Abstract
A model of measuring the level of a viscous incompressible liquid in a tank as based on the liquid level in a measuring tube is investigated. The tank is in the field of gravity, and the tank liquid level varies according to some law. As a result, a Dirichlet boundary value problem for a nonlinear integrodifferential equation of parabolic type is obtained. A global existence and uniqueness theorem is proved for a weak solution of the problem. In the case of a tank level decreasing linearly with time, it is shown numerically that the liquid level in the measuring tube oscillates with a decaying amplitude with respect to the tank level.
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References
S. V. Vallander, Lectures on Fluid Mechanics (Sankt-Peterb. Univ., St. Petersburg, 2005).
O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics (Nauka, Moscow, 1973; Springer- Verlag, New York, 1985).
V. P. Mikhailov, Partial Differential Equations (Nauka, Moscow, 1976; Mir, Moscow, 1978).
V. M. Uroev, Equations of Mathematical Physics (Yauza, Moscow, 1998) [in Russian].
B. C. Vladimirov, Equations of Mathematical Physics (Marcel Dekker, New York, 1971; Nauka, Moscow, 1981).
A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Nauka, Moscow, 1977; Dover, New York, 2011).
A. N. Bogolyubov and M. D. Malykh, “On a class of nonlocal parabolic equations,” Comput. Math. Math. Phys. 51 (6), 987–993 (2011).
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Original Russian Text © O.P. Filatov, 2016, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2016, Vol. 56, No. 12, pp. 2115–2124.
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Filatov, O.P. A model of liquid level measurements. Comput. Math. and Math. Phys. 56, 2084–2093 (2016). https://doi.org/10.1134/S0965542516120095
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DOI: https://doi.org/10.1134/S0965542516120095