A free-convection boundary-layer model for the centrifugal etching of an axisymmetric cavity

The etching of an axisymmetric cavity under the influence of an artificial acceleration field generated inside a centrifuge is considered. It is shown that the etching process is governed by a thin convective-diffusive boundary layer along the curved cavity wall. To describe this boundary layer, local coordinates are introduced near the wall, with arc length s being one of these. It is shown that the resulting boundary-layer model can be solved explicitly, whence an exact representation of mass transport from the wall follows. The solution is then substituted in the moving-boundary condition. This results in a highly nonlinear hyperbolic differential equation for the wall position as a function of s and time t. It is further shown that this equation admits a family of similarity solutions in terms of the variable = st –4/5. The position variables are now functions of only and a highly nonlinear system of ordinary differential equations results. This system is subjected to a series of transformations, until a system suitable for numerical integration is obtained and a family of similarity curves, i.e. cavity shapes, can be generated. It is remarkable that the integration stops at a finite value of beyond which, apparently, the similarity concept is no longer tenable.