Abstract.
We show that in an unsteady Poiseuille flow of a Navier–Stokes fluid in an infinite straight pipe of constant cross-section, σ, the flow rate, F(t), and the axial pressure drop, q(t), are related, at each time t, by a linear Volterra integral equation of the second type, where the kernel depends only upon t and σ. One significant consequence of this result is that it allows us to prove that the inverse parabolic problem of finding a Poiseuille flow corresponding to a given F(t) is equivalent to the resolution of the classical initial-boundary value problem for the heat equation.
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G. P. Galdi: Partially supported by the NSF grant DMS–0404834.
K. Pileckas: Supported by EC FP6 MCToK program SPADE2, MTKD–CT–2004–014508
A. L. Silvestre: Supported by FCT-Project POCI/MAT/61792/2004
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Galdi, G.P., Pileckas, K. & Silvestre, A.L. On the unsteady Poiseuille flow in a pipe. Z. angew. Math. Phys. 58, 994–1007 (2007). https://doi.org/10.1007/s00033-006-6114-3
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DOI: https://doi.org/10.1007/s00033-006-6114-3