We consider rotationally driven nonlinear polymeric fiber jet, whose centerline is in an inclined plane, in the presence of gravity force. An empirical viscosity model is used for the polymeric fluid flow to investigate properties of the rotating inclined polymeric fiber jet. The aim of the study is to understand properties of such inclined fiber jet shape, which can be due to orientation change of an orifice in a rotating spinneret that generates such jet. Perturbation, asymptotic, scaling and numerical techniques are used to determine the nonlinear steady solutions for the jet quantities for different values of the parameters due to gravity, rotation, viscosity, surface tension and relaxation time. In contrast to the horizontal jet case, presence of gravity and jet inclination increase values of the jet speed, strain rate, stretching rate and the centerline curvature and decrease the value of the jet radius and more so with increasing the arc length, gravity, rotation rate and the relaxation time. However, a non-inclined jet with no imposed restriction on its shape and in presence of gravity leads to smaller fiber radius and larger speed as compared to the ones for inclined jet case.

In this computational paper, we focused on the efficient numerical implementation of semi-implicit methods for models in materials science. In particular, we were interested in a class of nonlinear higher-order parabolic partial differential equations. The Cahn-Hilliard (CH) equation was chosen as a benchmark problem for our proposed methods. We first considered the Cahn-Hilliard equation with a convexity-splitting (CS) approach coupled with a backward Euler approximation of the time derivative and tested the performance against the bi-harmonic-modified (BHM) approach in terms of accuracy, order of convergence, and computation time. Higher-order time-stepping techniques that allow for the methods to increase their accuracy and order of convergence were then introduced. The proposed schemes in this paper were found to be very efficient for 2D computations. Computed dynamics in 2D and 3D are presented to demonstrate the energy-decreasing property and overall performance of the methods for longer simulation runs with a variety of initial conditions. In addition, we also present a simple yet powerful way to accelerate the computations by using MATLAB built-in commands to perform GPU implementations of the schemes. We show that it is possible to accelerate computations for the CH equation in 3D by a factor of 80, provided the hardware is capable enough.