Two and three dimensional flows in the non-N ewtonian fluids

Abstract

The non-Newtonian fluids find increasing practical appUcations in the recent years. It is due to the fact that many of the fluids used in industry and engineering are significantly nonNewlonian. The flows of such fluids have special relevance in oil and gas well drilUng to well completion operations from industrial processes involving waste fluids, synthetic fibers, foodstuffs to extrusion of molten plastic and polymer solutions. The expression between shear stress and shear rate in the non-Newtonian fluids is non-linear. A distinct fea ture of nonNewtonian fluids from Newtonian fluid is tha t these cannot be described by a single constitutive equa tion. Mathernatical systems in the non-Newtonian fluids are of higher order and in general more complicated in comparison to the Newtonian fluids. These systems need additional initial/boundary conditions for a unique solution. The boundary layer flows of non-Newtonian fluids over a stretching sheet is important in a variety of contexts including extrusion process, glass fiber and paper production, hot rolling, wire drawing, crysta l growing in food processing and movements of biolog-jcaI fluids. Such flows add complexities to the governing equations for the dependence of physical quantities in the two and three dimensions. The heat transfer analysis in boundary layer flow with radiation is further important in electrical power generation, astrophysical flows, solal' power technology, space vehicle reentry and other industrial areas. Mass transfer in such flows is inspired for an interest in membrane separation process, microfiltration, and reverse osmosis, in electrocllemislTY and fiber industries. There are transport processes in industrial applica tions in which heat and mass transfer is a consequence of buoyancy effects caused by thermal and mass diffusion in the presence of chemical reaction. Such interaction is significant in the design of chemical processing equipment, nuclear reactor safety, combustion of solar system equi pment etc. Motivated by such practical applications, the present thesis is structured as follows. Chapter one provides the background and boundary layer equations for some models of nOI1- Newtonian fluids namely second grade, Maxwell, Jeffrey and micropolar fl uids. Brief idea of homotopy analysis method (HAM) is also given. The unsteady mixed convec tion boundary layer magnetohydrodynamic (MHD) flow of a second grade fl uid bounded by a stre tching surface has been addressed in chapter two. Both the stretching velocity and U1e surface tempera ture arc taken time-dependenl. Problem formulation is developed in the presence of thermal radiation. Governing nonlinear problem is solved by a homotopy analysis method. Convergence of derived solution is studied. The dependence of velocity and temperature profiles on various quantities is shown and discussed by plotting graphs. Numerical values of skin friction coefficient and local Nusselt number are tabula ted. It is noticed that velocity profiles are increasing functions of second grade parameter. The local Nusselt number also increases when the value of Prandtl number is increased. These observations have been published in "International Journal for Numerical Methods in FlUids, DOl: 10.100/f1d.2285". Chapter three in vestiga tes the w1Steady three-dimensional flow of an elastico-viscous fluid Over a stretching sheet. The mass transfer analysis is also studied. The governing bounda ry layer equa tions are reduced into the partial diffe rential equa tions by similarity transforma tion. The effects of embedded parameters in tile considered problem are examined in detail. NUmerical data for sUl'face shear sb'esses and surface mass transfer in steady case are also tabulated. Both cases of destructive/genera tive chemical reactions me analyzed. It is found that tile influence of viscoelastic parameter and the Harhllan n umber on tl1e sheat· stresses are quantitatively similar. Such results are published in "International Journal for Numerical Methods in Fluids, 00[: 10.lOO/f1 d.2252". 'fhe joule heating and thermophoresis effects on MHO flow of a Maxwell fluid in the presence of tl1ermal radia tion are studied in chapter four. The nonlinear ordinary differential sYStems obtained after employing similarity transformations have been solved and series soLutions are constructed. The local Nusselt and Sherwood numbers are further computed. The thermal boundary layer thickness increases with increase in the Prand tl number. The 10Cc'l1 Nusselt and Sherwood number increases when porosity pa rameter is increased. These conclusions have been published in t'International Journal of Heat and Mass Transfer, 53 (201.0) 4780-4788". T'he influence of heat transfer on the boundary layer flow of a Maxwell fluid over a moving per1.11eable surface in a pa rallel free stream is argued in chapte r five. Solution of the governing prOblem is developed by homotopy analysiS method. The results of velOCity, tempera ture and Nusselt number are presented and discussed for various emerging pa rameters. A comparative ii study is seen with the known numerical solution in a limiting sense and an excellent agreement is noted. It is also fou nd tha t velocity in the Maxwell fluid is less than the viscous fluid. It is revealed that the bowldary layer thickness decreases with the increasing va lues of Deborah number. The thermal bowldary laye r thickness decreases by increasing