Steady/Unsteady MHD Flow Over An Axisymmetric Shrinking Sheet

Abstract

Steady/Unsteady MHD Flow Over An Axisymmetric Shrinking Sheet. This study has been undertaken to analyze the steady/unsteady MHD fluid flow and boundary layer control using strong magnetic field for stretching and shrinking phenomena. Stretching of sheets is one of the most important mechanisms of boundary driven flows having immense applications in industry and technology. Its importance can be further viewed in that; it helps to understand the physics of fluid flow in Newtonian and non- Newtonian fluids. In order to control the boundary layer development and to avoid the bifurcation, various mechanisms have been introduced in the literature and implemented experimentally. These include wall suction/injection (Prandtl 1904), setting the surface in motion, introducing the favorable pressure gradients and using body forces. We will impose the strong magnetic field to explore the possibility of achieving this end effectively. In addition to stretching, shrinking of the sheets has gained importance due to mathematical challenge it presents in explaining complex behavior generated by it and the emergence of its useful applications in industry. Its interest further lies in that; the solution of such problem is non-existent, even for the steady state case, unless a strong suction is introduced on the boundaries. The situations in which the solution can be obtained thus remain a good question which will be answered through the application of strong magnetic field. Newtonian fluids represent a wide spectrum of the fluids and have a fundamental importance in understanding the physics of fluid flow. However, with the advent of more complex industrial fluids and the non-linear behavior shown by the natural fluids; the science of Newtonian fluids has entered into the realm of non- Newtonian fluids. A good number of mathematical models, based on strong empiricism have been proposed. However, further investigations and improvement in these models is continuously underway. The consideration of stretching/ shrinking phenomena in non- Newtonian fluids is industrial requirement and mathematical xii challenge. Wall suction and wall motion is found an efficient way to control the thickness of momentum boundary layer. Besides this, electrically conducting fluid flow can be controlled by electromagnetic forces. The behavior and flow of fluids with high electrical conductivity can be controlled by the application of classical MHD. However, in weakly conducting fluids, the current induced by an external magnetic field alone is too small, and an external electric field is applied to achieve an efficient flow control. Nano science has attained a tremendous significance in present times in both science and technology. The controlling behavior of Lorentz force over Riga plate in the presence of nanofluids takes care of the situation of imposed electric field and nanofluids together. A number of objectives are addressed in this thesis. Firstly, the unsteady MHD flow induced by linear stretching of the sheet has been investigated. The analytic solution is obtained using strong magnetic field. The steady and unsteady flows are considered for both stretching and shrinking of the surface. Strong magnetic field is applied as a strong measure to control the boundary layer. The study of shrinking sheet is then taken to the non-Newtonian second grade fluid. The concepts of imposed electrical field, shrinking and nanofluids are combined together. Similarity solutions are obtained analytically using asymptotic approximations. Analytic solutions are supported by numerical investigations. The first & second chapter of this thesis contains the history and literature related to flat surfaces problems and states the basic definitions and equations to be used in later chapters. The third chapter is about “unsteady boundary layer flow of an electrically conducting second grade fluid over a stretching surface. Asymptotic analytical solution of the governing non-linear equations is obtained for large magnetic field. The evolution of velocity field and skin friction with time is investigated in detail. The stability of the asymptotic solution is also discussed”. This work has been published in AIP Advances 5, 127140 (2015). The fourth chapter addresses “the magneto hydrodynamic (MHD) steady and unsteady axisymmetric flow of viscous fluid by a shrinking sheet. Analysis has been carried out in the presence of large magnetic field. The ensuing xiii problem is a singular perturbation problem in the limit of large magnetic field having infinite domain singularity. The secular term in the solution is removed and a two term uniformly valid solution is derived. The corresponding unsteady problem is solved asymptotically in the large magnetic field regime. The asymptotic solution is validated by comparing it with numerical solution. The results for velocity profile and skin friction are shown graphically to explore the physical features of the fluid flow. The momentum boundary layer is discussed and the stability analysis is made”. The contents of this chapter are accepted for publication in Chinese Physics B. In fifth chapter, the analysis of fourth chapter for Newtonian flow has been extended to steady/unsteady MHD 2nd grade fluid flow. Analytical and numerical solutions have been achieved. Numerical solution is obtained by means of Keller Box method. The main analytical and numerical results for the velocity and skin friction are discussed and compared. Some interesting features pertaining to the second grade fluid have been figured out. In the sixth chapter we studied the “mixed convection boundary layer flow of a nanofluid past a vertical Riga plate in the presence of strong suction. The mathematical model incorporates the Brownian motion and thermophoresis effects due to nanofluid and the Grinberg-term for the wall parallel Lorentz force due to Riga plate. The analytical solution of the problem is presented using the perturbation method for small Brownian and thermophoresis diffusion parameters. The numerical solution is also presented to ensure the reliability of the asymptotic results. The comparison of the two solutions shows an excellent agreement. The correlation expressions for skin friction, Nusselt number and Sherwood number are developed by performing linear regression on the obtained numerical data. The effects of nanofluid and the Lorentz force due to Riga plate, on the skin friction are discussed”. The contents of this chapter are published in Journal of Magnetism and Magnetic Materials 402, 44–48 (2016).