N. V. Burmasheva, E. Yu. Prosviryakov
EXACT SOLUTIONS FOR NATURAL CONVECTION OF LAYERED FLOWS OF A VISCOUS INCOMPRESSIBLE FLUID WITH SPECIFIED TANGENTIAL FORCES AND THE LINEAR DISTRIBUTION OF TEMPERATURE ON THE LAYER BOUNDARIES
DOI: 10.17804/2410-9908.2017.4.016-031 A new exact solution of the Oberbeck-Boussinesq equations for the convective flow of a viscous incompressible fluid is considered. Layered flows of a viscous incompressible fluid are investigated within the class of the Sidorov-Lin exact solutions, which generalizes the family of the Ostroumov-Birikh solutions. The use of an exact solution allows an overdetermined system of fluid motion equations to be solved. The fluid is heated by setting a heat source at both boundaries. The dimension of the studied boundary value problem cannot be lowered by the transformation of the rotation. The obtained exact solution describes the counterflow in the fluid. Thermocline and a boundary layer occur near one of the boundary layers in the fluid flow.
Acknowledgements: This work was supported by the Foundation for Assistance to Small Innovative Enterprises in Science and Technology (the UMNIK program); the agreement no. 12281 GU/2017. Keywords: layered flow, exact solution, counterflows Bibliography:
- Joseph D.D. Stability of fluid motions. Berlin, Heidelberg, New York, Springer–Verlag, 1976.
- Gershuni G.Z., Zhukhovitskii E.M. Convective Stability of Incompressible Fluids. Israel Program for Scientific Translations. Jerusalem, Keter Publishing House, 1976, 330 p.
- Shtern V. Counterflows. Paradoxical Fluid Mechanics Phenomena. Cambridge University Press, 2012, 469 p. DOI: 10.1017/CBO9781139226516
- Dorrepaal J.M. An exact solution of the Navier-Stokes equation which describes nonorthogonal stagnation-point flow in two dimensions. Journal of Fluid Mechanics, 1986, vol. 163, no. 1, pp. 141–147. DOI: 10.1017/s0022112086002240
- Stuart J.T. The viscous flow near a stagnation point when the external flow has uniform vorticity. Journal of the Aerospace Sciences, 1959, vol. 26, no. 2, pp. 124–125. DOI: 10.2514/8.7963
- Riesco-Chueca P., De la Mora J.F. Brownian motion far from equilibrium: a hypersonic approach. Journal of Fluid Mechanics, 1990, vol. 214, pp. 639–663. DOI: 10.1017/S0022112090000301
- Hiemenz K. Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder. Dingler’s Politech. J., 1911, vol. 326, pp. 321–324.
- Ekman V.W. On the Influence of the Earth’s Rotation on Ocean-Currents. Ark. Mat. Astron. Fys., 1905, vol. 2, no. 11, pp. 1–52.
- Aristov S.N., Knyazev D.V., Polyanin A.D. Exact solutions of the Navier-Stokes equations with the linear dependence of velocity components on two space variables. Theoretical Foundations of Chemical Engineering, 2009, vol. 43, iss. 5, pp. 642–661. DOI: 10.1134/S0040579509050066
- Aristov S.N., Prosviryakov E.Yu. Nonuniform convective Couette flow. Fluid Dynamics, 2016, vol. 51, no. 5, pp. 581–587. DOI: 10.1134/S001546281605001X.
- Gorshkov A.V., Prosviryakov E.Yu. Layered B´enard-Marangoni convection during heat transfer according to the Newton's law of cooling. Kompyuternye Issledovaniya i Modelirovanie, 2016, vol. 8, no. 6, pp. 927–940. (In Russian).
- Gorshkov A.V., Prosviryakov E.Yu. Complex stationary convection with third-kind boundary conditions at the boundaries of a fluid layer. Diagnostics, Resource and Mechanics of Materials and Structures, 2016, iss. 2, pp. 34–47. DOI: 10.17804/2410-9908.2016.2.034-047 Available at: http://dream-journal.org/issues/2016-2/2016-2_81.html (accessed: 01.10.2017).
- Gorshkov A.V., Prosviryakov E.Yu. Analytic solutions of stationary complex convection describing a shear stress field of different signs. Trudy Inst. Mat. i Mekh. UrO RAN, 2017, vol. 23, no. 2, pp. 32–41. (In Russian). DOI: 10.21538/0134-4889-2017-23-2-32-4114
- Aristov S.N., Prosviryakov E.Yu., Spevak L.F. Nonstationary laminar thermal and solutal Marangoni convection of a viscous fluid. Vychislitelnaya Mekhanika Sploshnykh Sred, 2015, vol. 8, no. 4, pp. 445–456. (In Russian).
