"The objective of this work is to study the instabilities of the contraction flows at high Deborah numbers of various polymers. The following topics will be discussed: 1. Linear and nonlinear stability analysis of isothermal fiber spinning; 2. Linear stability analysis of nonisothermal fiber spinning; 3. Linear and nonlinear stability analysis of contraction flow; [and] 4. Propagation of reservoir instabilities in capillary. The analysis of these flow problems requires solution of the closed set of PDE's (or ODE's), consisting of equations for conservation of mass and momentum, along with an adequate viscoelastic constitutive equation, with appropriate inital/boundary conditions. The goal of this work is to demonstrate a procedure for determining the critical regime beyond which the process becomes unstable and also to determine weather [sic] the process is stable when the disturbances grow to a finite size. Linear and non-linear stability theories have been used to describe the fluctuation of fiber spinning and contraction flow. Linear stability analysis determines the onset of the instabilities of the process while nonlinear analysis establishes the complete range of the stable and unstable conditions. The melt fiber spinning is the most common of polymer fiber processing. Finding critical process conditions and the stabilizing effect of the cooling is described in this work. The critical draw ratio is established using linear stability analysis and the effect of the finite size imposed disturbances is studied through nonlinear stability analysis. Contraction flow is one of the benchmark problems in computational polymer fluid mechanics and polymer processing. In this modeling, the whole flow region is divided in naturally introduced sub-regions with well-known and highly simplified types of flow. Thus, the model analyzes the entire flow region in a simplified geometric manner with properly matched conditions between adjacent sub-regions. The propagation of the disturbances formed in the reservoir region has been analyzed. Employing the isothermal 'Jet approach' followed by linearized perturbation approximation of the governing equations for finding the onset of the instabilities supplies information about the stability of the contraction flow which has been used to describe the mechanism of propagation of the disturbances into capillary up to the die exit, and its numerical implementation."--Abstract.