Geodesic
The flow of a viscous fluid with a predetermined pressure gradient through periodic structures
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 2, pp. 222-243.

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In the Stokes approximation, the problem of viscous fluid flow through two-dimensional and three-dimensional periodic structures is solved. A system of thin plates of a finite width is considered as a two-dimensional structure, and a system of thin rods of finite length is considered as a three-dimensional structure. Plates and rods are periodically located in space with certain translation steps along mutually perpendicular axes. On the basis of the procedure developed earlier, the authors constructed an approximate solution of the equations for fluid flow with an arbitrary orientation of structures relative to a given vector of pressure gradient. The solution is sought in a finite region (cells) around inclusions in the class of piecewise smooth functions that are infinitely differentiable in the cell, and at the cell boundaries they satisfy the continuity conditions for velocity, normal and tangential stresses. Since the boundary value problem for the Laplace equation is solved, it is assumed that the solution found is unique. The type of functions allows us to separate the variables and to reduce the problem's solution to the solution of ordinary differential equations. It is found that the change in the flow rate of a fluid through a characteristic cross section is determined mainly by the geometric dimensions of the cells of the free liquid in such structures and is practically independent of the size of the plates or rods.
Mots-clés : viscous fluid, pressure gradient
Keywords: periodic structures, periodic solution. 
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M. S. Deryabina; S. I. Martynov. The flow of a viscous fluid with a predetermined pressure gradient through periodic structures. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 2, pp. 222-243. http://geodesic.mathdoc.fr/item/SVMO_2019_21_2_a5/

[1] I. F. Efremov, Periodic colloidal structures, Chemistry, L., 1971, 192 pp. (In Russ.)

[2] P. Habdas, E. R. Weeks, “Video microscopy of colloidal suspensions and colloidal crystals”, Current Opinion in Colloid and Interface Science., 2002, no. 7, 196–203 | DOI

[3] P. N. Pusey, W. van Megen, S. M. Underwood, P. Bartlett and R. H. Ottewill, “Colloidal fluids, crystals and glasses”, Physica A: Statistical Mechanics and its Application, 176:1 (1991), 16–27 | DOI

[4] P. N.Pusey, W. van Megen, “Phase behaviour of concentrated suspensions of nearly hard colloidal spheres”, Nature, 320 (1986), 340–342 | DOI

[5] A. V. Zhukov, “Crystal structure and rheology of highly concentrated ferromagnetic suspensions”, Proceedings of the Russian Academy of Sciences. Mechanics of Fluid and Gas, 2006, no. 5, 122–134 (In Russ.) | Zbl

[6] A. P. Hynninen, M. Dijkstra, R. van Roij, “Effect of triplet interactions on the phase diagram of suspensions of charged colloids”, Journal of Physics: Condensed Matter, 15:48 (2003), S3549–S3556 | DOI

[7] A. P. Hynninen, M. Dijkstra, R. van Roij, “Effect of three-body interactions on the phase behavior of charge-stabilized colloidal suspensions”, Physical Review E, 69 (2004), 061407-1–061407-8 | DOI

[8] J. Dobnikar, R. Rzehak, H. H. von Grünberg, “Effect of many-body interactions on the solid-liquid phase behavior of charge-stabilized colloidal suspensions”, Europhysics Letters, 61:5 (2003), 695–701 | DOI

[9] J. C. Maxwell, Atreatise on electricity and magnetism, Clarendon Press, Oxford, 1873, 500 pp. | MR

[10] J. W. Rayleigh, “On the influence of obstacles arrenged in rectengular order upon the properties of a medium”, The London, Edinburg, and Dublin Philosophical Magazine and Journal of Science, 34:211 (1892), 481-502 | DOI

[11] N. S. Bakhvalov, “The averaged characteristics of bodies with a periodic structure”, Akad. Nauk USSR, 218:5 (1974), 1046–1048 (In Russ.)

[12] A. L. Berdichevsky, “Spatial averaging of periodic structures ”, Akad. Nauk USSR, 222:3 (1975), 565–567 (In Russ.)

[13] A. L. Berdichevsky A.L., “On the flow past a viscous liquid of a periodic lattice of spheres”, Proceedings of the Russian Academy of Sciences. Mechanics of fluid and gas, 1981, no. 4, 37–-44 (In Russ.)

[14] S. I. Martynov, “Motion of a viscous fluid through a periodic lattice of spheres”, Proceedings of the Russian Academy of Sciences. Mechanics of Fluid and Gas, 2002, no. 6, 48–54 (In Russ.) | Zbl

[15] S. I. Martynov, A. O. Syromyasov, “Symmetry of the periodic lattice of particles and the flow of a viscous fluid in the Stokes approximation”, Proceedings of the Russian Academy of Sciences. Mechanics of fluid and gas, 2007, no. 3, 7–20 (In Russ.) | Zbl

[16] I. Happel, H. Brenner, Low Reynolds number hydrodinamics, Prentice-Hall, Englewood Giffs, 1965, 553 pp. | MR

[17] S. I. Aristov, A. D. Polyanin, “Exact solutions of three-dimensional nonstationary Navier-Stokes equations”, Akad. Nauk USSR, 427:1 (2009), 35–40 (In Russ.) | MR | Zbl

[18] M. S. Deryabina, S. I. Martynov, “Periodic flow of a viscous fluid with a predetermined pressure and temperature gradient”, Nonlinear Dynamics, 14:1 (2018), 81–97 (In Russ.) | MR