Nonlinear wave propagation in coaxial shells filled with viscous liquid
DOI:
https://doi.org/10.7242/1999-6691/2017.10.2.15Keywords:
nonlinear waves, viscous incompressible liquid, elastic cylinder shellAbstract
The investigation of deformation wave behavior in elastic shells is one of the significant trends in contemporary wave dynamics. There are wave motion mathematical models of infinitely long geometrically and physically nonlinear shells with viscous incompressible liquid inside. They are based on the coupled hydroelastic problems, described by the equations of dynamics of shells and the equations of viscous incompressible fluid in the form of generalized KdV equations. Mathematical models of wave processes in infinitely long geometrically nonlinear coaxial cylindrical shells are obtained by means of the small parameter perturbation method. The problems differ from the already known ones by the consideration of viscous incompressible liquid. The system of generalized KdV equations is obtained on the basis of coupled hydroelastic problems, described by the equations of dynamics of shells and fluid equations with corresponding boundary conditions. The article deals with investigating the model describing wave phenomena in two geometrically and physically nonlinear elastic coaxial cylindrical Kirchhoff-Love type shells, containing viscous incompressible liquid both between and inside them. The difference Crank-Nicholson type schemes aimed at investigating equations systems with the consideration of liquid impact are obtained with the help of Gröbner basis construction. To generate these difference schemes, basic integral difference correlations, approximating the initial equation system, are used. The use of Gröbner basis techniques makes it possible to generate the schemes allowing one to obtain discrete preservation law analogues to initial differential equations. To do this, equivalent transformations were made. On the basis of computational algorithm the program complex allowing one to construct graphs and to obtain Cauchy problem numerical solution was developed using the exact solutions of the system of coaxial shell dynamics equations as an initial condition.
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References
Gromeka I.S. K teorii dvizenia zidkosti v uzkih cilindriceskih trubah / Sobr. soc. - M.: Izd-vo AN SSSR, 1952. - 296 s. - S. 149-171.
2. Kondratov D.V., Mogilevic L.I. Matematiceskoe modelirovanie processov vzaimodejstvia dvuh cilindriceskih obolocek so sloem zidkosti mezdu nimi pri otsutstvii torcevogo istecenia v usloviah vibracii // Vestnik SGTU. - 2007. - T. 3, No 2(27). - S. 15-23.
3. Kondratov D.V., Kondratova U.N., Mogilevic L.I. Issledovanie amplitudnyh castotnyh harakteristik kolebanij uprugih stenok truby kol’cevogo profila pri pul’siruusem dvizenii vazkoj zidkosti v usloviah zestkogo zasemlenia po torcam // Problemy masinostroenia i nadeznosti masin. - 2009. - No 3. - S. 15-21. DOI
4. Paidoussis M.P., Nguyen V.B., Misra A.K. A theoretical study of the stability of cantilevered coaxial cylindrical shells conveying fluid // J. Fluids Struct. - 1991. - Vol. 5, no. 2. - P. 127-164. DOI
5. Amabili M., Garziera R. Vibrations of circular cylindrical shells with nonuniform constraints, elastic bed and added mass; Part III: Steady viscous effects on shells conveying fluid // J. Fluids Struct. - 2002. - Vol. 16, no. 6. - P. 795-809. DOI
6. Amabili M. Nonlinear vibrations and stability of shells and plates. - Cambridge University Press, 2008. - 374 p. DOI
7. Mogilevic L.I., Popov V.S. Dinamika vzaimodejstvia uprugogo cilindra so sloem vazkoj neszimaemoj zidkosti // MTT. - 2004. - No 5. - S. 179-190.
8. Bockarev S.A. Sobstvennye kolebania vrasausejsa krugovoj cilindriceskoj obolocki s zidkost’u // Vycisl. meh. splos. sred. - 2010. - T. 3, No 2. - S. 24-33. DOI
9. Lekomcev S.V. Konecno-elementnye algoritmy rasceta sobstvennyh kolebanij trehmernyh obolocek // Vycisl. meh. splos. sred. - 2012. - T. 5, No 2. - S. 233-243. DOI
10. Bockarev S.A., Matveenko V.P. Ustojcivost’ koaksial’nyh cilindriceskih obolocek, soderzasih vrasausijsa potok zidkosti // Vycisl. meh. splos. sred. - 2013. - T. 6, No 1. - S. 94-102. DOI
11. Erofeev V.I., Zemlanuhin A.I., Katson V.M., Sesenin S.F. Formirovanie solitonov deformacii v kontinuume Kossera so stesnennym vraseniem // Vycisl. meh. splos. sred. - 2009. - T. 2, No 4. - S. 67-75. DOI
12. Bagdoev A.G., Erofeev V.I., Sekoan A.V. Linejnye i nelinejnye volny v dispergiruusih splosnyh sredah. - M.: Fizmatlit, 2009. - 320 s.
