Analytic solutions of a two-fluid hydrodynamic model
Abstract
We investigate a one dimensional flow described with the non-compressible coupled Euler and non-compressible Navier-Stokes equations in the Cartesian coordinate system. We couple the two fluids through the continuity equation where different void fractions can be considered. The well-known self-similar Ansatz was applied and analytic solutions were derived for both velocity and pressure field as well.
Keyword : self-similar solution, two-fluid model
How to Cite
Barna, I. F., & Mátyás, L. (2021). Analytic solutions of a two-fluid hydrodynamic model. Mathematical Modelling and Analysis, 26(4), 582-590. https://doi.org/10.3846/mma.2021.13637
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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L. Tisza. The theory of liquid helium. Physical review journals archive, 72:838– 854, Nov 1947. https://doi.org/10.1103/PhysRev.72.838
G.E. Volovik. The Universe in a Helium Droplet. Clarendon Press, 2003. https://doi.org/10.1093/acprof:oso/9780199564842.001.0001
Ya.B. Zel’dovich and Yu.P. Raizer. Physics of Shock Waves and High Temperature Hydrodynamic Phenomena. Academic Press, New York, 1966.
G.I. Baraneblatt. Similarity, Self-Similarity, and Intermediate Asymptotics. Consultants Bureau, New York, 1979.
I.F. Barna. Self-similar solutions of three-dimensional Navier-Stokes equation. Communications in Theoretical Physics, 56(4):745–750, 2011. https://doi.org/10.1088/0253-6102/56/4/25
I.F. Barna, G. Bognár, M. Guedda, L. Mátyás and K. Hriczó. Analytic selfsimilar solutions of the Kardar-Parisi-Zhang interface growing equation with various noise terms. Mathematical Modelling and Analysis, 24(2):241–256, 2020. https://doi.org/10.3846/mma.2020.10459
I.F. Barna, G. Bognár and K. Hriczó. Self-similar analytic solution of the two-dimensional Navier-Stokes equation with a non-Newtonian type of viscosity. Mathematical Modelling and Analysis, 21(1):83–94, 2016. https://doi.org/10.3846/13926292.2016.1136901
I.F. Barna, A.R. Imre, G. Baranyai and Gy. Ézsol. Experimental and theoretical study of steam condensation induced water hammer phenomena. Nuclear Engineering and Design, 240:146–150, 2010. https://doi.org/10.1016/j.nucengdes.2009.09.027
I.F. Barna, A.R. Imre, L. Rosta and F.Mezei. Two-phase flow model for energetic proton beam induced pressure waves in mercury target systems in the planned European Spallation Source. The European Physical Journal B, 66(4):419–426, 2008. https://doi.org/10.1140/epjb/e2008-00444-x
I.F. Barna and R. Kersner. Heat conduction: a telegraph-type model with self-similar behavior of solutions. Journal of Physics A: Mathematical and Theoretical, 43(37):375210, aug 2010. https://doi.org/10.1088/17518113/43/37/375210
I.F. Barna and L. Mátyás. Analytic self-similar solutions of the OberbeckBoussinesq equations. Chaos, Solitons & Fractals, 78:249–255, 2015. ISSN 0960-0779. https://doi.org/10.1016/j.chaos.2015.08.002
I.F. Barna, L. Mátyás and M.A. Pocsai. Self-similar analysis of a viscous heated Oberbeck–Boussinesq flow system. Fluid Dynamics Research, 52(1):015515, mar 2020. https://doi.org/10.1088/1873-7005/ab720c
G.W. Bluman and J.D. Cole. The general similarity solution of the heat equation. Indiana University Mathematics Journal, 18(11):1025–1042, 1969. https://doi.org/10.2307/24893142
D. Campos. Handbook on Navier-Stokes Equations, Theory & Applied Analysis. Nova Publishers, New York, 2017.
C.T. Crowe. Multiphase Flow Handbook. CRC Taylor and Francis, 2006. https://doi.org/10.1201/9781420040470
M. Ishii and T. Hibiki. Thermo-Fluid Dynamics of Two-Phase Flow. Springer, New York, 2011. https://doi.org/10.1007/978-1-4419-7985-8
I.M. Khalatnikov. An Introduction to the Theory of Superfluidity. Westview Press, Boulder Colorado, 2000.
N.I. Kolev. Multiphase Flow Dynamics. Springer, 2006.
L. Landau. Theory of the superfluidity of helium II. Physical review journals archieve, 60:356–358, Aug 1941. https://doi.org/10.1103/PhysRev.60.356
R. Menikoff and B.J. Plohr. The Riemann problem for fluid flow of real materials. Reviews of modern physics, 61:75–130, Jan 1989. https://doi.org/10.1103/RevModPhys.61.75
A.B. Migdal. Superfluidity and the moments of inertia of nuclei. Nuclear Physics, 13(5):655– 674, 1959. ISSN 0029-5582. https://doi.org/10.1016/00295582(59)90264-0
F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W Clark. The NIST handbook of mathematical functions. Cambridge University Press, New York, NY, 2010.
L. Pitaevskii and S. Stringari. Bose-Einstein Condensation and Superfluidity. Oxford Science Publications, 2015. https://doi.org/10.1093/acprof:oso/9780198758884.001.0001
A. Scmidth. Introduction to Superfluidity. Springer, Cham, 2015. https://doi.org/10.1007/978-3-319-07947-9
L.I. Sedov. Similarity and Dimensional Methods in Mechanics. Mechanics CRC Press, 1993.
H.B. Stewart and B. Wendroff. Two-phase flow: Models and methods. Journal of Computational Physics, 56(3):363–409, 1984. https://doi.org/10.1016/0021-9991(84)90103-7
I. Tiselj and S. Petelin. Modelling of two-phase flow with second-order accurate scheme. Journal of Computational Physics, 136(2):503–521, 1997. https://doi.org/10.1006/jcph.1997.5778
L. Tisza. Green’s theory of liquid helium. Nature, 163:102–103, 1938. https://doi.org/10.1038/163102a0
L. Tisza. The theory of liquid helium. Physical review journals archive, 72:838– 854, Nov 1947. https://doi.org/10.1103/PhysRev.72.838
G.E. Volovik. The Universe in a Helium Droplet. Clarendon Press, 2003. https://doi.org/10.1093/acprof:oso/9780199564842.001.0001
Ya.B. Zel’dovich and Yu.P. Raizer. Physics of Shock Waves and High Temperature Hydrodynamic Phenomena. Academic Press, New York, 1966.