- Burmasheva N.V., Prosviryakov E.Yu. A large-scale layered stationary convection of an incompressible viscous fluid under the action of shear stresses at the upper boundary. Velocity field investigation. Vestn. Samar. Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki, 2017, vol. 21, no. 1, pp. 180–196. (In Russian). DOI: 10.14498/vsgtu152716
- Burmasheva N.V., Prosviryakov E.Yu. Exact solutions for layered large-scale convection induced by tangential stresses specified on the free boundary of a fluid layer. IOP Conference. Series: Materials Science and Engineering, 2017, vol. 208, conf. 1. DOI: 10.1088/1757-899X/208/1/012010
- Ostroumov G.A. Free convection under the condition of the internal problem. Washington, NACA Technical Memorandum 1407, National Advisory Committee for Aeronautics, 1958.
- Birikh R. V. Thermocapillary convection in a horizontal layer of liquid. J. Appl. Mech. Tech. Phys., 1966, vol. 7, no. 3, pp. 43–44.
- Sidorof A.F. On two classes of solutions of the equations of fluid and gas mechanics and their relation to the theory of traveling waves. Prikl. Mech. i Tekhnich. Fizika, 1989, no. 2, pp. 34–40. (In Russian).
- Aristov S.N., Prosviryakov E.Yu. A new class of exact solutions for three-dimensional thermal diffusion equations. Theor. Found. Chem. Eng., 2016, vol. 50, no. 3, pp. 286–293. DOI: 10.1134/S0040579516030027
- Aristov S.N., Frik P.G. Nonlinear effects of the Ekman layer on the dynamics of largescale eddies in shallow water. J. Appl. Mech. Tech. Phys., 1991, vol. 32, no. 2, pp. 189–194.
- Ingel L.Kh., Aristov S.N. The class of exact solutions of nonlinear problems on thermal circulation associated with volumetric heat release in the atmosphere. Tr. In-ta Eksperim. Meteorol., 1996, no. 27 (162), pp. 142–157. (In Russian).
- Aristov S.N., Shvarts K.G. Vikhrevye Techeniya Advektivnoy Prirody vo Vrashchayushchemsya Sloe Zhidkosti [Vortical Flows of the Advective Nature in a Rotating Fluid Layer]. Perm, Perm State Univ. Publ., 2006, 155 pp. (In Russian).
- Aristov S.N., Shvarts K.G. Vikhrevye Techeniya v Tonkikh Sloyakh Zhidkosti [Vortical Flows in Thin Fluid Layers]. Kirov, Vyatka State Univ. Publ., 2011, 207 pp. (In Russian).
- Aristov S.N., Shvarts K.G. Advective flow in a rotating liquid film. J. Appl. Mech. Tech. Phys., 2016, vol. 57, no. 1, pp. 188–194. DOI: 10.1134/S0021894416010211
- Aristov S.N., Prosviryakov E.Yu. On laminar flows of planar free convection. Nelin. Dinam., 2013, vol. 9, no. 4, pp. 651–657. (In Russian). DOI: 10.20537/nd1304004
- Andreev V.K. Resheniya Birikha uravneniy konvektsii i nekotorye ego obobshcheniya [Birikh Solutions to Convection Equations and Some of its Extensions]. Krasnoyarsk, IBM SO RAN Publ., 2010, 68 p. (In Russian).
- Aristov S.N., Prosviryakov E.Yu., Spevak L.F. Unsteady-state Bénard–Marangoni convection in layered viscous incompressible flows. Theor. Found. Chem. Eng., 2016, vol. 50, no. 2, pp. 132–141. DOI: 10.1134/S0040579516020019.
- Andreev V.K., Gaponenko Ya.A., Goncharova O.N., Pukhnachev V.V. Mathematical Models of Convection. Berlin, Boston, De Gryuter Publ., 2012, 417 p.
- Pukhnachev V.V. Group-theoretical methods in the convection problems. In: Application of Mathematics in Technical and Natural Sciences, M.D. Todorov and C.I. Christov, eds., American Institute of Physics, CP 1404, Melwille, NY, 2011, pp. 31–42.
- Pukhnachev V.V. Non-stationary analogues of the Birikh solution. Izvestiya AltGU, 2011, no. 1–2, pp. 62–69. (In Russian).
