Decay Estimates in Steady Pipe Flow

In this paper St. Venant type results are derived for the flow of viscous fluid in a pipe of arbitrary cross section. In the spirit of earlier work of Horgan and Wheeler [SIAMJ. Appl. Math., 35 (1978), pp. 97–116], the decay to fully developed flow as a function of distance from the entry section is investigated.Here, it is not assumed that the flow is fully developed at the exit section. Weighted energy inequalities are derived that lead to estimates for the “energy” associated with the velocity field represented by the difference between the entrance flow and the fully developed flow in a portion of the pipe near the exit section. The analysis is based on a variety of differential inequality techniques and Payne’s investigation of uniqueness criteria for steady-state solutions of the Navier–Stokes equations [Simpos. Internaz. Appl. Fis. Mat., 1965, pp. 130–153].