On linearized theory for compressible viscous planar flow with slip

Abstract

Steady compressible supersonic planar flow of a viscous heat-conducting fluid is examined under linearization about freestream conditions. Although the results are derived here for a large-Prandtl-number fluid, Wu has shown that the conclusions hold for an order-unity-Prandtl-number fluid in the limits of large and small Reynolds numbers. First the compressible correction to the fundamental solution for flow past a singular flat plate (a point source of momentum directed antiparallel to the uniform freestreaming) is examined. Results valid near the point source are obtained by contour integration. Then these results are superposed to form an integral equation describing linearized slip flow past a finite flat plate at zero angle of attack. The large-slip limit for a short plate is characterized as a regular perturbation limit while the small-slip limit has singular-perturbation characteristics. For a long flat plate the integral equation can be approximated as Poisson type and is easily solved; for a short flat plate the integral equation can be approximated as a Carleman equation of the second kind and is less readily solved. However, an approximate solution for the shear near the leading edge of a short flat plate with small slip is given through the Wiener-Hopf technique. The integral equation describing thermal slip at a finite flat plate is seen to be like that describing velocity slip.