- Andreev V.K., Bekezhanova V.B. Stability of non-isothermal fluids (Review). J. Appl. Mech. Tech. Phys., 2013, vol. 54, no. 2, pp. 171–184. DOI: 10.1134/S0021894413020016
- Andreev V.K., Stepanova I.V. Unidirectional flows of binary mixtures within the framework of the Oberbeck–Boussinesq model. Fluid Dyn., 2016, vol. 51, no. 2, pp. 136–147. DOI: 10.1134/S0015462816020022
- Goncharova O.N., Kabov O.A. Gravitational-thermocapillary convection of fluid in the horizontal layer in co-current gas flow. Dokl. Phys., 2009, vol. 54, no. 5, pp. 242–247. DOI: 10.1134/S1028335809050061
- Goncharova O.N., Rezanova E.V. Example of an exact solution of the stationary problem of two-layer flows with evaporation at the interface. J. Appl. Mech. Tech. Phys., 2014, vol. 55, no. 2, pp. 247–257. DOI: 10.1134/S0021894414020072
- Birikh R.V., Pukhnachev V.V. An axial convective flow in a rotating tube with a longitudinal temperature gradient. Dokl. Phys., 2011, vol. 56, no. 1, pp. 47–52. DOI: 10.1134/S1028335811010095
- Birikh R.V., Pukhnachev V.V., Frolovskaya O.A. Convective flow in a horizontal channel with non-Newtonian surface rheology under time-dependent longitudinal temperature gradient. Fluid Dyn., 2015, vol. 50, no. 1, pp. 173–179. DOI: 10.1134/S0015462815010172
- Ryzhkov I.I. Termodiffuziya v Smesyakh: Uravneniya, Simmetrii, Resheniya i ikh Ustoychivost [Thermodiffusion in Mixtures: Equations, Symmetries, Solutions and their Stability]. Novosibirsk, SB RAS Publ., 2013, 200 p. (In Russian).
Н. В. Бурмашева, Е. Ю. Просвиряков
ТОЧНЫЕ РЕШЕНИЯ ДЛЯ ЕСТЕСТВЕННОЙ КОНВЕКЦИИ СЛОИСТЫХ ТЕЧЕНИЙ ВЯЗКОЙ НЕСЖИМАЕМОЙ ЖИДКОСТИ ПРИ ЗАДАНИИ ТАНГЕНЦИАЛЬНЫХ СИЛ И ЛИНЕЙНОГО РАСПРЕДЕЛЕНИЯ ТЕМПЕРАТУРЫ НА ГРАНИЦАХ СЛОЯ
В статье рассмотрено новое точное решение уравнений Обербека-Буссинеска для конвективного течения вязкой несжимаемой жидкости. Слоистые течения вязкой несжимаемой жидкости исследуются в рамках класса точных решений Сидорова-Линя, обобщающего семейство решений Остроумова-Бириха. Использование точного решения позволяет разрешить переопределенную систему уравнений движения жидкости. Нагрев жидкости осуществляется посредством задания источника тепла на обеих границах. Размерность изучаемой краевой задачи не может быть понижена преобразованием поворота. Полученное точное решение описывает противотечения в жидкости. При течении жидкости возникает термоклин и пограничный слой вблизи одной из границ слоя.
Благодарности: Работа выполнена при поддержке фонда содействия развитию малых форм предприятий в научно-технической сфере (программа УМНИК), договор № 12281ГУ/2017. Ключевые слова: слоистое течение, точное решение, противотечения Библиография:
- Джозеф Д. Д. Устойчивость движений жидкости. – М. : Мир, 1981.
- Gershuni G. Z., Zhukhovitskii E. M. Convective Stability of Incompressible Fluids. Israel Program for Scientific Translations. – Jerusalem : Keter Publishing House, 1976. – 330 p.
- Shtern V. Counterflows. Paradoxical Fluid Mechanics Phenomena. – Cambridge : Cambridge University Press, 2012. – 469 p. – DOI: 10.1017/CBO9781139226516
- Dorrepaal J. M. An exact solution of the Navier-Stokes equation which describes nonorthogonal stagnation-point flow in two dimensions // Journal of Fluid Mechanics. – 1986. – Vol. 163, no. 1. – P. 141–147. – DOI: 10.1017/s0022112086002240
- Stuart J. T. The viscous flow near a stagnation point when the external flow has uniform vorticity // Journal of the Aerospace Sciences. – 1959. – Vol. 26, no. 2. – P. 124–125. – DOI: 10.2514/8.7963
- Riesco-Chueca P., De la Mora J. F. Brownian motion far from equilibrium: a hypersonic approach // Journal of Fluid Mechanics. – 1990. – Vol. 214. – P. 639–663. – DOI: 10.1017/S0022112090000301
- Hiemenz K. Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder // Dingler’s Politech. J. – 1911. – Vol. 326. – P. 321–324.
