Exact Solutions to the Oberbeck–boussinesq Equations for Convective Stokes Flows

L. S. Goruleva; I. I. Obabkov; E. Yu. Prosviryakov

doi:10.17804/2410-9908.2024.2.036-049

L. S. Goruleva, I. I. Obabkov, E. Yu. Prosviryakov

EXACT SOLUTIONS TO THE OBERBECK–BOUSSINESQ EQUATIONS FOR CONVECTIVE STOKES FLOWS

DOI: 10.17804/2410-9908.2024.2.036-049

TTo study large-scale convective flows (fluid motion in a thin layer), it is possible, for initial studies, to consider the Stokes approximation when integrating the Oberbeck–Boussinesq equation. The convective derivative in the momentum transfer equations and in the heat conduction equation is in this case assumed to be identically equal to zero. The paper discusses several approaches to constructing exact solutions for slow (creeping) flows of a non-uniformly heated fluid. Formulas for three-dimensional flows in the Lin–Sidorov–Aristov class are given for steady flows. The hydrodynamic fields are described by polynomials. Exact solutions are given for the velocity field nonlinearly depending on two spatial coordinates (longitudinal, or horizontal) with coefficients of nonlinear forms, which depend on the third coordinate. The study shows how it is possible to automate the computation of unknown coefficients for the formation of hydrodynamic fields (velocities and temperatures).

Keywords: exact solution, Navier–Stokes equation, Oberbeck–Boussinesq equation, Stokes approximation, convection

References:

Ostroumov, G.A. Free convection under the condition of the internal problem. Technical Memorandum Ser., 1407, National Advisory Committee for Aeronautics, Washington, 1958.
Gershuni, G.Z. and Zhukhovitskii, E.M. Convective Stability of Incompressible Liquid, Wiley, Keter Press, Jerusalem, 1976, 330 p.
Gershuni, G.Z., Zhukhovitskii, E.M., and Nepomnyashchii, A.A. Ustoychivost konvektivnykh techeniy [Stability of Convective Flows]. Nauka Publ., Moscow, 1989, 318 p. (In Russian).
Getling, A.V. Formation of spatial structures in Rayleigh–Bénard convection. Sov. Phys. Usp., 1991, 34 (9), 737–776. DOI: 10.1070/PU1991v034n09ABEH002470.
Andreev, V.K., Gaponenko, Ya.A., Goncharova, O.N., and Pukhnachev, V.V. Mathematical Models of Convection, De Gruyter Studies in Mathematical Physics Ser., Walter De Gruyter, Berlin, 2012, 417 p. DOI: 10.1515/9783110258592.
Aristov, S.N. and Schwarz, K.G. Vikhrevye techeniya v tonkkih sloyakh zhidkosti [Vortex Flows in Thin Layers of Liquid]. FGBOU VPO VyatGU Publ., Kirov, 2011, 206 p. (In Russian).
Birikh, R.V. Thermocapillary convection in a horizontal layer of liquid. J. Appl. Mech. Tech. Phys., 1966, 7, 43–44. DOI: 10.1007/bf00914697.
Shliomis, M.I. and Yakushin, V.I. The convection in a two-layer binary system with evaporation. Uchenye Zapiski Permskogo Gosuniversiteta, Seriya Gidrodinamika, 1972, 4, 129–140. (In Russian).
Gershuni, G.Z. On the stability of plane convective motion of a fluid. Zh. Tekh. Fiz., 1953, 23 (10), 1838–1844. (In Russian).
Batchelor, G.K. Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures. Quart. Appl. Math., 1954, 12 (3), 209–233. DOI: 10.1090/qam/64563.
Schwarz, K.G. Plane-parallel advective flow in a horizontal incompressible fluid layer with rigid boundaries. Fluid Dyn., 2014, 49 (4), 438–442. DOI: 10.1134/ S0015462814040036.
Knyazev, D.V. Two-dimensional flows of a viscous binary fluid between moving solid boundaries. J. Appl. Mech. Tech. Phys., 2011, 52 (2), 212–217. DOI:10.1134/ S0021894411020088.
Aristov, S.N. and Prosviryakov, E.Yu. On laminar flows of planar free convection. Nelineynaya Dinamika, 2013, 9 (4), 651–657. (In Russian).
Aristov, S.N., Prosviryakov, E.Yu., and Spevak, L.F. Unsteady-state Bénard–Marangoni convection in layered viscous incompressible flows. Theoretical Foundations of Chemical Engineering, 2016, 50 (2), 132–141. DOI: 10.1134/S0040579516020019.
Aristov, S.N., Prosviryakov, E.Yu., and Spevak, L.F. Nonstationary laminar thermal and solutal Marangoni convection of a viscous fluid. Vychislitelnaya Mekhanika Sploshnykh Sred, 2015, 8 (4), 445–456. (In Russian). DOI: 10.7242/1999-6691/2015.8.4.38.
Lin, C.C. Note on a class of exact solutions in magnetohydrodynamics. Archive for Rational Mechanics and Analysis, 1958, 1, 391–395. DOI:10.1007/BF00298016.
Sidorov, A.F. Two classes of solutions of the fluid and gas mechanics equations and their connection to traveling wave theory. Journal of Applied Mechanics and Technical Physics, 1989, 30 (2), 197–203. DOI: 10.1007/BF00852164.
Aristov, S.N. Vikhrevye techeniya v tonkikh sloyakh zhidkosti [Eddy Currents in Thin Liquid Layers: Synopsis of Doctoral Thesis]. Vladivostok, 1990, 303 p. (In Russian).
Aristov, S.N. and Shvarts, K.G. Convective heat transfer in a locally heated plane incompressible fluid layer. Fluid Dynamics, 2013, 48, 330–335. DOI: 10.1134/S001546281303006X.
Aristov, S.N. and Prosviryakov, E.Yu. A new class of exact solutions for three-dimensional thermal diffusion equations. Theoretical Foundations of Chemical Engineering, 2016, 50 (3), 286–293. DOI: 10.1134/S0040579516030027.
Burmasheva, N.V. and Prosviryakov, E.Yu. Exact solutions to the Navier–Stokes equations for describing the convective flows of multilayer fluids. Rus. J. Nonlin. Dyn., 2022, 18 (3), 397–410. DOI: 10.20537/nd220305.
Burmasheva, N.V. and Prosviryakov, E.Yu. Exact solutions to the Navier–Stokes equations describing stratified fluid flows. Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki, 2021, 25 (3), 491–507. DOI: 10.14498/vsgtu1860.
Goruleva, L.S. and Prosviryakov, E.Yu. A new class of exact solutions to the Navier–Stokes equations with allowance for internal heat release. Optics and Spectroscopy, 2022, 130 (6), 365–370. DOI: 10.1134/S0030400X22070037.
Burmasheva, N., Ershkov, S., Prosviryakov, E., and Leshchenko, D. Exact solutions of Navier–Stokes equations for quasi-two-dimensional flows with Rayleigh friction. Fluids, 2023, 8 (4), 123. DOI: 10.3390/fluids8040123.
