Similar and Nonsimilar Solutions of the Fully Viscous Flow in Slender Channels

摘要

Under conditions of low Reynolds numbers, the viscous effects in nozzles are no longer confined to a thin region near the wall, but can extend to the nozzle centerline, forming a fully viscous flow. An integral approach was developed to this problem, based upon fourth-order velocity and total temperature profiles and the use of the constant mass flow through the nozzle. Two-dimensional and axisymmetric, convergent, and divergent channels are examined in the incompressible, and compressible regimes. The incompressible results indicate that straight-walled channels yield similar solutions in the two-dimensional case, but for the axisymmetric nozzle an exponential contour is necessary to obtain similar velocity profiles. Comparisons of these results with other theories is good. Measurements of the nonsimilar velocity profiles of a 5 deg half-angle divergent nozzle operated at mass flow rates below .00001 slugs/sec and centerline velocities near 100 ft/sec are compared with the theoretical profiles. Centerline velocity distribution is well represented along the 10 in channel; the shape of the radial velocity profiles at four stations, although showing correct trends with mass flow rate, is not represented with the same precision. Much of the discrepancy is attributed to the difficulty of measuring static and total pressures of 1 to 10 microns Hg. (Author)