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On Small Motions of Hydrodynamic Systems Containing a Viscoelastic Fluid

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Abstract

This paper is devoted to small motions problem of composite hydrodynamic systems containing a viscoelastic fluid. Hydrodynamic systems of two or more viscoelastic fluids are considered, as well as partially dissipative systems containing ideal fluids or a barotropic gas (in addition to a viscoelastic fluid). The listed initial-boundary-value problems are investigated using the operator approach developed by N. D. Kopachevsky. This approach makes it possible to pass from the initial-boundary-value problem to the Cauchy problem for an operator-differential equation in the sum of Hilbert spaces and to prove the correct solvability theorem.

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REFERENCES

  1. J. C. Maxwell, ‘‘On the dynamical theory of gases,’’ Philos. Trans. Roy. Soc. London 157, 49–88 (1867).

    Article  Google Scholar 

  2. J. C. Maxwell, ‘‘On the dynamical theory of gases,’’ Philos. Mag. London 35, 129–145 (1868).

    Article  MATH  Google Scholar 

  3. W. Thomson (Kelvin), ‘‘On the theory of viscoelastic fluids,’’ Math. A. Phys. Pap. 3, 27–84 (1875).

    Google Scholar 

  4. W. Voight, ‘‘Uber die innere Reibung der fasten Korper, inslesondere der Krystalle,’’ Gottingen Abh. 36, 3–47 (1889).

    Google Scholar 

  5. W. Voight, ‘‘Uber innex Reibung fester Körper, insbesondere der Metalle,’’ Ann. Phys. 47, 671–693 (1892).

    Article  Google Scholar 

  6. J. G. Oldroyd, ‘‘On the formulation of rheological equations of state,’’ Proc. R. Soc. London, Ser. A 200, 523–541 (1950).

    Article  MathSciNet  MATH  Google Scholar 

  7. J. G. Oldroyd, ‘‘The elastic and viscous properties of emulsions and suspensions,’’ Proc. R. Soc. London, Ser. A 218, 122–137 (1953).

    Article  MATH  Google Scholar 

  8. J. G. Oldroyd, ‘‘Non-Newtonian flow of liquids and solids,’’ Rheol.: Theory Appl. 1, 653–682 (1956).

    Google Scholar 

  9. J. G. Oldroyd, ‘‘Non-linear stress, rate of strain relations at finite rate of shear in co-called ’linear elastico-viscous liquids’,’’ in Proceedings of the International Symposium on Second Order Effects in Elasticity, Plasticity and Fluid Dynamics, 1962 (1964), pp. 520–529.

  10. F. R. Eirich, Rheology: Theory and Applications (Academic, New York, 1956).

    MATH  Google Scholar 

  11. M. Reiner, Deformation Strain and Flow: An Elementary Introduction to Rheology (Interscience, New York, London, 1960).

    Book  Google Scholar 

  12. M. Reiner, Ten Lectures on Theoretical Rheology (H. K. Lewis, London, 1943).

  13. R. M. Christensen, Theory of Viscoelasticity (Academic, New York, 1982).

    Google Scholar 

  14. A. P. Oskolkov, ‘‘Initial-boundary-value problems for equations of motion of Kelvin-Voight fluids and Oldroyd fluids,’’ Tr. Mat. Inst. Steklov 179, 126–164 (1988).

    MathSciNet  Google Scholar 

  15. N. D. Kopachevsky and S. G. Krein, Operator Approach to Linear Problems of Hydrodynamics, Vol. 2: Nonself-Adjoint Problems for Viscous Fluids, Vol. 146 of Operator Theory: Advances and Applications (Springer, Basel, 2003).

  16. A. Cotsiolis and A. P. Oskolkov, ‘‘On the dynamical system generated b’ the equations of motion of Oldroyd fluids,’’ J. Sov. Math. 41, 967–970 (1988).

    Article  MathSciNet  Google Scholar 

  17. N. A. Karazeeva, A. A. Cotsiolis, and A. P. Oskolkov, ‘‘Dynamical systems generated by initial-boundary-value problems for equations of motion of linear viscoelastic fluids,’’ Proc. Steklov Inst. Math. 188, 73–108 (1991).

    MATH  Google Scholar 

  18. N. A. Karazeeva, ‘‘Initial boundary value problems for linear viscoelastic flows generated by integrodifferential equations,’’ J. Math. Sci. 127, 1869–1874 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  19. V. P. Orlov and P. E. Sobolevskii, ‘‘On mathematical models of a viscoelasticity with a memory,’’ Diff. Int. Equat. 4, 103–115 (1991).

