Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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The qualitative analysis of a dynamical system modeling the formation of two-layer scales on pure metalsHTML articles powered by AMS MathViewer

by R. L. Baker
Proc. Amer. Math. Soc. 123 (1995), 1373-1378 Request permission

Abstract:

F. Gesmundo and F. Viani have modeled the growth rates of two-oxide scales by the system: $\frac {{d{q_1}}}{{dt}} = m\frac {{{K_1}}}{{2{q_1}}} - \frac {{m - 1}}{m} \frac {{{K_2}}}{{2{q_2}}},\qquad \frac {{d{q_2}}}{{dt}} = - m\frac {{{K_1}}}{{2{q_1}}} + \frac {{{K_2}}}{{2{q_2}}}.$ We provide a complete qualitative analysis of (1.1) by making use of known results about the general n-dimensional dynamical system: $\frac {{d{p_i}}}{{dt}} = - \sum \limits _{j = 1}^n {\frac {{{a_{ij}}}}{{{p_j}}},\quad {p_i}(t) > 0,\qquad i = 1, \ldots ,n.}$ We show that for $m > 1$, the Gesmundo-Viani system admits a unique parabolic solution ${q_i}(t) = {c_i}\sqrt t ,{c_i} > 0$. This parabolic solution attracts all other solutions. Every solution extends uniquely to a solution on $[0, + \infty )$, such that the extended solution is eventually monotonically increasing. Finally, the trajectory of any solution coincides with a trajectory of the following linear system: $\frac {{d{q_1}}}{{dt}} = - \frac {{m - 1}}{m}{\mkern 1mu} \frac {{{K_2}}}{2}{q_1} + m\frac {{{K_2}}}{2}{q_2},\qquad \frac {{d{q_2}}}{{dt}} = \frac {{{K_2}}}{2}{q_1} + m\frac {{{K_1}}}{2}{q_2}.$
References
• H. C. Akuezue, R. L. Baker, and M. W. Hirsch, The qualitative analysis of a dynamical system modeling the formation of multilayer scales on pure metals, SIAM J. Math. Anal. 25 (1994), no. 4, 1167–1175. MR 1278897, DOI 10.1137/S003614109119853X
• F. Gesmundo and F. Viani, The formation of multilayer scales in the parabolic oxidation of pure metals, J. Corrosion Sci. 18 (1978), 217-230.
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