On a difficulty in the formulation of initial and boundary conditions for eigenfunction expansion solutions for the start-up of fluid flow

TL;DR

This paper identifies a flaw in the traditional eigenfunction expansion method for fluid flow start-up problems, especially for non-Newtonian fluids, and proposes a corrected formulation that ensures causality and accurate solutions.

Contribution

It introduces a rigorous eigenfunction expansion approach based on Duhamel's principle to correctly handle initial conditions in start-up flow problems.

Findings

Traditional methods can produce erroneous solutions for non-Newtonian fluids.

The new formulation aligns with solutions obtained via Laplace transform, ensuring causality.

Corrected approach resolves issues in initial-boundary value problems for fluid flow.

Abstract

Most mathematics and engineering textbooks describe the process of "subtracting off" the steady state of a linear parabolic partial differential equation as a technique for obtaining a boundary-value problem with homogeneous boundary conditions that can be solved by separation of variables (i.e., eigenfunction expansions). While this method produces the correct solution for the start-up of the flow of, e.g., a Newtonian fluid between parallel plates, it can lead to erroneous solutions to the corresponding problem for a class of non-Newtonian fluids. We show that the reason for this is the non-rigorous enforcement of the start-up condition in the textbook approach, which leads to a violation of the principle of causality. Nevertheless, these boundary-value problems can be solved correctly using eigenfunction expansions, and we present the formulation that makes this possible (in essence, an application of Duhamel's principle). The solutions obtained by this new approach are shown to agree identically with those obtained by using the Laplace transform in time only, a technique that enforces the proper start-up condition implicitly (hence, the same error cannot be committed).