Prosviryakov, E.Yu. New class of exact solutions of Navier-Stokes equations with exponential dependence of velocity on two spatial coordinates. Theoretical Foundations of Chemical Engineering, 2019, 53 (1), 107–114. DOI: 10.1134/S0040579518060088.
Baranovskii, E.S., Burmasheva, N.V., and Prosviryakov, E.Yu. Exact solutions to the Navier–Stokes equations with couple stresses. Symmetry, 2021, 13 (8), 1355. DOI: 10.3390/sym13081355.
Privalova, V.V. and Prosviryakov, E.Yu. A new class of exact solutions of the Oberbeck–Boussinesq equations describing an incompressible fluid. Theor. Found. Chem. Eng., 2022, 56 (3), 331–338. DOI: 10.1134/S0040579522030113.
Goruleva, L.S. and Prosviryakov, E.Yu. Nonuniform Couette–Poiseuille shear flow with a moving lower boundary of a horizontal layer. Technical Physics Letters, 2022, 48, 258–262, DOI: 10.1134/S1063785022090024.
Goruleva, L.S. and Prosviryakov, E.Yu. A new class of exact solutions for magnetohydrodynamics equations to describe convective flows of binary liquids. Khimicheskaya Fizika i Mezoskopiya, 2023, 25 (4), 447–462. (In Russian). DOI: 10.15350/17270529.2023.4.39.
Goruleva, L.S. and Prosviryakov, E.Yu. Exact solutions to the Navier–Stokes equations for describing inhomogeneous isobaric vertical vortex fluid flows in regions with permeable boundaries. Diagnostics, Resource and Mechanics of materials and structures, 2023, 1, 41–53. DOI: 10.17804/2410-9908.2023.1.041-053. Available at: http://dream-journal.org/issues/2023-1/2023-1_393.html
Goruleva, L.S. and Prosviryakov, E.Yu. Unidirectional steady-state inhomogeneous Couette flow with a quadratic velocity profile along a horizontal coordinate. Diagnostics, Resource and Mechanics of materials and structures, 2022, 3, 47–60. DOI: 10.17804/2410-9908.2022.3.047-060. Available at: http://dream-journal.org/issues/2022-3/2022-3_367.html
Goruleva, L.S. and Prosviryakov, E.Yu. Exact solutions for the description of nonuniform unidirectional flows of magnetic fluids in the Lin–Sidorov–Aristov class. Diagnostics, Resource and Mechanics of materials and structures, 2023, 5, 39–52. DOI: 10.17804/2410-9908.2023.5.039-052. Available at: http://dream-journal.org/issues/2023-1/2023-1_393.html
Privalova, V.V. and Prosviryakov, E.Yu. Steady convective Coutte flow for quadratic heating of the lower boundary fluid layer. Nelineinaya Dinamika, 2018, 14 (1), 69–79. (In Russian). DOI: 10.20537/nd1801007.
Vlasova, S.S. and Prosviryakov, E.Yu. Parabolic convective motion of a fluid cooled from below with the heat exchange at the free boundary. Russian Aeronautics, 2016, 59 (4), 529–535. DOI: 10.3103/S1068799816040140.
Aristov, S.N., Privalova, V.V., and Prosviryakov, E.Yu. Stationary nonisothermal Couette ﬂow. Quadratic heating of the upper boundary of the ﬂuid layer. Nelineynaya Dinamika, 2016, 12 (2), 167–178. (In Russian). DOI: 10.20537/nd1602001.
Aristov, S.N., Privalova, V.V., and Prosviryakov, E.Yu. Planar linear Bénard-Rayleigh convection under quadratic heating of the upper boundary of a viscous incompressible liquid layer. Vestnik Kazanskogo Gosudarstvennogo Tekhnicheskogo Universiteta im. A.N. Tupoleva, 2015, 2, 6–13. (In Russian).
Aristov, S.N. and Prosviryakov, E.Yu. Exact solutions of thermocapillary convection during localized heating of a flat layer of viscous incompressible liquid. Vestnik Kazanskogo gosudarstvennogo tekhnicheskogo universiteta im. A.N. Tupoleva, 2014, 3, 7–12. (In Russian).
Aristov, S.N. and Prosviryakov, E.Yu. On one class of analytic solutions of the stationary axisymmetric convection Bénard–Maragoni viscous incompressible fluid. Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2013, 3 (32), 110–118. (In Russian). DOI: 10.14498/vsgtu1205.
Privalova, V.V. and Prosviryakov, E.Yu. Exact solutions for a Couette–Hiemenz creeping convective flow with linear temperature distribution on the upper boundary. Diagnostics, Resource and Mechanics of materials and structures, 2018, 2, 92–109. DOI: 10.17804/2410-9908.2018.2.092-109. Available at: http://dream-journal.org/issues/2018-2/2018-2_170.html
Privalova, V.V. and Prosviryakov, E.Yu. Couette–Hiemenz exact solutions for the steady creeping convective flow of a viscous incompressible fluid, with allowance made for heat recovery. Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, 22 (3), 532–548. DOI: 10.14498/vsgtu1638.

