On Exact Solutions of the Navier-Stokes Equations for Uni-directional Flows

TL;DR

This paper derives exact time-dependent solutions for uni-directional flows in channels and pipes, challenging traditional steady-state assumptions and explaining classical experimental results through flow evolution.

Contribution

It presents exact spatio-temporal solutions to Navier-Stokes equations for uni-directional flows, showing classical steady flows are approximations of these solutions.

Findings

Classical steady flows are time-independent approximations.

Flow profiles evolve over time and are not inherently steady.

Poiseuille's law can be explained via flow evolution, not steady-state assumptions.

Abstract

In the present note, we show that the uni-directional flows in a rectangular channel and in a circular pipe are exact spatio-temporal solutions of the Navier-Stokes equations over a short time interval. We assert that the classical plane Poiseuille-Couette flow and Hagen-Poiseuille flow are time-independent approximations of the exact solutions if an appropriate initial velocity distribution at starting location is specified. Conceptually, there do not exist absolute steady flows starting from unspecified initial data. The classic experimental measurements by Poiseuille can be explained in terms of the evolutional solutions. In particular, the pipe flow does not have a time-independent characteristic velocity. The orthodox notion that the parabolic profile exists for arbitrary Reynolds numbers is unwarranted.