جنگ، جہاد اور قتل

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Analysis of Newtonian/Non-Newtonian Fluids by Lie Group Theoretic Methods

Analysis of Newtonian/Non-Newtonian Fluids by Lie Group Theoretic Methods The advent of efficient computation techniques has made it possible to solve nonlinear differential equations governing the fluid flow problems. Nevertheless, the possibility of obtaining exact or approximate analytical solutions is always preferred to understand the physics of fluid flow and to establish the reliability of both numerical and analytical results. This has always been an intellectual challenge for the mathematicians and engineers to find the analytical solutions of the nonlinear differential equations. Lie group analysis provides an analytic approach to find the solution of nonlinear problems and gives an insight for the possible exact and analytical solution techniques that may emerge as a consequence of this analysis. Further, Lie group analysis is a tool to simplify the problem systematically by applying the symmetries obtained through it. The main objective of this thesis is to establish the occurrence of various stretching velocities and to analyze complicated fluid flow problems using Lie group analysis. This thesis presents a Lie group analysis of partial differential equations elucidating the steady simple flow problems, mixed convection problems and combined heat and mass transfer problems for Newtonian and non-Newtonian fluids. The novelty of the work lies in using the generalized boundary conditions and to deduce the appropriate conditions that are invariant under the infinitesimal generator. These boundary conditions include the power-law stretching and exponential stretching models that have great applications in polymer and glass industries. Chapter 1 provides a brief history and literature survey covering the study of the present thesis. Chapter 2 contains some preliminaries, the basic equations of fluid flow and heat and mass transfer and general physical quantities that appear in the subsequent chapters. ixIn Chapter 3, we consider the Lie group analysis of mixed convection flow of Newtonian fluid with mass transfer over a stretching surface. We propose generalized forms of the wall stretching velocity, wall temperature and wall concentration and show the possibility of only two types of stretching velocities; namely the polynomial stretching and the exponential stretching. The similarity transformations are established and those available in the literature are extracted as special cases of our problem. Lie group analysis of non-Newtonian power-law fluid along a stretching surface is performed in Chapter 4. The application of infinitesimal generator on the generalized surface stretching conditions for non-Newtonian power-law fluid leads to the possibility of power-law and exponential stretching velocities. To author’s knowledge, exponential stretching in the flow of power-law fluid is not available in the literature. An exact analytical solution of the nonlinear similarity equation for new found exponential stretching is developed for shear thinning fluid with power-law index n = 1/2. Making use of the perturbation technique, analytical solutions are extended to a wider class of shear thinning fluids (0.1 ≤ n ≤ 0.9). The numerical solution for shear thinning fluid is also presented. An excellent agreement is found between the two solutions. The solution for the case of shear thickening fluid is obtained using the numerical technique namely; Keller box method. In Chapter 5, we investigate the flow and heat transfer of a non-Newtonian Powell- Eyring fluid over a stretching surface. Using Lie group analysis, the symmetries of the equations are found. The application of infinitesimal generator to the generalized boundary conditions leads us to the possibility of two types of surface condition that are in contrast to the findings in the last two chapters. Firstly, the surface is moving with constant velocity and surface temperature is either of exponential form or constant. Secondly, the surface is stretching with velocity proportional to x 1/3 (x is the distance along the plate) and the surface temperature is of power-law form. The latter case is discussed in this chapter and the similarity transformations are derived with help of the symmetries. The governing system of partial differential equations is transformed to a system of ordinary differential equations by using these similarity transformations. These equations are solved numerically using Keller box method. A xcomparison of the results thus obtained is made with the analytical and numerical solutions available in the literature and an excellent agreement is found. The effect of governing physical parameters on velocity and temperature profiles, skin friction and local Nusselt number is also analyzed and discussed. Chapter 6 is devoted to study the flow and heat transfer of Powell-Eyring fluid over a stretching surface in a parallel free stream. Geometry of the problem differs from the preceding problems, where the flow is caused solely by the stretching of the surface. The stretching velocity is proportional to x 1/3 and the free stream velocity is in terms of a generalized function. The governing equations are transformed to a system of nonlinear ordinary differential equations by using a special form of Lie group of transformations, namely scaling group of transformations. It is noted that self- similarity in the problem is possible only if free stream velocity is also proportional to x 1/3 . Numerical results are obtained by means of the Keller box method and the special cases of the problem are compared with the previous work giving good agreement. The effect of governing physical parameters on flow properties including their physical significance is also discussed. Steady three dimensional flow and heat transfer of viscous fluid on a rotating disk stretching in radial direction is investigated in Chapter 7. This problem is an extension of the traditional Von Karman flow problem to the configuration with stretchable rotating disk. Using Lie group theory the similarity transformations for nonlinear power-law stretching are derived. Exact analytical solutions are presented for pure stretching for stretching index n = 3. Numerical solutions, showing combined effects of stretching and rotation, are found using Keller box method. An excellent agreement is found between the two solutions for pure stretching problem. The quantities of physical interest, such as azimuthal and radial skin friction and Nusselt number are presented and discussed. Chapter 8 provides a concluding discourse of the thesis.