- Ekman V. W. On the Influence of the Earth’s Rotation on Ocean-Currents // Ark. Mat. Astron. Fys. – 1905. – Vol. 2, no. 11. – P. 1–52.
- Aristov S. N., Knyazev D. V., Polyanin A. D. Exact solutions of the Navier-Stokes equations with the linear dependence of velocity components on two space variables // Theoretical Foundations of Chemical Engineering. – 2009. – Vol. 43, iss 5. – P. 642–661. – DOI: 10.1134/S0040579509050066
- Aristov S. N., Prosviryakov E. Yu. Nonuniform convective Couette flow // Fluid Dynamics. – 2016. – Vol. 51, no. 5. – P. 581–587. – DOI: 10.1134/S001546281605001X
- Горшков А. В., Просвиряков Е. Ю. Слоистая конвекция Бенара-Марангони при теплообмене по закону Ньютона-Рихмана // Компьютерные исследования и моделирование. – 2016. – Т. 8, № 6. – С. 927–940.
- Gorshkov A. V., Prosviryakov E. Yu. Complex stationary convection with third-kind boundary conditions at the boundaries of a fluid layer // Diagnostics, Resource and Mechanics of Materials and Structures. – 2016. iss. 2. – P. 34–47. – DOI: 10.17804/2410-9908.2016.2.034-047. – URL: http://dream-journal.org/issues/2016-2/2016-2_81.html
- Gorshkov A. V., Prosviryakov E. Yu. Analytic solutions of stationary complex convection describing a shear stress field of different signs // Trudy Inst. Mat. i Mekh. UrO RAN. – 2017. – Vol. 23, no. 2. – P. 32–41. – DOI: 10.21538/0134-4889-2017-23-2-32-41
- Аристов С. Н., Просвиряков Е. Ю., Спевак Л. Ф. Нестационарная слоистая тепловая и концентрационная конвекция Марангони вязкой жидкости // Вычисл. мех. сплошн. сред. – 2015. – Т. 8, № 4. – C. 445–456.
- Бурмашева Н. В., Просвиряков Е. Ю. Крупномасштабная слоистая стационарная конвекция вязкой несжимаемой жидкости под действием касательных напряжений на верхней границе. Исследование поля скоростей // Вестн. Самарск. гос. техн. унив. Сер. физ.-мат. науки. – 2017. – Т. 21, № 1. – C. 180–196. – DOI: 10.14498/vsgtu1527
- Burmasheva N. V., Prosviryakov E. Yu. Exact solutions for layered large-scale convection induced by tangential stresses specified on the free boundary of a fluid layer // IOP Conference Series: Materials Science and Engineering. – 2017. – Vol. 208, conf. 1. – DOI: 10.1088/1757- 899X/208/1/012010
- Ostroumov G. A. Free convection under the condition of the internal problem. – Washington : NACA Technical Memorandum 1407, National Advisory Committee for Aeronautics, 1958.
- Birikh R. V. Thermocapillary convection in a horizontal layer of liquid // J. Appl. Mech. Tech. Phys. – 1966. – Vol. 7, no. 3. – P. 43–44.
- Сидоров А. Ф. О двух классах решений уравнений механики жидкости и газа и их связи с теорией бегущих волн // Прикл. механика и технич. физика. – 1989. – № 2. – С. 34–40.
- Aristov S. N., Prosviryakov E. Yu. A new class of exact solutions for three-dimensional thermal diffusion equations // Theor. Found. Chem. Eng. – 2016. – Vol. 50, no. 3. – P. 286–293. – DOI: 10.1134/S0040579516030027.
- Aristov S. N., Frik P. G. Nonlinear effects of the Ekman layer on the dynamics of largescale eddies in shallow water // J. Appl. Mech. Tech. Phys. – 1991. – Vol. 32, no. 2. – P. 189–194.
- Ингель Л. Х., Аристов С. Н. Класс точных решенийнелинейных задач о термических циркуляциях, связанных с объемным тепловыделением в атмосфере // Тр. ин-та эксперим. метеорол. – 1991. – № 27 (162). – С. 142–157.
