Fundamental solutions of the Stokes system in quaternion analysis

Doan Cong Dinh

doi:10.37863/umzh.v74i11.6510

Doan Cong Dinh School Appl. Math. and Informatics, Hanoi Univ. Sci. and Technology, Vietnam

DOI: https://doi.org/10.37863/umzh.v74i11.6510

Анотація

УДК 532.5

Фундаментальні розв’язки системи Стокса у кватерніонному аналізі

Метод кватерніонного аналізу в механіці рідин був розроблений кількома поколіннями математиків, і було отримано багато важливих результатів. У даній статті додано один невеликий результат у цьому напрямку. Так, наведено нове переформулювання фундаментальних розв’язків системи Стокса в рамках кватерніонного аналізу та побудовано інтегральні зображення її розв’язків.

Посилання

L. Bers, Theory of pseudo-analytic functions, New York Univ., New York (1953).

S. Bock, K. Gürlebeck, D. Legatiuk, H. M. Nguyen, $Psi$-hyperholomorphic functions and a Kolosov–Muskhelishvili formula, Math. Methods Appl. Sci., 38, No. 18, 5114–5123 (2015). DOI: https://doi.org/10.1002/mma.3431

S. Bock, On monogenic series expansions with applications to linear elasticity, Adv. Appl. Clifford Algebras, 24, 931–943 (2014). DOI: https://doi.org/10.1007/s00006-014-0490-0

T. T. Cong, J. Blake, General solution of the Stokes flow equations, J. Math. Anal. and Appl., 90, 72–84 (1982). DOI: https://doi.org/10.1016/0022-247X(82)90045-2

Y. Grigor'ev, Three-dimensional analogue of Kolosov–Muskhelishvili formulae, Modern Trends in Hypercomplex Analysis, Trends Math., Birkhäuser, Cham (2016). DOI: https://doi.org/10.1007/978-3-319-42529-0_11

Yu. Grigor'ev, K. Gürlebeck, D. Legatiuk, A. Yakovlev, On quaternionic functions for the solution of an ill-posed Cauchy problem for a viscous fluid, AIP Conf. Proc., 2116, No. 1, Article 160005 (2019). DOI: https://doi.org/10.1063/1.5114149

Yu. Grigor'ev, Quaternionic functions and their applications in a viscous fluid flow, Complex Anal. and Oper. Theory, 12, 491–508 (2018). DOI: https://doi.org/10.1007/s11785-017-0715-z

K. Gürlebeck, W. Sprössig, Quaternionic and Clifford calculus for physicists and engineers, Math. Methods Practice, Wiley, Chichester (1997).

K. Gürlebeck, W. Sprössig, Quaternionic analysis and elliptic boundary value problems, Birkhäuser, Basel and Akademie-Verlag, Berlin (1989). DOI: https://doi.org/10.1515/9783112576182

V. V. Kravchenko, Applied quaternionic analysis, Birkhäuser, Basel (2009).

O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, 2nd english ed., Gordon and Breach, New York etc. (1969) (translated from Russian).

W. E. Langlois, M. O. Deville, Slow viscous flow, second ed., Springer Int. Publ., Switzerland (2014). DOI: https://doi.org/10.1007/978-3-319-03835-3

L.-W. Liu, H.-K. Hong, Clifford algebra valued boundary integral equations for three-dimensional elasticity, Appl. Math. Model., 54, 246–267 (2018). DOI: https://doi.org/10.1016/j.apm.2017.09.031

P. M. Naghdi, C. S. Hsu, On the representation of displacements in linear elasticity in terms of three stress functions, J. Math. and Mech., 10, 233–245 (1961). DOI: https://doi.org/10.1512/iumj.1961.10.10016

B. S. Padmaathi, G. P. R. Sekhar, T. Amaranath, A note on complete general solutions of Stokes equations, Quart. J. Mech. and Appl. Math., 51, No. 3, 383–388 (1998). DOI: https://doi.org/10.1093/qjmam/51.3.383

W. Sprössig, K. Gürlebeck, On the treatment of fluid problems by methods of Clifford analysis, Math. and Comput. Simulation, 44, No. 4, 401–413 (1997). DOI: https://doi.org/10.1016/S0378-4754(97)00066-9

W. Sprössig, Quaternionic analysis in fluid mechanics, Birkhäuser, Boston, MA (2000). DOI: https://doi.org/10.1007/978-1-4612-1374-1_3

I. N. Vekua, Generalized analytic functions, Pergamon Press, London (1962).

D. Weisz-Patrault, S. Bock, K. Gürlebeck, Three-dimensional elasticity based on quaternion-valued potentials, Int. J. Solids and Structures, 51, No. 19, 3422–3430 (2014). DOI: https://doi.org/10.1016/j.ijsolstr.2014.06.002

M. Zabarankin, Cauchy integral formula for generalized analytic functions in hydrodynamics, Proc. Roy. Soc. A: Math., Phys. and Eng. Sci., 468, 3745–3764 (2012). DOI: https://doi.org/10.1098/rspa.2012.0335

M. Zabarankin, The framework of $k$-harmonically analytic functions for three-dimensional Stokes flow problems, Part I, SIAM J. Appl. Math., 89, No. 3, 845–880 (2008). DOI: https://doi.org/10.1137/080715913