Two and three-dimensional flows of second grade and micropolar fluids

Abstract

There is a wide use of non-Newtonian fluids in the industrial and teclmological processes. The heat transfer has a central role in these processes. No doubt, the behavior of non-Newtonian fluids calmot be analyzed through implementation of the classical Navier-Stokes equations. It has been recognized that the complex behavior of non-Newtonian fluids in nature cannot be predicted by one constitutive relationship between shear stress and rate of strain. Hence many constitutive relationships have been obtained for such fluids. The second grade is one of these fluids which have essence to predict the normal stress effects. There are fluids like polymeric liquid, crystals, animal blood etc which can be well analyzed using Eringen's theory of rnicropolar fluids. In such case, the local effects due to the microstructure and microrotations of the fluid have substantial impact on the flow. Bearing all such issues in mind, the present thesis is structured as follows. Chapter one briefly reviews the existing literature survey related to stretched flows. The equations of motion for axisymmetric flow of micropolar fluid are also included here. The two-dimensional flow of non-Newtonian fluid in convergent/divergent chalmel is explored in chapter two. Rheological expressions of second grade fluid are employed in the development of nonlinear differential equation. Resulting nonlinear mathematical problem containing continuity and momentum equations is solved by homotopy analysis method (HAM). Convergence interval of the derived solution is explicitly identified. Numerical values of skin friction coefficient are tabulated. Results showed that the flow quantities of interest are influenced greatly by embedded parameters. Main observations of this chapter are published in "Canadian Journal of Physics 88 (2010) 911". Chapter three examines the salient features of heat transfer in flow configuration studied in chapter two. For this purpose, the energy equation is solved in dimensionless fom1. Related convergence analysis is performed. Temperature distribution is analyzed for various pertinent parameters. Especially, the effects of angle of inclination between channel walls and Prandtl and Eckert numbers are given due attention. The obtained results are published in "International Journal for Numerical Methods in fluids 64 (2010) 761". Chapter four addresses the thermal-diffusion and diffusion-thermo effects in the axisymmetric flow of second grade fluid. The fluid is electrically conducting in the presence of a constant applied magnetic field. Further, the effects of louie heating and first order chemical reaction are taken into account. The mathematical statement of the problem is derived employing four fundamental laws namely the conversations of mass, linear momentum, energy and concentration. Transformation procedure reduces the partial differential equations into the ordinary differential equations. Homotopic solutions for physical quantities are constructed. Skin friction, Nusselt and Sherwood numbers are computed and analyzed in details. It is observed that the shear stresses on the surface of stretching sheet increases with an increase in magnetic field strength and non-Newtonian parameter of the fluid. Heat and diffusion flu xes are increased by increasing Reynolds, Hartman, Schamidt, Soret and Pnindtl numbers. The contents of this chapter have been published in "International Journal of Heat and Mass Transfer 54 (2011) 3031". Hall and ion-slip effects on three-dimensional flow of second grade fluid over a stretching surface have been shown in chapter five. Mathematical formulation is caITied out for small magnetic Reynolds number and constant material properties of fluid. Effects of various physical parameters on the dimensionless velocity components are examined by graphs. Variation of skin fi:iction coefficients for different involved parameters is seen through tabulated values. Skin friction coefficients are found to increase when Hartman number and second grade parameter are increased. On the other hand there is a decrease in skin friction coefficient when ion slip parameter is increased. These conclusions ar e published in "International J ournal for Numerical Methods in fluids 66 (2011) 183". Chapter six extends the work ofchapter five in the regime of heat transfer process. Viscous dissipation and Joule heating in the energy equation are considered. Effects of Prandtl number, local Eckert number, Hall parameter, ion-slip parameter and Hartman number on the dimensionless temperature are analyzed in particular. A comparative study between the present and existing limiting results is carefully made. Convergence regarding the obtained solution of temperature is shown. Nusselt number is analyzed for various values of sundry parameters. The results have been published in "Zeitschrift Naturforchung 56a (2010) 683". Soret and Dufour effects on the mixed convection three-dimensional flow of a second grade fluid over a vertical stretching have been considered in chapter seven. Matl1ematical analysis is presented in the presence of Hall and ion-slip cUITents. In order to get clear insight of the considered problem, the dimensionless velocities, temperature and concentration fields are displayed and numerical computations are caITied out for various values of embedded flow parameters. It is observed tl1at boundary layer iliickness can be controlled through Hartman number, convection parameters, ion-slip and Hall slip parameters. A comparative study between ilie present and previous limiting results is carefully seen. The contents of this chapter are published in "International Journal for Numerical Methods in fluids 67(9) (2011) 1073". Chapter eight discusses magnetohydrodynamic flow of a micropolar fluid between ilie radially stretching sheets in ilie presence of constant magnetic field. A uniform magnetic field is applied in the transverse direction to flow. The governing partial differential equations are transformed into ilie ordinary differential equations using similarity transformations. The nonlinear problem is computed. Convergence of obtained solutions is checked. The velocity profiles are discussed for the pertinent parameters. The values of skin friction and wall couple stress coefficients are obtained for various values of Reynolds number, Hartman number and micropolar fluid parameter. These findings have been published in "Zeitschrift Naturfor chung 66a (2011) 53". The work of chapter eight in the presence of heat transfer is extended in chapter nine. In other words, axisymmetric flow of magneto hydrodynamic micropolar fluid between two radially stretching sheets with heat transfer is explored. Viscous dissipation, micropolar heat conduction and Joule heating are present. Ertergy equation is transformed into the ordinary differential equation by appropriate variables. The resulting nonlinear problem is solved for the series solution. It is observed that the rate of heat transfer in micropolar fluid is higher than that in a Newtonian fluid. From practical point of view, micropolar fluid can be used instead of Newtonian fluid if some one is interested to increase the rate of heat transfer from surface into the fluid. This is significant when certain temperature is required to improve the quality of product in manufacturing process. The contents of this chapter have been published in "Journal of Mechanics, 27 (2011) 607". Dufour and Soret effects on axisymmetric two-dimensional flow of an incompressible micropolar fluid between radially stretching sheets are studied in chapter ten. The fluid is taken electrically conducting. Joule heating and chemical reaction are considered. The energy and concentration laws develop the mathematical formulation. The related problems are solved and validity of series solutions is verified through residual errors. Dimensionless temperature and concentration are discussed through graphs. Behaviors of sundry parameters on skin friction coefficient, wall couple stress coefficient, Nusselt and Sherwood numbers are analyzed. The contents of chapter ten have been submitted in "Computers & Fluids". Chapter eleven explores the MHD time-dependent flow problem of a micropolar fluid between two radially stretching sheets. Both cases of strong and weak interactions of microelements are considered. The equations along with the boundary conditions are solved. The variation of micropolar parameter on axial and radial velocity component is opposite to that of Hartmann number. The effect of unsteadiness parameter on axial, radial and angular velocity is discussed. The variations of Hartman number and micropolar parameter on angular velocity in weak and strong concentrations are opposite. Such results are published in "Applied Mathematics and Mechanics; English Edition 32(3) (2011) 361".