Miccal T. Matthews

A note on the boundary layer equations with linear slip boundary condition

Abstract In a recent article the authors study the effect of replacing the standard no-slip boundary condition with a nonlinear Navier boundary condition for the boundary layer equations. The resulting equations contain an arbitrary index parameter, denoted by n , and it is found that the case n = 1 corresponding to linear Navier boundary condition must be excluded. In this article the authors remedy this situation and show that the case n = 1 corresponds to a particular similarity solution, not included in the previous work by the authors.

Asymptotic phenomena in pressurized thin films

An asymptotic study of the wrinkling of a pressurized circular thin film is performed. The corresponding boundary-value problem is described by two non-dimensional parameters; a background tension μ and the applied loading P. Previous numerical studies of the same configuration have shown that P tends to be large, and this fact is exploited here in the derivation of asymptotic descriptions of the elastic bifurcation phenomena. Two limiting cases are considered; in the first, the background tension is modest, while the second deals with the situation when it is large. In both instances, it is shown how the wrinkling is confined to a relatively narrow zone near the rim of the thin film, but the mechanisms driving the bifurcation are different. In the first scenario, the wrinkles are confined to a region which, though close to the rim, is asymptotically separate from it. By contrast, when μ is larger, the wrinkling is within a zone that is attached to the rim. Predictions are made for the value of the applied loading P necessary to generate wrinkling, as well as details of the corresponding wrinkling pattern, and these asymptotic results are compared to some direct numerical simulations of the original boundary-value problem.

On using planar microcontractions for extensional rheometry

Planar contractions within a microfluidic environment have shown some potential to be used as a simple, cheap extensional rheometer. In this article, flow within a planar contraction is analyzed using an energy minimization calculation with a measure of slip included. The effect of slip on the calculated flow-field is used to show how the apparent extensional viscosity compares with the true extensional viscosity. In particular, it was found that the apparent planar extensional viscosity would be difficult to determine reliably at either low or high Weissenberg numbers but could be distinguished to some extent at moderate Weissenberg numbers between 0.25 and 1. It was also shown, however, that an unknown extent of slip at the vortex boundary would cause an uncertainty of a factor of 2 in any apparent measure of rheological properties.

An analytical and numerical study of unsteady channel flow with slip

A theoretical investigation of the unsteady flow of a Newtonian fluid through a channel is presented using an alternative boundary condition to the standard no-slip condition, namely the Navier boundary condition, independently proposed over a hundred years ago by both Navier and Maxwell. This boundary condition contains an extra parameter called the slip length, and the most general case of a constant but different slip length on each channel wall is studied. An analytical solution for the velocity distribution through the channel is obtained via a Fourier series, and is used as a benchmark for numerical simulations performed utilizing a finite element analysis modified with a penalty method to implement the slip boundary condition. Comparison between the analytical and numerical solution shows excellent agreement for all combinations of slip lengths considered. doi:10.1017/S1446181112000272

Complex Mapping of Aerofoils--A Different Perspective

In this article an application of conformal mapping to aerofoil theory is studied from a geometric and calculus point of view. The problem is suitable for undergraduate teaching in terms of a project or extended piece of work, and brings together the concepts of geometric mapping, parametric equations, complex numbers and calculus. The Joukowski and Karman–Trefftz aerofoils are studied, and it is shown that the Karman–Trefftz aerofoil is an improvement over the Joukowski aerofoil from a practical point of view. For the most part only a spreadsheet program and pen and paper is required, only for the last portion of the study of the Karman–Trefftz aerofoils a symbolic computer package is employed. Ignoring the concept of a conformal mapping and instead viewing the problem from a parametric point of view, some interesting mappings are obtained. By considering the derivative of the mapped mapping via the chain rule, some new and interesting analytical results are obtained for the Joukowski aerofoil, and numerical results for the Karman–Trefftz aerofoil.