Results of a theoretical investigation of an acceleration liquid flow in a microcylinder beginning sudden rotation are presented. The problem on this flow was solved analytically with the use of the Laplace transform and the method of groups of symmetries as well as numerically using the method of Boltzmann lattices. The nonstationary flow-velocity profiles obtained analytically were compared with the results of a numerical solution. It is shown that the flow in the microcylinder develops by the asymptotic law and that the time of stabilization of this flow increases with increase in the Knudsen number. An analytical expression for the friction coefficient of the indicated flow has been obtained.
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M. Calvert and J. Baker, Thermal conductivity and gaseous microscale transport, J. Thermophys. Heat Transf., No. 12, 138–145 (1998).
M. Gad-el-Hak, The fluid mechanics of microdevices, ASME J. Fluid. Eng., No. 121, 5–33 (1999).
G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, Oxford (1994).
J. M. Haile, Molecular Dynamics Simulation, Wiley and Sons, New York (1992).
G. Karniadakis, A. Beskok, and N. Aluru, Microflows and Nanoflows Fundamentals and Simulation, Springer, New York (2005).
B. J. N. Wylie, Application of Two-Dimensional Cellular Automaton Lattice-Gas Models to the Simulation of Hydrodynamics, University of Edinburgh, Edinburgh (1990).
J. B. Maxwell, Lattice Boltzmann Methods for Interfacial Wave Modelling, University of Edinburgh, Edinburgh (1997).
X. Shan and H. Chen, Lattice Boltzmann model for simulating flows with multiple phases and components, Phys. Rev. E, No. 47, 1815–1819 (1993).
X. Shan and H. Chen, Simulation of nonideal gases and liquid–gas phase transitions by the lattice Boltzmann equation, Phys. Rev. E, No. 49, 2941–2948 (1994).
X. He and L. Luo, Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann eqution, Phys. Rev. E, No. 56, 6811–6817 (1997).
Q. Zou and X. He, On pressure and velocity boundary conditions for the lattice Boltzmann BGK model, Phys. Fluids, 9, No. 6, 1591–1596 (1997).
F. Szymansky, Quelques solution exactes des équations de l′hydrodynamique de fluide visqueux dans le cas d′un tube cylindrique, J. Math. Pures Appl., 97, No. 11, 67–107 (1932).
W. Müller, Zum Problem der Anlaufströmung einer Flüssigkeit im geraden Rohr mit Kreisring- und Kreisquerschnitt, ZAMM, No.16, 227–238 (1936).
W. Gerberts, Zur instationären, laminaren Strömung einer inkompressiblen zähen Flüssigkeit in kreiszylindrischen Rohren, Z. Angew. Phys., No. 3, 267–271 (1951).
H. Schlichting and K. Gersten, Boundary Layer Theory, 8th edn., Springer, Berlin (2000).
A. A. Avramenko and A. V. Kuznetsov, Start-up flow in a channel or pipe occupied by a fluid-saturated porous medium, J. Porous Media, 12, No. 4, 361–367 (2009).
H. Lamb, Hydrodynamics, Cambridge Univ. Press (1993).
A. A. Avramenko, A. I. Tyrinov, and I. V. Shevchuk, Start-up slip flow in a microchannel with a rectangular cross section, Theor. Comput. Fluid Dyn., 29, No. 5, 351–371 (2015).
A. A. Avramenko, A. I. Tyrinov, and I. V. Shevchuk, An analytical and numerical study on the start-up flow of slightly rarefied gases in a parallel-plate channel and a pipe, Phys. Fluids, 27, Issue 4, 042001-1–042001-18 (2015).
P. Olver, Applications of Lie Groups to Differential Equations, Springer (2000).
M. Abramowitz and I. A. Stegun, Reference Book on Special Functions [Russian translation], Nauka, Moscow (1979).
A. I. Tyrinov, Simulation of microflows by the method of Boltzmann lattices, Prom. Teplotekh., No. 2, 11–19 (2011).
A. I. Tyrinov, A. A. Avramenko, B. I. Basok, and B. V. Davydenko, Modeling of flows in a microchannel on the Boltzmann lattice equation, J. Eng. Phys. Thermophys., 85, No. 1, 65–72 (2012).
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Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 91, No. 6, pp. 1526–1536, November–December, 2018.
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Avramenko, A.A., Dmitrenko, N.P., Kravchuk, A.B. et al. Hydrodynamics of a Nonstationary Flow in a Microcylinder Beginning Sudden Rotation. J Eng Phys Thermophy 91, 1452–1461 (2018). https://doi.org/10.1007/s10891-018-1880-2
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DOI: https://doi.org/10.1007/s10891-018-1880-2