Л. С. Горулева, И. И. Обабков, Е. Ю. Просвиряков

ТОЧНЫЕ РЕШЕНИЯ ДЛЯ УРАВНЕНИЙ ОБЕРБЕКА – БУССИНЕСКА ДЛЯ КОНВЕКТИВНЫХ ТЕЧЕНИЙ СТОКСА

При изучении конвективных крупномасштабных течений (движение жидкости в тонком слое) можно для первоначальных исследований рассматривать приближение Стокса при интегрировании уравнения Обербека – Буссинеска. В этом случае конвективную производную в уравнениях переноса импульса и в уравнении теплопроводности полагают тождественно равной нулю. В статье рассмотрено несколько подходов к построению точных решений для медленных (ползущих) течений неоднородно нагретой жидкости. Для установившихся течений приведены формулы для трехмерных течений в классе Линя – Сидорова – Аристова. Гидродинамические поля описываются полиномами. Приведены точные решения для поля скоростей, нелинейно зависящего от двух пространственных координат (продольных, или горизонтальных) с коэффициентами нелинейных форм, зависящими от третьей координаты. Показано, как можно автоматизировать вычисления неизвестных коэффициентов для формирования гидродинамических полей (скоростей и температуры).

Ключевые слова: точное решение, уравнение Навье – Стокса, уравнение Обербека – Буссинеска, аппроксимация Стокса, конвекция

Библиография:

