The thesis is concerned with the flow of polymeric fluids and their response to deformations. The current state of research into the rheology of polymers is reviewed and an introduction to non-Newtonian fluid mechanics is given. A novel numerical algorithm for simulating the flow of non-Newtonian fluids is presented, in which the fluid is represented by a set of Lagrangian particles embedded in it. Each particle carries a velocity and a stress with it as it moves through the flow geometry. The velocity gradient tensor and the divergence of the stress tensor are calculated for each particle by averaging over the neighbouring particles. The velocity is changed according to a discretised form of the equation of conservation of momentum and the stress is updated according to the constitutive equation for the fluid. Extra algorithms are presented to deal with the boundary conditions. The simulation is used to study the two-dimensional flow of a co-rotational Maxwell fluid past an array of cylinders between two walls. In the second part of the thesis, a computer simulation is developed which will allow the constitutive equation for a polymer melt to be replaced by a numerical version. Previous computer simulations of polymers are reviewed and a new, real space, reptation model is presented. This model is shown to have the correct behaviour for a reptating chain and is used to study the stress response to a step shear deformation. The long-term behaviour agrees with reptation theory, but the short-time behaviour is also found.

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