase. The opposite trend is observed when the stretching ratios are negative and increase. As expected, suction increases all these quantities and the effect of injection is just the opposite. The effect of the crucial parameter ‘c’ which characterises different three?dimensional bodies indicates that the skin?friction and heat?transfer values are less for a three?dimensional body when compared with the corresponding values for an axi?symmetric body. The heat transfer is observed to be increasing with the Prandtl number. Locally self?similar solution exhibits trends similar to those observed in the semi?similar case.
Chapter five presents the analysis of the study of unsteady forced convection flow of a viscous incompressible fluid over a rotating disk. An infinite disk is considered to be rotating with time?dependent velocity. The free stream, parallel to the plane of the disk, is also assumed to have time?dependent velocity. The translational motion breaks the symmetry in the flow. The resultant flow due to the rotation of the fluid and the translational motion consists of two types of flows namely, (1) the basic flow due to the rotating disk which is axi?symmetric and (2) the secondary flow due to the translational motion of the free stream. By using suitable transformations, separate equations are obtained which govern the primary and secondary flows. Semi?similar solution has been obtained for accelerating and decelerating nature of the flows. Effect of mass transfer on primary and secondary velocity fields and thermal field are considered. The secondary boundary layers are observed to be thicker than the basic boundary layer. The suction and injection have the usual effects of increasing and decreasing the gradients of the secondary velocity profiles. The skin?friction parameter in the x?direction for the primary flow is observed to be decreasing both for suction and injection. The effect of mass transfer on the skin?friction parameter in the y?direction for the primary flow is more prominent as compared to that in the x?direction. The primary flow is less affected by the secondary flow. The heat transfer is observed to be increasing with the Prandtl number.
In solving all the above problems, the non?linear partial differential equations governing the flow have been transformed into dimensionless form using suitable transformations. The non?dimensional equations obtained have been linearised using quasilinearisation technique. The resulting linear equations are discretised using implicit finite?difference scheme with constant step size. The equations reduce to a system of linear algebraic equations, which can be expressed in the matrix form with a block tri?diagonal structure, which is solved using Varga’s algorithm. In all the problems, the step sizes are optimised by studying the effect of variation in the step sizes using Richardson’s extrapolation method.
References are listed in alphabetical order at the end of the thesis. Figures and tables relevant to each chapter are presented at the end of the chapter. The various symbols used are defined as and when they arise, but for the sake of convenience all the symbols are listed at the end of the thesis.
Papers based on the work reported in the thesis will be communicated for publication shortly. | |
