Fluid Mechanics: Consistent Analytical, Numerical and Laboratory Models of Stratified Flows

Authors: Chashechkin Yu.D. Published: 03.12.2014
Published in issue: #6(57)/2014  

Category: Mechanics  
Keywords: fundamental set, full solution, stratification, laboratory experiment, fine structure, dynamics

The results of the consistent analytical, numerical and laboratory modeling of dynamics and internal structure of flows are presented. A fundamental set of equations of the inhomogeneous fluids mechanics is mathematical basis. This set includes the differential equations of continuity, momentum balance, energy, diffusion components and closing equation of state. This simultaneous equations is analyzed under accounting of compatibility and observability conditions of incoming physical quantities. Symmetries of the fundamental set correspond to the basic principles of physics in contrast to many reduced and constitutive models. A complete mathematical classification for the components of periodic large- and small-scales flows is given. As an example, the full solution of definition problem for two-dimensional flows induced by diffusion on the topography and the linearized theory of periodic internal waves are considered. We discuss the physical and mathematical content of the concepts of "mechanical motion" and the "fluid flow", following requirements for measurement technique and methodology in order to ensure the fulfillment of the condition for the experiment efficiency.


[1] Descartes R. Principia philosophiae. Amsterdam, Louis Elzevir, 1644 (in Latin). (Russ. Ed.: Dekart R. Pervonachala filosofii. V 21. [Engl. Ed.: Descartes R. Principles of Philosophy. Gardners Books, 2013]. Moscow, Mysl’ Publ., 1989, vol. 1, pp. 297654).

[2] Leybnits G.V. Kratkoe dokazatel’stvo primechatel’noy oshibki Dekarta i drugikh, otnosyashcheysya k vvodimomu imi i primenyaemomu v mekhanike estestvennomu zakonu, soglasno kotoromu Bog khranit vsegda odno i tozhe kolichestvo dvizheniya. V kn.: "Sochineniya", v 4 t., t. 1, pp. 118-125. [Leibniz G.W. A short proof of a remarkable error of Descartes and others, relating to restrictions imposed by them and mechanics employed in the natural law, which says that God always keeps the same amount of traffic]. Moscow, Mysl’ Publ., 1981.

[3] D’Alembert J.-R. Re flexions sur la cause generale des vents [Reflections on the general cause of the winds]. Paris, 1747. 372 p.

[4] Euler L. Principles of the motion of fluids. Izv. Akad. Nauk, Mekh. Zhidk. Gaza [Fluid Dyn.], 1999, no. 6, pp. 26-54 (in Russ.). (Fr. Ed.: Principes generaux du mouvement des fluides. Memoires de l’Acad’emie royale des sciences et belles lettres. Berlin, 1755. vol. 11, pp. 274-315). Available at: http://www.bbaw.de/bibliothek/digital/tiff/02-hist/1755/tif/00000282.tif (accessed 27.07.2014).

[5] Navier C.-L.-M.-H. M’emoire sur les Lois du Mouvement des Fluids [Memory on the Laws of Movement Fluids]. Mem. d l’Acad. des Sciences, 1822, vol. 6, pp. 389-417.

[6] Stokes G.G. On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic bodies. Trans. Cam. Phil. Soc., 1845, vol. 8, pp. 287-305.

[7] Landau L.D., Lifshits E.M. Teoreticheskaja fizika. V 10 t. T. 6. Gidrodinamika [Theoretical physics. Ten-volume set. Vol. 6. Fluid mechanic]. Moscow, Nauka Publ., 1989. 752 p. (Eng. Ed.: Landau L.D., Lifshits E.M. Fluid Mechanics. Vol. 6 (Course of Theoretical Physics). Second Ed. Oxford, New York, Pergamon Press, 1987.)

[8] Mendeleev D.I. Ob uprugosti gazov [Elasticity of gases]. SPb, 1875. 262 p.

[9] Mendeleev D.I. Issledovanie vodnykh rastvorov po udel’nomu vesu [The study of aqueous solutions of specific weight]. SPb, 1887. 520 p.

[10] Chashechkin Yu.D., Bardakov R.N., Shabalin V.V. The regular fine structure of flows in a drying drop of a suspension of quartz nanoparticles. Dokl. Akad. Nauk [Doklady Physics, vol. 56, iss. 1, pp. 62-64], 2011, vol. 436, no. 3, pp. 336-338 (in Russ.). DOI: 10.1134/S1028335810901161

[11] Maxwell J.C. Remarks on the Mathematical Classification of Physical Quantities. Proc. L. Math. Soc., 1871, vol. 3, s. 1-3, pp. 224-233.

