A finite-series solution for the creeping flow in an exit region adjacent to a straight open tube exit

The flows in the straight round tube with open exit and the successive half-infinite region are approximately solved in the analytical manner by taking account of the flow in the exit region adjacent to the tube exit. The whole flow field is divided into four regions, namely the fully-developed flow region, the exit region and the half-innite region composed of the inner region and the outer one. The inner region is the closed one composed of the exit and the hemisphere with the origin at the exit center and the exit radius, while the outer region is the successive half-infinite one. Navier-Stokes equations performed the Stokes approximation are approximately solved by series expansions. The solution in the exit region is patched at the upstream and downstream patching surfaces with the solutions of the fully-developed flow and the inner region one. The outer region solution is obtained under the pressure and velocity conditions which are those in the inner region solution over the hemisphere. In consequence, not only the stress tensors in both the exit region and the inner one are completely patched at the exit but also the coefficients in the solution are determined self-consistently. Finally, the pressure loss coefficient at the exit is given in the concrete form.