Bulletin of the American Physical Society
68th Annual Meeting of the APS Division of Fluid Dynamics
Volume 60, Number 21
Sunday–Tuesday, November 22–24, 2015; Boston, Massachusetts
Session L3: General Fluid Dynamics: Theory
4:05 PM–6:41 PM,
Monday, November 23, 2015
Chair: Serafim Kalliadasis, Imperial College, UK
Abstract ID: BAPS.2015.DFD.L3.10
Abstract: L3.00010 : On a difficulty in eigenfunction expansion solutions for the start-up of fluid flow*
6:02 PM–6:15 PM
Preview Abstract Abstract
Ivan C. Christov
(Theoretical Division \& Center for Nonlinear Studies, Los Alamos National Laboratory)
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).
*Supported, in part, by NSF Grant DMS-1104047 and the U.S. DOE (Contract No.\ DE-AC52-06NA25396) through the LANL/LDRD Program.
To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2015.DFD.L3.10
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