Blasius Boundary Layer Solution With Slip Flow Conditions

As the number of applications of micro electro mechanical systems, or MEMS, increase, the variety of flow geometries that must be analyzed at the micro-scale is also increasing. To date, most of the work on MEMS scale fluid mechanics has focused on internal flow geometries, such as microchannels. As applications such as micro-scale flyers are considered, it is becoming necessary to consider external flow geometries. Adding a slip-flow condition to the Blasius boundary layer allows these flows to be studied without extensive computation. BOUNDARY LAYER WITH SLIP The Blasius boundary layer solution for flow over a flat plate is among the best know solutions in fluid mechanics [1]. The boundary layer equations assume the following: (1) steady, incompressible flow, (2) laminar flow, (3) no significant gradients of pressure in the x-direction, and (4) velocity gradients in the xdirection are small compared to velocity gradients in the y-direction. Only the last assumption is questionable for MEMS scale flows. No-Slip Boundary Layer Equations The simplified Navier-Stokes Equations based on these assumptions, known as the boundary layer equations, are given as: du dv — + — = 0 (1) dx dy du du d u u — +v — =v — (2) dx dy dy where u and v are the x and y components of the velocity, and \) is the kinematic viscosity of the fluid. In the Blasius solution, a non-dimensional position T| combines both the x and y position: _ y (**) (x/L) ( where x* and y* are non-dimensional coordinates, u0 is the free stream velocity, and L is an arbitrary length scale that cancels itself out. The non-dimensional velocities u* and v* are then functions of the nondimensional stream function/ CP585, Rarefied Gas Dynamics: 22 International Symposium, edited by T. J. Bartel and M. A. Gallis © 2001 American Institute of Physics 0-7354-0025-3/01/$18.00 518