    MathSciNet  MATH  Google Scholar 

  20. V. P. Orlov, ‘‘On the Oldroid model of a viscoelastic fluid,’’ Funct. Anal. Appl. 33, 72–75 (1999).

    Article  MathSciNet  Google Scholar 

  21. V. P. Orlov, ‘‘On the strong solutions of a regularized model of a nonlinear visco-elastic medium,’’ Math. Notes 84, 224–238 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  22. V. G. Zvyagin and M. V. Turbin, ‘‘The study of initial-boundary-value problems for mathematical models of the motion of Kelvin–Voigt fluids,’’ J. Math. Sci. 168, 157–308 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  23. V. G. Zvyagin and V. P. Orlov, ‘‘On strong solutions of a fractional nonlinear viscoelastic Voigt-type model,’’ Russ. Math. 63, 96–100 (2019).

    Article  MATH  Google Scholar 

  24. V. G. Zvyagin and A. V. Zvyagin, ‘‘Optimal feedback control for a thermoviscoelastic model of the motion of water polymer solutions,’’ Sib. Adv. Math. 29, 137–152 (2019).

    Article  MATH  Google Scholar 

  25. V. V. Vlasov and N. A. Rautian, ‘‘Integrodifferential equations in viscoelasticity theory,’’ Russ. Math. 56 (6), 48–51 (2012).

    Article  MATH  Google Scholar 

  26. V. V. Vlasov and N. A. Rautian, ‘‘Spectral analysis of linear models of viscoelasticity,’’ J. Math. Sci. 230, 668–672 (2018).

    Article  MATH  Google Scholar 

  27. V. V. Vlasov and N. A. Rautian, ‘‘Well-posedness and spectral analysis of integrodifferential equations arising in viscoelasticity theory,’’ J. Math. Sci. 233, 555–577 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  28. A. V. Davydov and Y. A. Tikhonov, ‘‘Study of Kelvin–Voigt models arising in viscoelasticity,’’ Diff. Equat. 54, 1620–1635 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  29. A. V. Davydov and Y. A. Tikhonov, ‘‘On properties of the spectrum of an operator pencil arising in viscoelasticity theory,’’ Math. Notes 103, 841–845 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  30. M. A. Artemov and E. S. Baranovskii, ‘‘Mixed boundary-value problems for motion equations of a viscoelastic medium,’’ Electron. J. Differ. Equat. 252, 1–9 (2015).

    MathSciNet  MATH  Google Scholar 

  31. M. A. Artemov and E. S. Baranovskii, ‘‘Global existence results for Oldroyd fluids with wall slip,’’ Acta Appl. Math. 147, 197–210 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  32. H. Essaouini and P. Capodanno, ‘‘Mathematical study of the small oscillations of a pendulum completely filled with a viscoelastic fluid,’’ Russ. J. Nonlin. Dyn. 16, 309–324 (2020).

    MathSciNet  MATH  Google Scholar 

  33. H. Essaouini and P. Capodanno, ‘‘Analysis of the small oscillations of two nonmixing fluids, the upper inviscid, the lower viscoelastic, in a fixed open container,’’ J. Math. Fluid Mech. 22 (4), 14 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  34. G. Amendola, M. Fabrizio, and J. M. Golden, Thermodynamics of Materials with Memory (Springer, New York, 2012).

    Book  MATH  Google Scholar 

  35. G. Amendola, M. Fabrizio, and J. M. Golden, ‘‘Free energies and minimal states for scalar linear viscoelasticity,’’ J. Elast. 123, 97–123 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  36. O. A. Ladyzhenskaya, The Boundary Value Problem of Mathematical Physics (Springer, Berlin, 1985), p. 322.

    Book  MATH  Google Scholar 

  37. A. I. Miloslavskii, ‘‘Spectral properties of a class of quadratic operator pencils,’’ Funct. Anal. Appl. 15, 142–144 (1988).

    Article  MathSciNet  Google Scholar 

  38. A. I. Miloslavskii, ‘‘Stability of a viscoelastic isotropic medium,’’ Dokl. Math. 33, 300–301 (1988).

    MathSciNet  Google Scholar 

  39. A. I. Miloslavskii, ‘‘The spectrum of small oscillations of a viscoelastic hereditary medium,’’ Dokl. Math. 40, 538–541 (1990).

