Corner singularity

The analysis of the corner singularity is delicate. We refer to the recent works of Hinch [36] and Renardy [37,38], who have contructed a matched asymptotic expansion for the steady solution to a Maxwell fluid flow near the corner. 

Unlike prisms, in this class of bodies uniqueness requires knowledge of the density. This theorem was proved by P. Novikov. The simplest example of starshaped bodies is a spherical mass. Of course, prisms are also star-shaped bodies but due to their special form, that causes field singularities at corners, the inverse problem is unique even without knowledge of the density. It is obvious that these two classes of bodies include a wide range of density distributions besides it is very possible that there are other classes of bodies for which the solution of the inverse problem is unique. It seems that this information is already sufficient to think that non-uniqueness is not obvious but rather a paradox. 

The interesting physics of this diffusion problem results from the existence of a singularity at the corner of the coating (b-q, 0). At this point, the flux approaches infinity and much of the flux from the occluded portion may "pour" around the corner and flow through the hole. To study corner flow, we define the normalized corner flow function, Z,... 

A typical result of numerical analysis is an estimate of the error U — Uh between the solution U of the continuous problem (. e., the solution of the initial boundary value problem) and the solution Uh of the discrete problem (also called approximate problem). In what follows the error estimates are obtained with the assumption that U is sufficiently regular. In many realistic situations the geometry of the flow has singularities (corners for example), the solution U is not regular, and these results do not apply. (As a matter of fact existence of a solution has not been shown yet in those singular situations.)... 

Indeed, to exploit effectively the viscoelasticity of fluids for chaotic flow instability, and thus mixing, sharper and smaller geometries should be employed. Stress singularities developed at such corners have been the source of elastic instabilities in many macroscale experiments [3], while rounded comers tend to suppress elastic behavior. From a practical standpoint, it is necessary to understand the rheological nature of such flow in order to optimize the use of viscoelastic effects in microfluidic mixing applications. The complex interplay that arises between the elasticity and viscosity of the fluids, and the ratio of contraction of the channel is the key to efficient mixing of fluid streams in microfluidic channels. 

Most microfiuidic chips for electrokinetic dispensing applications have a channel layout containing 90-degree corners, which will cause singularity problems at the corners in the applied electrical potential field and the EDL potential field. This is because the direction of the applied potential field (the direction of the potential gradient) at that corner changes dramatically (90 degree from x-direction to y-direction) and the potential is not differen-... 

When both materials are compliant, singular stresses can also occur. The situation is more complicated because now there may be multiple singularities and they depend on the elastic properties of each material in addition to the corner angles in each material (Fig. 18). There are several ways to present the state of stress near a bimaterial comer. One common approach is given below. 

Corner cracks may also initiate under fatigue loading. In fact this may be the most common form of nucleation. Nonetheless, this problem seems to have received relatively little attention in the open literature. Lefebvre and Dillard [26,27] considered an epoxy wedge on an aluminum beam under cyclic loading. They chose corner angles (55°, 70° and 90°) that resulted in one singularity. A stress intensity factor based fatigue initiation envelope was then developed. 

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