Abstract
A class of exact solutions of hydrodynamic equations with additional Korteweg stresses is obtained which is characterized by a linear dependence of part of the velocity components on the space variable. In this class, exact solutions of two problems of binary fluid flow between moving flat solid boundaries was found. A family of particular exact solutions is obtained for the problem of viscous fluid flow between planes which approach or move away from each other according to a special law.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. W. Cahn and J. E. Hilliard, “Free energy of a nonuniform system. 1. Interfacial free energy,” J. Chem. Phys., 28, No. 2, 258–267 (1958).
V. N. Starovoitov, “Model of the motion of a two-component liquid with allowance of capillary forces,” J. Appl. Mech. Tech. Phys., 35, No. 6, 891–897 (1994).
L. K. Antanovskii, “A phase-field model of capillarity,” Phys. Fluids, 7, 747–753 (1995).
D. J. Korteweg, “Sur la forme que prennent leséquations du mouvement des fluides si l’on tient compte des forces capillaires causées par des variations de densité considérables mais connues et sur la théorie de la capillarité dans l’hypothèse d’une variation continue de la densité,” Arch. Néerland. Sci. Exact. Naturell., 6, 1–24 (1901).
J. E. Dunn and J. B. Serrin, “On the thermomechanics of interstitial working,” Arch. Rational Mech. Anal., 88, 95–133 (1985).
D. M. Anderson, G. B. McFadden, and A. A. Wheeler, “Diffuse interface methods in fluid mechanics,” Ann. Rev. Fluid Mech., 30, 139–165 (1998).
S. Aristov, N. Bessonov, Y. Gaponenko, and V. Volpert, “Interfacial waves in binary fluids,” in: Patterns and waves: Theory and applications, Proc. of the Int. Conf. (St. Petersburg, Russia, July 8–12, 2002), AcademPrint, St. Petersburg (2003), pp. 79–97.
Y. A. Gaponenko, V. A. Vol’pert, S. M. Zenkovskaya, and D. A. Pojman, “Effect of high-frequency vibration on convection in miscible fluids,” J. Appl. Mech. Tech. Phys., 47, No. 2, 190–198 (2006).
J. A. Pojman, Yu. Chekanov, V. Wyatt, et al., “Numerical simulations of convection induced by Korteweg stresses in a miscible polymer-monomer system: Effects of variable transport coefficients, polymerization rate and volume changes,” Microgravity Sci. Technol., 21, 225–237 (2009).
C. C. Lin, “Note on a class of exact solutions in magneto-hydrodynamics,” Arch. Rational Mech. Anal., 1, 391–395 (1958).
A. F. Sidorov, “On a class of solutions of the equations of gas dynamics and natural convection,” in: Numerical and Analytical Methods for Solving Problems of Continuum Mechanics (collected scientific papers) [in Russian], Ural. Scientific Center of USSR Academy of Sciences, Sverdlovsk (1981), pp. 101–117.
V. V. Kuznetsov and V. V. Pukhnachev, “New family of exact solutions of the Navier-Stokes equations,” Dokl. Ross. Akad. Nauk, 429, No. 1, 40–44 (2009).
S. N. Aristov, D. V. Knyazev, and A. D. Polyanin, “Exact solutions of the Navier-Stokes equations with a linear dependence of the velocity components on two spatial variables,” Teoret. Osnovy Khim. Tekhnol., 43, No. 5, 547–566 (2009).
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 52, No. 2, pp. 66–72, March–April, 2011.
Rights and permissions
About this article
Cite this article
Knyazev, D.V. Two-dimensional flows of a viscous binary fluid between moving solid boundaries. J Appl Mech Tech Phy 52, 212–217 (2011). https://doi.org/10.1134/S0021894411020088
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0021894411020088