SOME FLOW PROBLEMS IN NON-NEWTONIAN FLUIDS

Abstract

las thesis consists of a study of some flaw problems of certain aon-Hewtonian fluids. It rune into seven chapters. Hie first chapter ie introductory and covers, in the first instance, the development of the iundameatal concepts '"•^s ^w™mBBwme*wp^as* * aaaip MrjsscasuBawPmp sa ^f ernes ew ^MBfwFs»tBMs^p iWHPasWP*' wg"sws> jghex v*e* ^<^s*b *m^mwssa^s#aejfc w^ vggwF wsa^g^wflF^esswBP a«e* ^P •WB^m •*e*ap^w vmMvapasassm osssss ep^maavmwsm sm» wpsPdfc* fluids. It also provides the transformation, to different coordinate systems (used In the thesla), of the fundamental equations of motion and oonstitutivs equation of second order fluid Ins subssqusnt eontsnt of the thesis forma the main contribution, chapters IX-VI embody the steady aad unsteady j1^ problems in second order fluids aad the oonoludlnm chanter VII contains the problem of aperiodic wave propagation in a compressible • lastloo-viscous fluid. las brief outlines of the chapters are as follows i Chapter 11 comprises the study of the steady flow of an incompressible second order fluid in the aanulus of tmo rotating coaxial cylinders with suction sad Injection on their surfaces and with the relative linear action of the cylinders, solution has been found in terms of a series of ascending powers of suction parameter which la assumed email* She slsstico-viscous aad cross-viscous effects are found to be governed by two non11. dimensional parameters « (• j*-*) and B(« t*£) reepectively, ^1* /a2f/u3 •*• *b**01|lon visoceity, elsstico-viscosity exd croae-viaccelty coefficients and U0, Dare, reepeotively, the characteristic velocity sad characteristic length. Besides other Interacting results, it is found that the axial component desxsasea wails toroidal component increasea with the increase in ek opposite ie toe phenomenon when the order of auction aad injection ia reversed. It ie further observed that axial ecapaaeat decresses with the inoreass la cross-viscosity, this parameter hem no offsot oa toroidal component of velseity. Behaviour of a circular oscillating cylinder, suspended in aa lnfinlts expense of Incompressible second order fluid, is examined in chapter III, She expressions for velocity compensate aad the drag, sxperisaoad by the cylinder, are obtained by using asymptotic ssrlee for Modified Bessel Functions of first ordsr and second kind. It ia found out that the presence of elasticity ia the fluid reduees the virtual mesa of the oyUnder end increases the damping fcress. It is alec found that the magnitude of damping foroes deereesai ea we increase the else of the cylinder. Chapter XV embodies the study of the behaviour of sphere oeolUating la ea infinite expanse of incompressible second ordsr f laid. Jfixprsaelons of the drag experienced by the ephere and the velocity oospenente of the motion induced In the fluid are found analytieally. It is concluded that the drag increasea as the viscous or the elestle forces of the fluid are incrssxsd. ill. Aa effects of the else of the sphere on the drag ars aleo studied sad found that the magnitude of damping forces, acting oa the sphere,decreases whan the radios of toe sphere is increases. Xt la interesting to note that the cross-viscous foroee, la chapter III as asll as In chapter IV, do aot affect the flow pattern and the drag. la chapter V, we have diacuseed the unsteady flow through the circular pipe, when the aeoond order fluid la act in motion, from rest, impulsively by applying a constant pressure gradisat across the pipe. Xhe solution ie obtained, aumsrioally, by using finite difference method. She elestioovlsoous effects are found to be governed by a non-dimensional parameter «(• -*~) where £ is the density of the fluid and a' ia the radius of the pipe. It is found out that the preeenoe of elasticity listens the parabolic shape of the velocity profile, the cross-viscosity doss act affect the velocity profile. la chapter VX, Stokes *a firet aad second problems axe studies ia second order fluids. She problem ie attacked by operational sethods and inverss laplaes trsnsform is obtained, numerically, by using Poet-adder*e formula. Stokes *e results ere eoaflrmad when the viscous fluid is considered. Ins viscous aad clastic effects are found to be governed by iteynoldvs numbei istie length, it is found that with the decrease in viscosity. lv. the depth of penetration, in both the problems, deoreeece. Elasticity is found to reduce the region of induced motion la the fluid. Values of velocity profile for varioue times ars alao calculated for both the viscous and ssesad order fluids. She concluding chapter VX1 is devoted to the problem of the propagation of a non-periodic wave in a ooapreaalble slsstico-visecus fluid of oLdroyd^s model B, occupying a eemi-infinite epaoe. Shis problem ie almo salved, aumsriaally, with the help of Post-vidder'e formula. Ins viscoua and elastic effects are governed by tmo non-dimensional psremeters l(. 4ytsg ) and «(- ** ) whsrs /^ , ot X% are, reepeotively, 3^ozfa o the coefficient of viscosity, density end relaxation tins of the fluid, while o and f0 are the velocity of sound in an invieeid f laid and a ebaraoterlstle time, Xt la interesting to acta that viscous and elastic effects are opposing each other* the viscosity has diepsreivs offeet on the wave front while cleetioo-vieoeaity has contractive effect. It has also been found that, with the passage of time, thie opposition Ihe entire numerical work has been carried out on I.B.M. 1620 computer Installed at Structural Sngineering Rsssaroh Centre,Roorkee and an ball 360 installed at Delhi university. »e resulte are summarised at ths sad of each Xhe work of WmWm% throe ohsptera has been acosptsd for publication aa follows: (i) "Steady Flam of Second Order Fluid Ihrough ths Annulua of Parses Coaxial Cyliadere", accepted for publication in Indian Journal of Pare and Applied Mathematics. (ii) "Drag on a Circular Cylinder Ceaillatlag in Second Order Fluid", published in the Proceedings of First Annual CASMACU Research Symposium, Calcutta* (ill) Drag oa a Sphere Oscillating in Second Order I laid", accepted for publication in Journal of batheaatloal aad Physical Solencee, iacres. lotet Sensorial notations with ths following oonvsntlon are ased throughout ths thesis. (1) 9m cub-suffix following the symbol deastea ths sovariaat character of the entity while supsr-suffix exhibits the eontrsvarlsnt character of the entity, (ii) A auffix appearing both at the apper end lower 3?laces ia a certain torn iapUsa summation over lte values. (ill) suffix following 'comma' denotes the covariant differentiation. (Iv) gjj and g1* are, reepectlvely, the metric tensor and