13. Erofeev V.I., Kazaev V.V., Pavlov I.S. Neuprugoe vzaimodejstvie i rasseplenie solitonov deformacii, rasprostranausihsa v zernistoj srede // Vycisl. meh. splos. sred. - 2013. - T. 6, No 2. - S. 140-150. DOI
14. Zemlanuhin A.I., Bockarev A.V. Metod vozmusenij i tocnye resenia uravnenij nelinejnoj dinamiki sred s mikrostrukturoj // Vycisl. meh. splos. sred. - 2016. - T. 9, No 2. - S. 182-191. DOI
15. Blinkova A.U., Blinkov U.A., Mogilevic L.I. Nelinejnye volny v soosnyh cilindriceskih obolockah, soderzasih vazkuu zidkost’ mezdu nimi, s ucetom rasseania energii // Vycisl. meh. splos. sred. - 2013. - T. 6, No 3. - S. 336-345. DOI
16. Blinkova A.U., Ivanov S.V., Kovalev A.D., Mogilevic L.I. Matematiceskoe i komp’uternoe modelirovanie dinamiki nelinejnyh voln v fiziceski nelinejnyh uprugih cilindriceskih obolockah, soderzasih vazkuu neszimaemuu zidkost’ // Izv. Sarat. un-ta. Nov. ser. Cep. Fizika. - 2012. - T. 12, No 2. - S. 12-18.
17. Blinkova A.U., Blinkov U.A., Ivanov S.V., Mogilevic L.I. Nelinejnye volny deformacij v geometriceski i fiziceski nelinejnoj vazkouprugoj cilindriceskoj obolocke, soderzasej vazkuu neszimaemuu zidkost’ i okruzennoj uprugoj sredoj // Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mehanika. Informatika. - 2015. - T. 15, No 2. - S. 193-202. -
18. Blinkov U.A., Kovaleva I.A., Mogilevic L.I. Modelirovanie dinamiki nelinejnyh voln v soosnyh geometriceski i fiziceski nelinejnyh obolockah, soderzasih vazkuu neszimaemuu zidkost’ mezdu nimi // Vestnik RUDN. Seria: Matematika, informatika, fizika. - 2013. - T. 3. - S. 42-51.
19. Blinkov U.A., Mesanzin A.V., Mogilevic L.I. Matematiceskoe modelirovanie volnovyh avlenij v dvuh geometriceski nelinejnyh uprugih soosnyh cilindriceskih obolockah, soderzasih vazkuu neszimaemuu zidkost’ // Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mehanika. Informatika. - 2016. - T. 16, No 2. - S. 184-197.
20. Kauderer G. Nelinejnaa mehanika. - M.: Inostrannaa literatura, 1961. - 778 c.
21. Lojcanskij L.G. Mehanika zidkosti i gaza. - M.: Drofa, 2003. - 840 c.
22. Vallander S.V. Lekcii po gidroaeromehanike. - L.: Izd-vo LGU, 1978. - 296 c.
23. Vol’mir A. S. Nelinejnaa dinamika plastinok i obolocek. - M.: Nauka, 1972. - 432 c.
24. Vol’mir A. S. Obolocki v potoke zidkosti i gaza: zadaci gidrouprugosti. - M.: Nauka, 1979. - 320 c.
25. Slihting G. Teoria pogranicnogo sloa. - M.: Nauka, 1974. - 712 c.
26. Civilihin S.A., Popov I.U., Gusarov V.V. Dinamika skrucivania nanotrubok v vazkoj zidkosti // DAN. - 2007. - T. 412, No 2. - S. 201-203. DOI
27. Popov I.U., Rodygina O.A., Civilihin S.A., Gusarov V.V. Soliton v stenke nanotrubki i stoksovo tecenie v nej // PZTF. - 2010. - T. 36, No 18. - S. 48-54. DOI
28. Gerdt V.P., Blinkov U.A. O strategii vybora nemultiplikativnyh prodolzenij pri vycislenii bazisov Zane // Programmirovanie. - 2007. - T. 33, No 3. - S. 34-43. DOI
29. Blinkov U.A., Gerdt V.P. Specializirovannaa sistema komp’uternoj algebry GINV // Programmirovanie. - 2008. - T. 34, No 2. - S. 67-80. DOI
30. Gerdt V.P., Blinkov Yu.A. Involution and difference schemes for the Navier-Stokes equations // Computer Algebra in Scientific Computing. CASC 2009. - Lecture Notes in Computer Science. - Vol. 5743. - Springer, 2009. - P. 94-105. DOI
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