- Аристов С. Н., Шварц К. Г. Вихревые течения адвективной природы во вращающемся слое жидкости. – Пермь : Перм. гос. ун-т., 2006. – 155 с.
- Аристов С. Н., Шварц К. Г. Вихревые течения в тонких слоях жидкости. – Киров : Вятск. гос. ун-т, 2011. – 207 с.
- Aristov S. N., Shvarts K. G. Advective flow in a rotating liquid film // J. Appl. Mech. Tech. Phys. – 2016. – Vol. 57, no. 1. – P. 188–194. – DOI: 10.1134/S0021894416010211
- Аристов С. Н., Просвиряков Е. Ю. О слоистых течениях плоской свободной конвекции // Нелин. динам. – 2013. – Т. 9, № 4. – C. 651–657. – DOI: 10.20537/nd1304004
- Андреев В. К. Решения Бириха уравнений конвекции и некоторые его обобщения // Красноярск : Ин-т вычисл. мех. СО РАН, препринт № 1–10, 2010. – 68 с.
- Aristov S. N., Prosviryakov E. Yu., Spevak L. F. Unsteady-state Bénard–Marangoni convection in layered viscous incompressible flows // Theor. Found. Chem. Eng. – 2016. – Vol. 50, no. 2. – P. 132–141. DOI: 10.1134/S0040579516020019
- Mathematical Models of Convection / V. K. Andreev, Ya. A. Gaponenko, O. N. Goncharova, V. V. Pukhnachev. – Berlin–Boston : De Gryuter Publ., 2012. – 417 p.
- Pukhnachev V. V. Group-theoretical methods in the convection problems // Application of Mathematics in Technical and Natural Sciences / ed. by M. D. Todorov and C. I. Christov. – Melwille, NY : American Institute of Physics, CP 1404, 2011. – P. 31–42.
- Пухначев В. В. Нестационарные аналогии решения Бириха // Известия АлтГУ. – 2011. – №. 1–2. – С. 62–69.
- Andreev V. K., Bekezhanova V. B. Stability of non-isothermal fluids (Review) // J. Appl. Mech. Tech. Phys. – 2013. – Vol. 54, no. 2. – P. 171–184. – DOI: 10.1134/S0021894413020016
- Andreev V. K., Stepanova I. V. Unidirectional flows of binary mixtures within the framework of the Oberbeck–Boussinesq model // Fluid Dyn. – 2016. – Vol. 51, no. 2. – P. 136–147. – DOI: 10.1134/S0015462816020022.
- Goncharova O. N., Kabov O. A. Gravitational-thermocapillary convection of fluid in the horizontal layer in co-current gas flow // Dokl. Phys. – 2009. – Vol. 54, no. 5. – P. 242–247. – DOI: 10.1134/S1028335809050061
- Goncharova O. N., Rezanova E. V. Example of an exact solution of the stationary problem of two-layer flows with evaporation at the interface // J. Appl. Mech. Tech. Phys. – 2014. – Vol. 55, no. 2. – P. 247–257. – DOI: 10.1134/S0021894414020072.
- Birikh R. V., Pukhnachev V. V. An axial convective flow in a rotating tube with a longitudinal temperature gradient // Dokl. Phys. – 2011. – Vol. 56, no. 1. – P. 47–52. – DOI: 10.1134/S1028335811010095
- Birikh R. V., Pukhnachev V. V., Frolovskaya O. A. Convective flow in a horizontal channel with non-Newtonian surface rheology under time-dependent longitudinal temperature gradient // Fluid Dyn. – 2015. – Vol. 50, no. 1. – P. 173–179. – DOI: 10.1134/S0015462815010172.
- Рыжков И. И. Термодиффузия в смесях: уравнения, симметрии, решения и их устойчивость. – Новосибирск : СО РАН, 2013. – 200 с.
Библиографическая ссылка на статью
Burmasheva N. V., Prosviryakov E. Yu. Exact Solutions for Natural Convection of Layered Flows of a Viscous Incompressible Fluid with Specified Tangential Forces and the Linear Distribution of Temperature on the Layer Boundaries // Diagnostics, Resource and Mechanics of materials and structures. -
2017. - Iss. 4. - P. 16-31. - DOI: 10.17804/2410-9908.2017.4.016-031. -
URL: http://dream-journal.org/issues/2017-4/2017-4_145.html (accessed: 23.11.2024).
|