A technique for calculating exponentially small terms beyond all orders in singularly perturbed difference equations is presented. The approach is based on the application of a WKBJ-type ansatz to the late terms in the naive asymptotic expansion and the identification of Stokes lines, and is closely related to the well-known Stokes line smoothing phenomenon in linear ordinary differential equations. The method is illustrated by application to examples and the results extended to time-dependent differential-difference problems.

In a previous paper the author and Demay advanced a model to explain the melt fracture instability observed when molten linear polymer melts are extruded in a capillary rheometer operating under the controlled condition that the inlet flow rate was held constant. The model postulated that the melts were a slightly compressible viscous fluid and allowed for slipping of the melt at the wall. The novel feature of that model was the use of an empirical switch law which governed the amount of wall slip. The model successfully accounted for the oscillatory behavior of the exit flow rate, typically referred to as the melt fracture instability, but did not simultaneously yield the fine scale spatial oscillations in the melt typically referred to as shark skin. In this note, a new model is advanced which simultaneously explains the melt fracture instability and shark skin phenomena. The model postulates that the polymer is a slightly compressible linearly viscous fluid but assumes no-slip boundary conditions at the capillary wall. In simple shear the shear stress τ and strain rate d are assumed to be related by d = Fτ, where F ranges between F2 and F1 > F2. A strain-rate dependent yield function is introduced and this function governs whether F evolves towards F2 or F1. This model accounts for the empirical observation that at high shears polymers align and slide more easily than at low shears, and explains both the melt fracture and shark skin phenomena.

Fibre drawing is an important industrial process for synthetic polymers and optical communications. In the manufacture of optical fibres, precise diameter control is critical to waveguide performance, with tolerances in the submicron range that are met through feedback controls on processing conditions. Fluctuations arise from material non-uniformity plague synthetic polymers but not optical silicate fibres which are drawn from a pristine source. The steady drawing process for glass fibres is well-understood (e.g. [11, 12, 20]). The linearized stability of steady solutions, which characterize limits on draw speed versus processing and material properties, is well-understood (e.g. [9, 10, 11]). Feedback is inherently transient, whereby one adjusts processing conditions in real time based on observations of diameter variations. Our goal in this paper is to delineate the degree of sensitivity to transient fluctuations in processing boundary conditions, for thermal glass fibre steady states that are linearly stable. This is the relevant information for identifying potential sources of observed diameter fluctuation, and for designing the boundary controls necessary to alter existing diameter variations. To evaluate the time-dependent final diameter response to boundary fluctuations, we numerically solve the model nonlinear partial differential equations of thermal glass fibre processing. Our model simulations indicate a relative insensitivity to mechanical effects (such as take-up rates, feed-in rates), but strong sensitivity to thermal fluctuations, which typically form a basis for feedback control.

The classical time-dependent drift-diffusion model for semiconductors is considered for small scaled Debye length (which is a singular perturbation parameter). The corresponding limit is carried out on both the dielectric relaxation time scale and the diffusion time scale. The latter is a quasineutral limit, and the former can be interpreted as an initial time layer problem. The main mathematical tool for the analytically rigorous singular perturbation theory of this paper is the (physical) entropy of the system.

Free convection along a vertical flat plate embedded in a porous medium is considered, within the framework of boundary layer approximations. In some cases, similarity solutions can be obtained by solving a boundary value problem involving an autonomous third-order nonlinear equation, depending on a parameter related to the temperature on the wall. The paper deals with existence and uniqueness questions to this problem, for every value of the parameter.