Spatial Decay Estimates for the Navier–Stokes Equations with Application to the Problem of Entry Flow

The development of velocity profiles in the inlet region of channels or pipes is a classic problem of laminar flow theory which has given rise to an extensive literature. Most previous work on this entry flow problem has involved some degree of simplification either in flow geometry or in the governing equations. Here we treat the flow development within the general framework of the Navier–Stokes equations governing the steady laminar flow of an incompressible viscous fluid in a cylindrical pipe of arbitrary cross-section. The problem to be treated is that of an “end effect” involving comparison between two distinct solutions of the Navier–Stokes equations. Thus we consider the spatial evolution of the difference between the base flow and the fully developed solution. The corresponding velocity difference clearly satisfies a condition of zero net inflow. In this way, we draw an analogy between the issue of concern here and the celebrated “Saint-Venant’s Principle” of elasticity theory involving the effect of self-equilibrated end loads. Our analysis is based on the modern researches on the latter topic, making particular use of the work of J. K. Knowles [4]. The main result is an explicit estimate which establishes the exponential spatial flow development. The work is in the spirit of mathematical treatments of the Navier–Stokes equations. In particular, a priori estimates of the type considered by L. E. Payne [10] in stability and uniqueness studies for the Navier–Stokes equations also play a major role in the present investigation.