Numerical studies of unsteady boundary layer flow problems

Abstract

This thesis presents the results of numerical investigations on certain unsteady laminar incompressible boundary?layer flow problems. It consists of an introductory chapter and four main chapters. The introductory chapter provides a brief introduction to boundary?layer theory. It mainly focuses on explaining various terminologies used in the text. A survey of various methods of solution of the boundary?layer equations is also presented. The remaining four chapters are concerned with the specific problems investigated by us. Each chapter begins with an introduction and a brief survey of the literature relevant to the problem considered, and the contributions made by the author are given. The mathematical formulation of the problem, analysis, and method of solution follow next. The effects of various parameters such as unsteadiness in wall velocities, stretching ratio, suction, magnetic field, rotation parameter, etc., on the skin friction, heat and mass?transfer coefficients, as well as on the velocity, temperature and concentration, have been studied in detail. The results, presented in graphical form and comparison tables, are analysed and discussed in detail. A brief conclusion summarizes each chapter. Although various symbols are defined as and when they occur, for easy reference a list of common symbols is given at the end of the thesis. The books and original papers referred to in the thesis are listed sequentially at the end of each chapter.

Chapter II Chapter II deals with the semi?similar solution for the unsteady, laminar, incompressible three?dimensional boundary?layer flow over a continuously stretching surface in two lateral directions in a fluid at rest. Numerical solutions have been obtained for nodal and saddle?point regions of flow when the velocity of the stretching surface varies arbitrarily with time. Both constant wall temperature/concentration and constant heat/mass?flux conditions at the stretching surface have been considered. The effect of mass transfer (suction) has also been included. The quasilinearisation method with an implicit finite?difference scheme has been used in the nodal?point region. This method fails in the saddle?point region due to the occurrence of reverse flow in the y?component of velocity. To overcome this difficulty, parametric differentiation with the implicit finite?difference scheme is used in the saddle?point region using the values at c = 0 as starting values. Results have been obtained for stretching velocities that accelerate and decelerate with time. Skin friction, heat and mass?transfer coefficients respond significantly to the time?dependent stretching velocity, stretching ratio, and mass?transfer parameter. Analysis of the stretching?ratio parameter c has been made in the range �< c < 1, which covers the entire solvable range. For c > 1, one simply interchanges the x? and *y?*axes, and for c < � the equations become insolvable as pointed out by Davey. Suction (A > 0) is found to be an important parameter in obtaining convergent solutions in the saddle?point region. The Prandtl and Schmidt numbers strongly affect the heat and mass transfer of the diffusing species respectively.

Chapter III In Chapter III, a numerical solution of the unsteady incompressible boundary?layer flow has been obtained when a lighter fluid impinges downward on a heavier, otherwise quiescent fluid. The boundary?layer equations yield semi?similar solutions for the upper (lighter) and lower (heavier) fluids. The problem is governed by a non?dimensional parameter ?, which denotes the lateral motion of the interface. Both two?dimensional and axisymmetric cases have been considered. Analysis is made for various situations such as accelerating and decelerating cases of the upper fluid. The effect of an applied magnetic field on various coefficients has also been studied. Results for the skin friction and convective heat transfer are presented. In the presence of an applied magnetic field, velocity overshoot occurs in the upper fluid and is strongly dependent on time and magnetic?field strength. The effect of interface velocity is more pronounced on skin friction than on heat transfer. For both upper and lower fluids, the thermal boundary layer becomes thinner for large Prandtl numbers. In all cases, the values of skin friction and heat?transfer coefficients are higher for the two?dimensional case than for the axisymmetric case. The flow problem considered here, without magnetic field, may find applications in flows such as air over water, spreading of oil on the sea surface, and梬ith magnetic field梚n metallurgical processes.

Chapter IV In Chapter IV, we present the unsteady, laminar incompressible boundary?layer flow caused by the stretching of a flat surface in a rotating fluid. It consists of two subdivisions, Part A and Part B. In Part A, a semi?similar solution has been obtained when both the stretching?surface velocity and the angular velocity of the rotating fluid vary arbitrarily with time. Numerical results have been obtained for both accelerating and decelerating stretching?velocity distributions. In Part B, it is shown that when the stretching?surface velocity and rotating?fluid angular velocity vary inversely as a linear function of time, a self?similar solution becomes possible. This is discussed in detail. Both the semi?similar and self?similar cases have been solved numerically using a finite?difference scheme combined with the quasilinearisation technique. The solution is found to depend on a parameter ?, which signifies the relative importance of rotation rate to stretching rate. The effects of the power?law variation of surface temperature and surface heat flux on heat?transfer characteristics have also been analysed. Results show that skin friction and heat?transfer coefficients are significantly influenced by stretching?velocity distributions, the rotation parameter, and surface?temperature conditions. Increasing rotation rate increases both skin friction and heat?transfer coefficients. The temperature parameter m and the Prandtl number Pr play important roles in determining heat?transfer characteristics. For the prescribed?wall?temperature case, there is no heat transfer between surface and rotating fluid for m = �in the steady case.

Chapter V The fifth and final chapter deals with the effect of large injection on unsteady laminar incompressible boundary?layer flow over a long thin cylinder. Here, unsteadiness arises from the free?stream velocity. The governing partial differential equations are transformed into dimensionless form using suitable transformations, resulting in nonsimilar equations involving three independent variables. These have been solved using an implicit finite?difference scheme combined with the quasilinearisation technique. The nonsimilarity primarily arises due to curvature of the surface. Computations have been carried out for three different unsteady free?stream velocity distributions: accelerating, oscillating, and decelerating flows. The effect of viscous?dissipation parameter has also been included. Results show that the nature of free?stream unsteadiness, mass?transfer parameter, curvature parameter, dissipation parameter, and Prandtl number exert strong influence on skin friction, heat transfer, velocity, and temperature profiles. For large injection rates, velocity and temperature profiles are more significant than skin?friction or heat?transfer parameters (which become very small). The dividing streamline shifts away from the boundary with increasing injection rates or curvature parameter. For large Prandtl and Eckert numbers, temperature overshoot occurs near the wall for moderate injection rates, implying that viscous dissipation can raise the fluid temperature near the wall above the wall temperature (initially higher), causing heating instead of cooling. Without viscous dissipation, no overshoot occurs for any Prandtl number. Boundary?layer thickness increases with either injection or curvature parameter. Papers based on the work reported in the thesis will be communicated for publication in international journals shortly.