Ostroumov G. A. Free convection under the condition of the internal problem. Ser. Technical Memorandum. – No. 1407. – Washington : National Advisory Committee for Aeronautics, 1958. DOI: 10.1007/s11003-015-9815-y.
Гершуни Г. З., Жуховицкий Е. М. Конвективная устойчивость несжимаемой жидкости. – М. : Наука, 1972. – 392 с.
Гершуни Г. З. Жуховицкий Е. М., Непомнящий А. А. Устойчивость конвективных течений. – М. : Наука, 1989. – 318 с.
Getling A. V. Formation of spatial structures in Rayleigh–Bénard convection // Sov. Phys. Usp. – 1991. – 34 (9). – P. 737–776. – DOI: 10.1070/PU1991v034n09ABEH002470.
Mathematical Models of Convection. Ser. De Gruyter Studies in Mathematical Physics / V. K. Andreev, Ya. A. Gaponenko, O. N. Goncharova, V. V. Pukhnachev. – Berlin : Walter De Gruyter, 2012. – 417 p. – DOI: 10.1515/9783110258592.
Аристов С. Н., Шварц К. Г. Вихревые течения в тонких слоях жидкости. – Киров : ФГБОУ ВПО "ВятГУ", 2011, 206 с.
Birikh R. V. Thermocapillary convection in a horizontal layer of liquid // J. Appl. Mech. Tech. Phys. – 1966. – No. 7. – P. 43–44. – DOI: 10.1007/bf00914697.
Шлиомис М. И., Якушин В. И. Конвекция в двухслойной бинарной системе с испарением // Ученые записки Пермского госуниверситета. Сер. Гидродинамика. – 1972. – № 4. – C. 129–140.
Гершуни Г. З. Об устойчивости плоского конвективного течения жидкости // Журнал технической физики. – 1953. – Т. 23, № 10. – С. 1838–1844.
Batchelor G. K. Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures // Quart. Appl. Math. – 1954. – Vol. 12, No. 3. – P. 209–233. – DOI: 10.1090/qam/64563.
Schwarz K. G. Plane-parallel advective flow in a horizontal incompressible fluid layer with rigid boundaries // Fluid Dynaics. – 2014. – Vol. 49, No. 4. – P. 438–442. – DOI: 10.1134/ S0015462814040036.
Knyazev D. V. Two-dimensional flows of a viscous binary fluid between moving solid boundaries // J. Appl. Mech. Tech. Phys. – 2011. – Vol. 52, No. 2. – P. 212–217. – DOI: 10.1134/ S0021894411020088.
Аристов С. Н., Просвиряков Е. Ю. О слоистых течениях плоской свободной конвекции // Нелинейная динамика. – 2013. – Т. 9, № 4. – С. 651–657.
Aristov S. N., Prosviryakov E. Y., Spevak L. F. Unsteady-state Bénard–Marangoni convection in layered viscous incompressible flows // Theoretical Foundations of Chemical Engineering. – 2016. – Vol. 50 (2). – P. 132–141. – DOI: 10.1134/S0040579516020019.
Аристов С. Н., Просвиряков Е. Ю., Спевак Л. Ф. Нестационарная слоистая тепловая и концентрационная конвекция Марангони вязкой несжимаемой жидкости // Вычислительная механика сплошных сред. – 2015. – Т. 8, № 4. – С. 445–456. – DOI: 10.7242/1999-6691/2015.8.4.38.
Lin C. C. Note on a class of exact solutions in magnetohydrodynamics // Archive for Rational Mechanics and Analysis. – 1958. – Vol. 1. – P. 391–395. – DOI: 10.1007/BF00298016.
Sidorov A. F. Two classes of solutions of the fluid and gas mechanics equations and their connection to traveling wave theory // Journal of Applied Mechanics and Technical Physics. – 1989. – Vol. 30, No. 2. – P. 197–203. – DOI: 10.1007/BF00852164.
Аристов С. Н. Вихревые течения в тонких слоях жидкости : автореф. дис. … докт. физ.-мат. наук : 01.02.05. – Владивосток, 1990. – 303 с.
Aristov S. N., Shvarts K. G. Convective heat transfer in a locally heated plane incompressible fluid layer // Fluid Dynamics. – 2013. – Vol. 48. – P. 330–335. – DOI: 10.1134/S001546281303006X.
Aristov S. N., Prosviryakov E. Yu. A new class of exact solutions for three-dimensional thermal diffusion equations // Theoretical Foundations of Chemical Engineering. – 2016. – Vol. 50 (3). – P. 286–293. – DOI: 10.1134/S0040579516030027.
Burmasheva N. V., Prosviryakov E. Yu. Exact solutions to the Navier–Stokes equations for describing the convective flows of multilayer fluids // Rus. J. Nonlin. Dyn. – 2022. – Vol. 18, No. 3. – P. 397–410. – DOI: 10.20537/nd220305.
Burmasheva N. V., Prosviryakov E. Yu. Exact solutions to the Navier–Stokes equations describing stratified fluid flows // Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki. – 2021. – Vol. 25 (3). – P. 491–507. – DOI: 10.14498/vsgtu1860.
Goruleva L. S., Prosviryakov E. Yu. A New class of exact solutions to the Navier-Stokes equations with allowance for internal heat release // Optics and Spectroscopy. – 2022. – Vol. 130, No. 6. – P. 365–370. – DOI: 10.1134/S0030400X22070037.
Exact solutions of Navier–Stokes equations for quasi-two-dimensional flows with Rayleigh friction / N. Burmasheva, S. Ershkov, E. Prosviryakov, D. Leshchenko // Fluids. – 2023. – Vol. 8 (4). – P. 123. – DOI: 10.3390/fluids8040123.
Prosviryakov E. Yu. New class of exact solutions of Navier–Stokes equations with exponential dependence of velocity on two spatial coordinates // Theoretical Foundations of Chemical Engineering. – 2019. – Vol. 53, No. 1. – P. 107–114. –10.1134/S0040579518060088.