suction parameter. This research is submitted f01" publication in "Ch inese Journal of Mechanics~Series A". In chapter six, we perform a study for heat and mass h'ansfer analysis in the presence of thermal radiation on the unsteady MHD flow of a micropoJar fluid. Series solutions for velocity, temperature and concentration fields a re derived and discussed. Plots for various interesting parame ters are re ported and analyzed . Numerical data for surface shear stress, Nusselt number and Sherwood numbers in steady cases are also computed. Compa rison between the present and previous limiting results in shown. The results of this chapter have been published in "Zeitschrift Naturforschung A, 64 (2010) 950-960". O lapter seven is prepared to analyze the heat and mass transfer characteristics for the steady mixed convection flow of an incompressible micropolar fluid. The relevan t system of the partial differential equations has been reduced into ordinary differential equa tions by employi.ng similarity transformation. Series solutions for velocity, temperature and concentration fields are developed by Llsing homotopy analysis method (HAM). Effects of various parameters un velocity, tempera ture and concentration fields are discussed by displaying graphs. Numerical va lues of skin friction coefficient, Nusselt number and Sherwood numbers are worked out. A comparison be tween the available numerical solutions in a limiting situation is seen. The velocity profile is found to decrease when the Prandtl number increases. Moreover, the buoyancy parameter decreases the thickness of thermal boundary layer. These conclusions have been accepted for publication in "International Journal for Numerical Methods in Fluids, DOl: lO.100?jfld.2424". Chapter eigh t examines the MHD flow and mass transfer characteristics in a Jeffrey fl uid bounded by a non-linearly stretchin g surface. The velocity and concentration fields are derived. Homotopy analysis proceduJe is adopted for computations of a set of coupled nonlinear ordinary differential equations. Effects of involved parameters on the velocity and concentra tions fields are examined carefully. Numerical values of mass h'ansfer coefficient are first tabulated and then investiga ted. As expected the concentration fields decreases by increasing the Schimidt number. 11K! surface mass transfer decreases when the Hartman number increases. The observations of this problem have been published in "Zeitschrift Naturforschung A, 64 (2010) 1111 - 1120" . In chapter nine, we have considered the effect of mass transfer on stagna tion point flow of a Jeffrey fluid bounded by a stre tching surface. Similarity transformations reduce the partial differential equations into the ordinary differential equations. Homotopy analysis method (HAM) is invoked for the development of solulions. Plots are prepared to illustrate the flow and mass transfer d laracteristics and their dependence on the physical parameters. The values of surface mass transfer and g-rad ient of mass transfer are computed and analyzed. It is observed that velocity field and boundary layer thickness are increasing fWlCtions of Deborah number. The fluid concentration increases with an increase in generative chemical reaction parameter and has opposite behavior for destructive chemical reaction when compa red with the situation in case of generative chernical reaction parameter. Such contexts have been submitted for publication in "Asia-Pacific Journal of Chemical Engineering". The wlsteady stagnation pOint flow of a second STade fluid with hea t transfer is discussed in chapter ten. The time-dependent free stream is considered. The equa tions of motion and energy are transformed ullO lhe ordinary difft:!rt!ntial equations by similarity transformations. Homotopy analysis method is used to find the solution of the governing problem. Graphical results are given in order to ill ustra te the deta ils of flow and heat transfer characteristics and their dependence upon the embedded parameters. Numerical values of skin- friction coefficients and Nusselt number are given and examined carefully. It is seen tha t velocity is greater for second grade fluid when compared with a Newtonian fl uid. The velocity and boundary layer thickness ulcreases Ul both cases of suction and injection as second grade pa rameter increases. However, in injection case such increase is la rger than that of suction. We further fowld that for fixed values of other parameters, the local Nusselt number increases when there is an increase Ul the second grade parameter. The findings of this chapter have been submitted for publication in "International Journal of Heat and Mass Transfer". Chapter eleven discllsses the steady mixed convection stagnation point flow of a micropolar flow towards a stretching sheet. Governing problems of flow, heat and mass transfer are solved by employulg homotopy analysis method (HAM). The skin friction coefficients, local Nusselt iv number and Sherwood number are computed . Comparison of the present solulion series solution is given to the corresponding numerical solution. A good agreement is achieved. When the stretching velocity of the surface is greater than the stagnation velocity of the external stream the flow has inverted boundary layer sh·ucture. The effect of the local buoyancy parameter on the velocity is fOWld similar to the material parameter. These observations have been submitted for publication in "Central Euorpean Journal of Physics".