Convection with Slip Boundary Conditions

Abstract

There is increasing interest in micro-electro-mechanical-systems (MEMS), and flow in microfluidic channels due to their applications in the electronics and related industries. In particular, at nanoscales there is increasing evidence that boundary conditions of type are needed rather than those of no-slip. In this chapter we describe work of linearized instability and global nonlinear stability for the problem of Benard convection, but allowing for the velocity at the boundary of the bounding planes to in the tangential directions. While the main concern of this book is with problems of thermal convection, within the field of microfluidics the problem of flow of a clear fluid in a channel is of great importance. In particular, the instability problem of flow in a channel occupied by a linearly viscous fluid, and subject to boundary conditions, has an interesting history. We discuss the problem of Poiseuille flow of a linear viscous, incompressible fluid in an infinite channel bounded by the planes \(y=\pm d\) with the flow driven in the \(x-\)direction by a constant pressure gradient \(\partial p/\partial x=-G<0\), but with boundary conditions of type. Furthermore, we consider the problem of instability of flow in a channel occupied by a sparse porous medium when the boundary conditions are those of type. We also analyse the problem of thermal convection with boundary conditions but with the addition of “temperature - slip boundary condtions. In microfluidics thermal - boundary conditions are believed very important. In gas dynamic flows velocity and thermal boundary conditions have been utilized for some time, and specific values for the thermal accommodation coefficient and the momentum accommodation coefficient which appear in the boundary conditions are known for several boundary materials and various gases. In microfluidics such boundary conditions are being advanced as necessary.