Solution of a problem of a seepage fields distribution into infinite porous massif with two circular inclusions

Yu. V. Obnosov

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Kazanskii Gosudarstvennyi Universitet. Uchenye Zapiski. Seriya Fiziko-Matematichaskie Nauki, Tome 148 (2006) no. 2, pp. 109-123

Résumé

A closed solution to the problem on 2-D seepage flow in a heterogeneous three-component porous medium is presented. The medium is composed of an infinite matrix and two isotropic circular inclusions, placed in the medium without intersections of the corresponding circumferences. The limiting cases of touching circumferences as well as the degeneration of one circumference into a straight line are studied. The solutions are obtained in the form of infinite series, whose convergence is investigated and the truncated terms are estimated. Summation of the series is carried out in the form of elementary functions for the limiting cases when the inclusion hydraulic conductivity is 0 or $\infty$.

Bibliographie

[1] Polubarinova-Kochina P. Ya., Teoriya dvizheniya gruntovykh vod, Nauka, M., 1977, 664 pp. | MR

[2] Budiansky B., Carrier G. F., “High shear stress in stiff-fiber composites”, J. Appl. Mech., 51 (1984), 733–735 | DOI | Zbl

[3] Goree J. G., Wilson H. B., “Transverse shear loading in an elastic matrix containing two circular cylindrical inclusions”, J. Appl. Mech., 1967, June, 511–513 | DOI

[4] Dhondt G., Köhl M., “On the force between a dielectric cylinder in a constant electric field and a conducting half space”, Quart. Appl. Math., 55:2 (1997), 337–359 | MR

[5] Emets Yu. P., Obnosov Yu. V., “Elektricheskoe pole v sloistom krugovom vklyuchenii s effektom Kholla”, Tekhn. elektrodinamika. AN Ukr., 1987, no. 3, 3–8

[6] Emets Yu. P., Obnosov Yu. V., “Vzaimnoe vliyanie neodnorodnykh vklyuchenii v polevykh zadachakh dispersnykh sred”, Dokl. AN UkrSSR. Ser. A, 1988, no. 2, 74–78 | Zbl

[7] Emets Yu. P., Obnosov Yu. V., “Kraevaya zadacha dlya sloistogo nekontsentricheskogo krugovogo vklyucheniya pri anizotropnoi provodimosti sredy”, Tekhn. elektrodinamika. AN Ukr., 1988, no. 1, 3–7 | MR

[8] Emets Yu. P., Obnosov Yu. V., “Tochno razreshimaya zadacha o vzaimnom vliyanii vklyuchenii v teorii geterogennykh sred”, PMTF, 1990, no. 1, 20–29 | MR

[9] Emets Yu. P., Obnosov Yu. V., Onofriichuk Yu. P., “Vzaimodeistvie mezhdu kasayuschimisya krugovymi dielektricheskimi tsilindrami v odnorodnom elektricheskom pole”, ZhTF, 63:12 (1993), 12–24

[10] Emets Yu. P., Obnosov Yu. V., Onofriichuk Yu. P., “Elektricheskie sily v dielektricheskom dvukhsloinom tsilindre s nekontsentricheskim raspolozheniem sloev”, PMTF, 37:1 (1996), 3–14 | MR | Zbl

[11] Kacimov A. R., Obnosov Yu. V., “Minimization of ground water contamination by lining of a waste repository”, Proc. Indian Nat. Sci. Acad. A, 60:6 (1994), 783–792

[12] Steif Paul S., “Shear stress concentration between holes”, J. Appl. Mech., 56 (1989), 719–721 | DOI

[13] Goluzin G. M., “Reshenie osnovnykh ploskikh zadach matematicheskoi fiziki dlya sluchaya uravneniya Laplace'a i mnogosvyaznykh oblastei, ogranichennykh okruzhnostyami (metod funktsionalnykh uravnenii)”, Matem. sb., 41:2 (1934), 236–276

[14] Goluzin G. M., “Reshenie ploskoi zadachi teploprovodnosti dlya mnogosvyaznykh oblastei, ogranichennykh okruzhnostyami, v sluchae nalichiya izoliruyuschego sloya”, Matem. sb., 42:2 (1935), 191–198

[15] Mityushev V. V., O reshenii obschei kraevoi zadachi lineinogo sopryazheniya dlya neskolkikh kontsentricheskikh okruzhnostei, Dep. v VINITI 29.04.83, No 2279-83, Rukopis predst. red. zhurn. “Vestsi AN BSSR”

[16] Mityushev V. V., “Plane problem for the steady heat conduction of material with circular inclusions”, Arch. Mech., 45:2 (1993), 201–215 | MR

[17] Mityushev V. V., “Transport properties of double-periodic arrays of circular cylinders”, Z. angew. Math. Mech., 77:2 (1997), 115–120 | DOI | MR | Zbl

[18] Obnosov Yu. V., “Filtratsionnaya refraktsiya na polukrugovoi linze, sopryazhennoi s dvumya poristymi massivami”, PMM, 62:5 (1998), 810–824 | MR | Zbl

[19] Mikhailov L. G., Novyi klass osobykh integralnykh uravnenii i ego prilozhenie k differentsialnym uravneniyam s singulyarnymi koeffitsientami, Dushanbe, 1963, 183 pp.

[20] Miln-Tompson L. M., Teoreticheskaya gidrodinamika, Mir, M., 1964, 655 pp.

[21] Lavrentev M. A., Shabat B. V., Metody teorii funktsii kompleksnogo peremennogo, Nauka, M., 1987, 688 pp. | MR

[22] Yarmitskii A. G., “Filtratsionnaya teorema o dvukh okruzhnostyakh”, Izv. AN SSSR. MZhG, 1986, no. 4, 76–82 | MR