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Equivalence of u– p and ζ–ψ formulations of the time‐dependent Navier–Stokes equations
Author(s) -
Guermond J.L.,
Quartapelle L.
Publication year - 1994
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.1650180504
Subject(s) - mathematics , vorticity , navier–stokes equations , stream function , compatibility (geochemistry) , equivalence (formal languages) , compressibility , mathematical analysis , boundary value problem , vortex , physics , mechanics , pure mathematics , geology , geochemistry
This paper deals with the non‐stationary incompressible Navier‐‐Stokes equations for two‐dimensional flows expressed in terms of the velocity and pressure and of the vorticity and streamfunction. The equivalence of the two formulations is demonstrated, both formally and rigorously, by virtue of a condition of compatibility between the boundary and initial values of the normal component of velocity. This condition is shown to be the only compatibility condition necessary to allow for solutions of a minimal regularity, namely H 1 for the velocity, as in most current numerical schemes relying on spatial discretizations of local type.