Investigation of Numerical Evaluation Improvement for Three-Dimensional Infinite Cylindrical Heat Conduction Problems

To estimate the thermal properties from transient data, a model is needed to produce numerical values with sufficient precision. Iterative regression or other estimation procedures must be applied to evaluate the model again and again. From this perspective, infinite or semi-infinite heat conduction problems are a challenge. Since the analytical solution usually contains improper integrals that need to be computed numerically, computer-evaluation speed is a serious issue. To improve the computation speed with precision maintained, an analytical method has been applied to three-dimensional (3D) cylindrical geometries. In this method, the numerical evaluation time is improved by replacing the integral-containing solution by a suitable finite body series solution. The precision of the series solution may be controlled to a high level and the required computer time may be minimized by a suitable choice of the extent of the finite body. The practical applications for 3D geometries include the line-source method for obtaining thermal properties, the estimation of thermal properties by the laser-flash method, and the estimation of aquifer properties or petroleum-field properties from well-test measurements. This paper is an extension of earlier works on one-dimensional (1D) and two-dimensional (2D) cylindrical geometries. In this paper, the computer-evaluation time for the finite geometry 3D solutions is shown to be hundreds of times faster than the infinite or semi-infinite solution with the precision maintained.

Polymer Brush Friction in Cylindrical Geometries

Polymer brushes are outstanding lubricants that can strongly reduce wear and friction between surfaces in sliding motion. In recent decades, many researchers have put great effort in obtaining a clear understanding of the origin of the lubricating performance of these brushes. In particular, molecular dynamics simulations have been a key technique in this scientific journey. They have given us a microscopic interpretation of the tribo-mechanical response of brushes and have led to the prediction of their shear-thinning behavior, which has been shown to agree with experimental observations. However, most studies so far have focused on parallel plate geometries, while the brush-covered surfaces might be highly curved in many applications. Here, we present molecular dynamics simulations that are set up to study the friction for brushes grafted on the exterior of cylinders that are moving inside larger cylinders that bear brushes on their interior. Our simulations show that the density distributions for brushes on the interior or exterior of these cylinders are qualitatively different from the density profiles of brushes on flat surfaces. In agreement with theoretical predictions, we find that brushes on the exterior of cylinders display a more gradual decay, while brushes on the interior of cylinders becomes denser compared to flat substrates. When motion is imposed, the density profiles for cylinder-grafted brushes adapt qualitatively differently to the shear motion than observed for the parallel plate geometry: the zone where brushes overlap moves away from its equilibrium position. Surprisingly, and despite all these differences, we observe that the effective viscosity is independent of the radius of the brush-grafted cylinders. The reason for this is that the viscosity is determined by the overlap between the brushes, which turns out to be insensitive to the exact density profiles. Our results provide a microscopic interpretation of the friction mechanism for polymer brushes in cylindrical geometries and will aid the design of effective lubricants for these systems.