    MathSciNet  Google Scholar 

  40. S. G. Krein and G. I. Laptev, ‘‘Motion of a viscous liquid in an open vessel,’’ Funct. Anal. Appl. 2, 38–47 (1968).

    Article  MATH  Google Scholar 

  41. S. G. Krein, Linear Equations in Banach Spaces (Birkhauser, Basel, 1982), p. 106.

    Book  MATH  Google Scholar 

  42. N. D. Kopachevsky and S. G. Krein, Operator Approach to Linear Problems of Hydrodynamics, Vol. 1: Self-Adjoint Problems for an Ideal Fluid, Vol. 128 of Operator Theory: Advances and Applications (Birkhäuser, Basel, 2001).

  43. N. D. Kopachevsky and E. V. Syomkina ‘‘Small movements of hydrosystem in stationary container,’’ Dinam. Sist. 7, 207–228 (2017).

    MATH  Google Scholar 

  44. N. D. Kopachevsky and E. V. Syomkina, ‘‘Formulas for orthogonal projectors generated by the problem on small motions of three viscoelastic fluids in a stationary container,’’ Tavrich. Vestn. Inform. Mat. 2, 48–61 (2017).

    Google Scholar 

  45. N. D. Kopachevsky, ‘‘To the problem on small motions of the System of two viscoelastic fluids in a fixed vessel,’’ Sovrem. Mat. Fundam. Napravl. 64, 547–572 (2018).

    MathSciNet  Google Scholar 

  46. N. D. Kopachevsky and E. V. Syomkina, ‘‘On small motions of hydraulic systems containing a viscoelastic fluid,’’ Itogi Nauki Tekh., Ser.: Sovrem. Mat. Pril. Tem. Obzory 172 (2), 48–90 (2019).

    Google Scholar 

  47. D. A. Zakora and N. D. Kopachevsky, ‘‘To the problem on normal oscillations of hydrosystem of two viscoelastic fluids in stationary container (a model problem),’’ Sovrem. Mat. Fundam. Napravl. 66, 186–208 (2020).

    Google Scholar 

  48. D. A. Zakora, ‘‘Small motions of a partially dissipative hydrodynamic system,’’ J. Math. Sci. 103, 252–255 (2001).

    Article  MathSciNet  Google Scholar 

  49. D. A. Zakora, ‘‘Structure and localization of the spectrum in a hydrodynamic problem,’’ J. Math. Sci. 107, 4448–4452 (2001).

    Article  MathSciNet  Google Scholar 

  50. D. A. Zakora, ‘‘Operator approach to the Ilyushin model for a viscoelastic body of parabolic type,’’ J. Math. Sci. 225, 345–381 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  51. D. A. Zakora, ‘‘Spectral analysis of a viscoelasticity problem,’’ Comput. Math. Math. Phys. 58, 1761–1774 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  52. D. A. Zakora, ‘‘Oldroyd model for compressible fluids,’’ J. Math. Sci. 239, 582–607 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  53. D. A. Zakora, ‘‘Asymptotics of solutions in the problem about small motions of a compressible Maxwell fluid,’’ Diff. Equat. 55, 1150–1163 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  54. D. A. Zakora, ‘‘Model of the Maxwell compressible fluid,’’ J. Math. Sci. 250, 593–610 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  55. N. D. Kopachevsky, ‘‘Abstract Green formulas for triples of Hilbert spaces and sesquilinear forms,’’ J. Math. Sci. 225, 226–264 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  56. M. S. Agranovich, ‘‘Remarks on potential spaces and Besov spaces in a Lipschitz domain and on Whitney arrays on its boundary,’’ Russ. J. Math. Phys. 15, 146–155 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  57. V. S. Rychkov, ‘‘On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains,’’ J. London Math. Soc. 60, 237–257 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  58. V. I. Voititsky, N. D. Kopachevsky, and P. A. Starkov, ‘‘Multicomponent conjugation problems and auxiliary abstract boundary-value problems,’’ J. Math. Sci. 170, 131–172 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  59. B. M. Vronskii, ‘‘On small motions of a ’liquid-gas’ system in a bounded domain,’’ Ukr. Math. J. 58, 1501–1511 (2006).

    Article  MathSciNet  Google Scholar 

  60. E. L. Gaziev and N. D. Kopachevsky, ‘‘Small motions and eigenoscillations of a ’fluid–barotropic gas’ hydrosystem,’’ J. Math. Sci. 192, 389–416 (2013).

    Article  MathSciNet  MATH  Google Scholar 

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  1. V.I. Vernadsky Crimean Federal University, 295007, Simferopol, Republic of Crimea, Russia

    E. V. Plokhaya

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Plokhaya, E.V. On Small Motions of Hydrodynamic Systems Containing a Viscoelastic Fluid. Lobachevskii J Math 42, 996–1013 (2021). https://doi.org/10.1134/S1995080221050140

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