Newtonian fluids like water, air, milk, glycerol, thin motor oil and alcohol and Non-Newtonian fluids such as paint, ketchup, blood, custard, toothpaste, shampoo and starch suspensions etc. vary tremendously in their properties and behaviors. It is immensely important to study the physical behavior of these fluids in order to enhance their performance in various indus-trial and manufacturing procedures. One of the pertinent non-Newtonian fluid nowadays is nanofluid which has extensive range of utility in numerous engineering problems e.g.,heat exchangers, chemical processes, cooling of electronic equipment, in nuclear reactors,safer surgery, cancer therapy, heat exchangers, micro-channel heat sinks, in designing the waste heat removal equipment, paper printing, polymer extrusion, rapid spray cooling,glass blowing, cooling of microelectronics, quenching in metal foundries and wire drawing. Thus this thesis emphasizes on the modeling of Newtonian and non-Newtonian fluids possessing distinct flow geometries and their solutions. The governing systems of equations for Newto-nian and non-Newtonian fluids are of higher orders, so the solutions are not easily attainable.Four different techniques, namely homotopy analysis method, optimal homotopy analysis method, shooting method and method of lines have been employed to solve these different flow geometries.The first chapter is based on the relevant literature review, some basic laws and definitions.Various methods employed in the thesis are also discussed briefly.The second chapter incorporates steady magnetohydrodynamic flow of nanofluid between two concentric circular cylinders with the consideration of heat generation/absorption effects.The flow is assessed with respect to constant surface temperature （CST） and constant heat flux（CHF） thermal boundary conditions. The governing nonlinear partial differential equations are remodeled into a dimensionless system of ordinary differential equations by means of suitable similarity transformations and solutions are obtained by employing homotopy analysis method. Comparison of computed solutions with existing results in the literature are displayed. The heat and mass transfer characteristics are analyzed for various values of relevant parameters by demonstrating and discussing the plots of velocity, temperature and concentration profiles. The numerical values of skin friction coefficient, Nusselt number and Sherwood number for both the boundary conditions are also computed.The third chapter is devoted to the flow of third grade nanofluid instigated by riga plate.The theory of Cattaneo-Christov is adopted to investigate the thermal and mass diffusions and the incorporation of newly eminent zero nanoparticles mass flux conditions yield im-portant results. The governing system of equations is nondimensionalized through relevant similarity transformations. The behavior of affecting parameters for velocity, temperature and concentration profiles is briefly examined and graphically indicated. The values of skin friction coefficient and Nusselt number with the relevant preliminary discussion have been recorded.In the fourth chapter, the influence of homogeneous heterogeneous reactions on the flow of single-wall and multi-wall carbon nanotube fluid along the surface of riga plate fixed in a porous medium is analyzed. The riga surface which is recognized as an electromagnetic drive consisting of a sequence of constant magnets and a span wise adjusted array of alternating electrodes mounted on a flat surface is of great importance in many demanding problems.Further, the problem is based on water and kerosene oil as two different base fluids and vis-cous dissipation is discussed as well. Numerical solutions for non-dimensionalized ordinary differential equations are assembled with the help of shooting technique and by employing the same procedure, the conduct of dominating parameters on velocity, temperature and concentration profiles is reported. The values of skin friction coefficient and Nusselt number are determined through tabular data.The last chapter deals with the capillary rise dynamics for magnetohydrodynamics（MHD） fluid flow through deformable porous material in the presence of gravity effects. The modeling is performed using the mixture theory approach and mathematical manipulation yield a nonlinear free boundary problem. Due to the capillary rise action the pressure gradient in the liquid generates a stress gradient which results in the deformation of porous substrate.The capillary rise process for MHD fluid slows down as compared to the Newtonian fluid case. Numerical solutions are obtained using the line approach. The graphical results are presented for important physical parameters and comparison is presented with the Newtonian fluid case.