Baranovskii E. S., Burmasheva N. V., Prosviryakov E. Yu. Exact solutions to the Navier–Stokes equations with couple stresses // Symmetry. – 2021. – Vol. 13 (8). – P. 1355. – DOI: 10.3390/sym13081355.
Privalova V. V., Prosviryakov E. Yu. A new class of exact solutions of the Oberbeck–Boussinesq equations describing an incompressible fluid // Theor. Found. Chem. Eng. – 2022. – Vol. 56, No. 3. – P. 331–338. – DOI: 10.1134/S0040579522030113.
Goruleva L. S., Prosviryakov E. Yu. Nonuniform Couette–Poiseuille shear flow with a moving lower boundary of a horizontal layer // Technical Physics Letters. – 2022. – Vol. 48. – P. 258–262. – DOI: 10.1134/S1063785022090024.
Горулева Л. С., Просвиряков Е. Ю. Новый класс точных решений уравнений магнитной гидродинамики для описания конвективных течений бинарных жидкостей // Химическая физика и мезоскопия. – 2023. – Т. 25, № 4. – С. 447–462. – DOI: 10.15350/17270529.2023.4.39.
Goruleva L. S., Prosviryakov E. Yu. Exact solutions to the Navier–Stokes equations for describing inhomogeneous isobaric vertical vortex fluid flows in regions with permeable boundaries // Diagnostics, Resource and Mechanics of materials and structures. – 2023. – Iss. 1. – P. 41–53. – DOI: 10.17804/2410-9908.2023.1.041-053. – URL: http://dream-journal.org/issues/2023-1/2023-1_393.html
Goruleva L. S., Prosviryakov E. Yu. Unidirectional steady-state inhomogeneous Couette flow with a quadratic velocity profile along a horizontal coordinate // Diagnostics, Resource and Mechanics of materials and structures. – 2022. – Iss. 3. – P. 47–60. – DOI: 10.17804/2410-9908.2022.3.047-060. – URL: http://dream-journal.org/issues/2022-3/2022-3_367.html
Goruleva L. S., Prosviryakov E. Yu. Exact solutions for the description of nonuniform unidirectional flows of magnetic fluids in the Lin–Sidorov–Aristov class // Diagnostics, Resource and Mechanics of materials and structures. – 2023. – Iss. 5. – P. 39–52. – DOI: 10.17804/2410-9908.2023.5.039-052. – URL: http://dream-journal.org/issues/2023-1/2023-1_393.html
Привалова В. В., Просвиряков Е. Ю. Стационарное конвективное течение Куэтта–Хименца при квадратичном нагреве нижней границы слоя жидкости // Нелинейная динамика. – 2018. – Т. 14, № 1. – С. 69–79. – DOI: 10.20537/nd1801007.
Vlasova S. S., Prosviryakov E. Y. Parabolic convective motion of a fluid cooled from below with the heat exchange at the free boundary // Russian Aeronautics. – 2016. – Vol. 59, No. 4. – P. 529–535. – DOI: 10.3103/S1068799816040140.
Аристов С. Н., Привалова В. В., Просвиряков Е. Ю. Стационарное неизотермическое течение Куэтта. Квадратичный нагрев верхней границы слоя жидкости // Нелинейная динамика. – 2016. – Т. 12, № 2. – С. 167–178. – DOI: 10.20537/nd1602001.
Аристов С. Н., Привалова В. В., Просвиряков Е. Ю. Плоская линейная конвекция Бенара-Рэлея при квадратичном нагреве верхней границы слоя вязкой несжимаемой жидкости // Вестник Казанского государственного технического университета им. А. Н. Туполева. – 2015. – № 2. – С. 6–13.
Аристов С. Н., Просвиряков Е. Ю. Точные решения термокапиллярной конвекции при локализованном нагреве плоского слоя вязкой несжимаемой жидкости // Вестник Казанского государственного технического университета им. А. Н. Туполева. – 2014. –№ 3. – С. 7–12.
Аристов С. Н., Просвиряков Е. Ю. Об одном классе аналитических решений стационарной осесимметричной конвекции Бенара–Марангони вязкой несжимаемой жидкости // Вестник Самарского государственного технического университета. Серия «Физ.-мат. науки». – 2013. – № 3 (32). – С. 110–118. – DOI: 10.14498/vsgtu1205.
Privalova V. V., Prosviryakov E. Yu. Exact solutions for a Couette–Hiemenz creeping convective flow with linear temperature distribution on the upper boundary // Diagnostics, Resource and Mechanics of materials and structures. – 2018. – Iss. 2. – Р. 92–109. – DOI: 10.17804/2410-9908.2018.2.092-109. – URL: http://dream-journal.org/issues/2018-2/2018-2_170.html
Privalova V. V., Prosviryakov E. Yu. Couette–Hiemenz exact solutions for the steady creeping convective flow of a viscous incompressible fluid, with allowance made for heat recovery // Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki. – 2018. – Vol. 22 (3). – P. 532–548. – DOI: 10.14498/vsgtu1638.

Библиографическая ссылка на статью

Goruleva L. S., Obabkov I. I., Prosviryakov E. Yu. Exact Solutions to the Oberbeck–boussinesq Equations for Convective Stokes Flows // Diagnostics, Resource and Mechanics of materials and structures. - 2024. - Iss. 2. - P. 36-49. -
DOI: 10.17804/2410-9908.2024.2.036-049. -
URL: http://dream-journal.org/issues/content/article_437.html
(accessed: 06.04.2025).