Approximate Solutions of Differential Equations of Non-Newtonian Fluids Flow Arising in the Study of Helical Screw Rheometer The thesis presents the theoretical analyses of extrusion process inside Helical Screw Rheometer (HSR). Efforts to obtain better insight into the process must be mainly theoretical rather than experimental. But the hope, of course, is that better insight than experimental so gained will provide practical benefits such as better control of the processing, optimize the processing process and improve the quality of production. The main objective of the study is to develop mathematical models in order to evaluate the velocity profiles, shear stresses and volume flow rates for isothermal flow of incompressible non-Newtonian fluids in HSR. The calculations of these values are of great importance during the production process. In this thesis, two types of geometries are considered. • In first geometry the Cartesian co-ordinates system is used to study the flow of third-grade fluid, co-rotational Maxwell fluid, Eyring fluid, Eyring-Powell fluid and Oldroyd 8-constant fluid models in HSR. The geometry of the HSR is simplified by unwrapping or flattening the channel, lands and the outside rotating barrel. A shallow infinite channel is considered by assuming the width of the channel large as compared to the depth. We also assumed that the screw surface, the lower plate, is stationary and the barrel surface, the upper plate, is moving across the top of the channel with a velocity at an angle to the direction of the channel. The phenomena xis same as, the barrel held stationary and the screw rotates. Solutions for velocity profiles, volume flow rates, average velocity, shear and normal stresses, shear stresses at barrel surface and shear forces exerted on the fluid are obtained using analytical techniques. Adomian decomposition method is used to obtain the solutions for third-grade fluid, Eyring-Powell fluid and Oldroyd 8-constant fluid and perturbation method for co-rotational Maxwell fluid, where exact solution is obtained for Eyring fluid model. The effects of the rheological parameters, pressure gradients and flight angle on the velocity distributions are investigated and discussed. The behavior of the shear stresses is also discussed with the help of graphs for different values of non-Newtonian parameters. • For better analysis cylindrical co-ordinates system is taken in second geometry, assuming that the outer barrel of radius r 2 is stationary and the screw of radius r 1 rotates with angular velocity Ω. Here we have used third-grade fluid model with and without flight angle and co-rotational Maxwell fluid model with nonzero flight angle in HSR. The analytical expressions for the velocities, shear and normal stresses and the shear stresses exerted by the fluid on the screw, volume flow rates and average velocity are derived using analytical techniques and the outcomes have been presented with the help of graphs. The effects of the rheological parameters and pressure gradients on the velocity distribution are investigated.