[12] Manturov O.V., Solntsev Yu.K., Sorkin Yu.I., Fedin N.G. Tolkovyy slovar’ matematicheskikh terminov [Explanatory dictionary of mathematical terms]. Moscow, Prosveshchenie Publ., 1964. 540 p.

[13] Newton I. Philosophiae Naturalis Principia Mathematica. London, "The Royal Society of London for the Improvement of Natural Knowledge", 1687 (in Latin). (Russ. Ed.: N’yuton I. Matematicheskie nachala natural’noy filosofii. Pod. red. Polaka L.S., per. s latinskogo Krylova A.N. Moscow, Nauka Publ., 1989. 688 p.).

[14] Helmholtz H. Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen [About integrals of the hydrodynamic equations, which correspond to the vortex motion]. J. fur die reine und angewandte Mathematik [J. of Pure and Applied Mathematics], 1858, vol. 55, pp. 25-55.

[15] Kalinichenko V.A., Chashechkin Yu.D. Structuring and restructuring of a homogeneous suspension in the standing-wave field. Izv. Akad. Nauk, Mekh. Zhidk. Gaza [Fluid Dyn.], 2012, no. 6, pp. 109-121 (in Russ.).

[16] Stepanova E.V., Chashechkin Yu.D. Marker transfer in a composite vortex. Izv. Akad. Nauk, Mekh. Zhidk. Gaza [Fluid Dyn.], 2010, no. 6, pp. 12-29 (in Russ.).

[17] Prokhorov V.E., Chashechkin Yu.D. Sound generation as a drop falls on a water surface. Akust. Zh. [Acoust. Phys.], 2011, vol. 57, no. 6, pp. 792-803 (in Russ.).

[18] Baydulov V.G., Chashechkin Yu.D. Invariant properties of systems of equations of the mechanics of inhomogeneous fluids. Prikl. Mat. i mekh. [J. Appl. Math. Mech.], 2011, vol. 75, no. 4, pp. 551-562 (in Russ.).

[19] Chashechkin Yu.D. Hierarchy of the models of classical mechanics of inhomogeneous fluids. Mors. Gidrof. Zh. [Phys. Oceanogr, 2011, vol. 230, iss. 5, pp. 317-324], 2010, no. 5, pp. 3-10 (in Russ.). DOI: 10.1007/s11110-011-9087-5

[20] Chashechkin Yu.D., Zagumennyy Ya.V. Structure of diffusion-induced flow on an inclined plate. Dokl. Akad. Nauk [Doklady Physics, vol. 57, iss. 5, p. 201-216], 2012, vol. 444, no. 2, pp. 165-171 (in Russ.).

[21] Kistovich A.V., Chashechkin Yu.D. Diffusion induced unsteady boundary flows in the sphenoid cavity. Prikl. Mat. i mekh. [J. Appl. Math. Mech.], 1998, vol. 62, no. 54, pp. 803-809 (in Russ.).

[22] Nayfeh Ali H. Introduction to Perturbation Techniques. 1st Ed. Wiley, 1981. 536 p. (Russ. Ed.: Nayfe A. Vvedenie v metody vozmushcheniy. Moscow, Mir Publ., 1984. 535 p.).

[23] Bardakov R.N., Vasil’ev A.Yu., Chashechkin Yu.D. Calculation and measurement of conical beams of three-dimensional periodic internal waves excited by a vertically oscillating piston. Izv. Akad. Nauk, Mekh. Zhidk. Gaza [Fluid Dyn.], 2007, no. 46, pp. 117-133 (in Russ.). DOI:10.1134/S0015462807040114

[24] Chashechkin Yu.D. Visualization of singular components of periodic motions in a continuously stratified fluid. J. Vis., 2007, vol. 10, no. 1, pp. 17-20.

[25] Chashechkin Yu.D., Prikhod’ko Yu.V. Regular and singular flow components for stimulated and free oscillations of a sphere in continuously stratified liquid Dokl. Akad. Nauk [Doklady Physics, vol. 52, iss. 5, p. 261], 2007, vol. 414, no. 1, pp. 4448 (in Russ.).