Some New Results in the Theory of Non- Newtonian Fluids. Exact Solutions and Energetic Balance

Abstract

The main aim of this thesis is to present some new and recent results from the theory of non-Newtonian fluids. Such results refer to different motions of generalized second grade and Oldroyd-B fluids and ordinary Maxwell fluids. Generally, the con- stitutive equations for generalized non-Newtonian fluids are obtained from those for non-Newtonian fluids by replacing the time derivatives of an integer order by the so called Rieman-Liouville fractional operators. In chapter 2, it is studied the rotational flow of a generalized second grade fluid between two infinite coaxial cylinders. The velocity field ω(r, t) and the shear stress τ (r, t), obtained by means of Laplace and Hankel transforms, are presented under series form in terms of generalized G functions. The obtained solutions can be spe- cialized to give the similar solutions for ordinary second grade and Newtonian fluids performing the same motion. Chapter 3 deals with the study of helical flow of gener- alized Oldroyd-B fluids in a single circular cylinder. The components of velocity field and their associated shear stresses have been found in terms of generalized G and R functions and are presented as sum of two terms, one of them is the similar solution for the Newtonian fluid. Chapter 4 contains some remarkable results regarding the energetic balance for the flow of Maxwell fluid due to a constantly accelerating plate. We have determined the dissipation, the power due to the shear stress at the wall and boundary layer thickness for this motion. The corresponding results for the similar flow of a Newtonian fluid are also recovered as special case. The specific features of both fluids are compared and discussed. viiAcknowledgements In the name of Allah, the Most Beneficent, the Most Merciful. Clearly, when the task of this magnitude is undertaken, many people are involved in some capacity and deserve to be recognized. Firstly, it has been my pleasure to thank Dr. Constantin Fetecau, my supervisor, a highly educated, capable and devoted person, and I do feel much pleased and grateful for the attention, he has given me and also for the keen interest which he has kindly shown towards the future prospect of this dissertation. I will remain thankful to him for his encouragement to me whenever I lost heart due to lengthy and complicated computations. In fact the work could never been completed without his lucid instructions, consistent help and watchful checking. I took advantage of his useful advices and guidance at each step of the completion of this work. It is so nice of Professor A. D. R. Chaudhary, the Director General, Abdus Salam School of Mathematical Sciences (ASSMS), who provided me a wonderful atmosphere for quality research and hard work at this institute. I admire his efforts that he succeeded to attract world renowned mathematicians who contributed a lot to the progress of this school. I am also thankful to Higher Education Commission and the Punjab Govern- ment for their financial support, which supported me very much in concentrating my research work. I humbly pray to Allah for my parents, for their prayers and guidance throughout my life. I would also like to thank my colleagues and official staff of ASSMS for their moral support and cooperation.