Some comments on a singular elliptic problem associated with a rectangle

Abstract

SynopsisThe construction of the approximate solution within a rectangle of a singular elliptic problem is discussed. It is found that, provided the boundary data satisfy certain continuity conditions at the corners of the rectangle, ordinary boundary layers and parabolic boundary layers only are necessary to describe the solution. A correction term, however, has to be added to the solution if the continuity conditions on the boundary data are not satisfied.

Application to magnetohydrodynamic duct flows of a singular elliptic problem associated with a rectangle

“…In a previous paper ( [8]) it was shown how an approximate solution could be constructed when the following Dirichlet boundary conditions applied:…”

“…with N a prescribed integer. For later convenience it is useful to quote the form given in [8] for the approximate solution. Provided the boundary functions satisfy certain continuity conditions we have where x = x/e, y = y/Ve and the error term R N is uniformly o(e N ) throughout D. with a = a/Ve (and we have a »1), the quantities A ; and B { may be expressed in the form:…”

“…An example which comes from magnetohydrodynamics was used in [8] to illustrate the construction. In that example, e was the reciprocal of the Hartmann number M and the duct walls were all taken to be nonconducting.…”

“…We therefore adopt the basic approximation given in [8] and this enables us to deduce values for each g f , g t in order that the boundary conditions (3.1b), (3.1c) are satisfied. The continuity conditions on the boundary functions are then taken into account so that a formal approximate solution may be established.…”

“…where we employ the notation used in [8], and we observe that the boundary conditions on Aj and B 1 may be put in the form…”