## Couette-Hiemenz exact solutions for the steady creeping convective flow of a viscous incompressible fluid with allowance made for heat recovery

**Authors:**Privalova V.V^{1}, Prosviryakov E.Y.^{1}^{,2}**Affiliations:**- Institute of Engineering Science, Urals Branch, Russian Academy of Sciences
- Ural Federal University named after the First President of Russia B. N. Yeltsin

**Issue:**Vol 22, No 3 (2018)**Pages:**532-548**Section:**Articles**URL:**https://journals.eco-vector.com/1991-8615/article/view/20605**DOI:**https://doi.org/10.14498/vsgtu1638- Cite item

# Abstract

In this paper, we study the steady creeping convective flow of a viscous incompressible fluid in the thin infinite layer. The study of the fluid flow is based on the exact solutions class for the Oberbeck-Boussinesq equations in the Stokes approximation using. The velocity field is described by the Hiemenz exact solution. The temperature field and the pressure field linearly depend on the horizontal (longitudinal) coordinate, it corresponds to the Ostroumov-Birich exact solutions class. The convective motion of a viscous incompressible fluid was induced by tangential stresses on the upper permeable (porous) boundary and thermal source definition at the lower boundary. In addition, the heat exchange according to the Newton-Richmann law takes into account at the upper boundary. The obtained exact solutions describe counterflows in fluids. The stagnant points number in the fluid layer does not exceed three. The formation of counterflows in the fluid is accompanied by sucking and injection of the fluid through the permeable boundary. The larger number of stagnant points presence forms a cellular structure of the streamlines. In addition, the velocity field, which obtained in the solution of the boundary value problem is characterized by localization of the flow near the boundary of the fluid layer (boundary layer). The exact solutions obtained in this paper can be used for the nonlinear Oberbeck-Boussinesq system solving. The Grashof number can take large values, which depends on the geometric anisotropy index for the linearized Oberbeck-Boussinesq system.

# Full Text

Introduction. The system consisting of the Boussinesq approximation of the Navier-Stokes equations (Oberbeck-Boussinesq equation system) and the incompressibility equation is one of the most common mathematical systems of equations which describes convective flows of a viscous incompressible fluid [1- 7]. These equations are extremely useful tools for the modeling of convective processes in a fluid [1-7]. The complexity of obtaining exact solutions for the Navier-Stokes equations and their modifications consists in the presence of a substantial (quadratic) nonlinearity of the equations. This nonlinearity is generated by the presence of a convective term in the total derivative. By now, a certain stock of exact solutions to the Oberbeck-Boussinesq equations has been accumulated [3-18]. Exact solutions describing convective flows of a viscous incompressible fluid allow the qualitative and quantitative characteristics of a moving flow to be re-evaluated. With certain assumptions for highly viscous fluids, the Oberbeck-Boussinesq system can be simplified by neglecting inertial effects due to the predominance of viscous forces (the Stokes approximation) [8, 10-19]. The first classes of exact solutions for the natural convection equations were considered by G. A. Ostroumov [20], R. V. Birich [21], and M. I. Schliomis [22]. Later on, the exact solutions presented in [20-22] were generalized in papers and monographs [3, 6-12, 18, 19, 23-30]. The classes of the exact solutions, based on the representation of velocities in the form of linearly dependent functions, were proposed by C. C. Lin for magnetic hydrodynamics [31]. The most complete list of exact solutions obtained in Lin’s class for isothermal fluids is found in the review [32]. The Hiemenz exact solution [33] belongs to the class of velocities, linear in part of the coordinates. It is characterized by the existence of a stagnant point in the flow of a viscous incompressible fluid. By now, there have appeared Hiemenz exact solution modifications for isothermal flows [34-36] and convective flows [37-40] of a viscous incompressible fluid. The proposed generalizations of the Hiemenz exact solution family allow one to investigate counterflows in a fluid that are induced by kinematic, dynamic, and thermal perturbations of the fluid flow (boundary conditions) [8, 9, 11-17, 37-40]. Additionally, papers studying the existence of stagnant points in the flow and flows near such points are worthy of note. The assignment of relations between hydrodynamic fields ignoring fluid evaporation and suction predominates among the diversity of boundary conditions on the free (interphase) boundary of the fluid layer [3, 12, 26-30]. Convective flows with fluid evaporation from a free surface can be simulated by replacing it with a permeable non-deformable boundary [7, 37]. This approach is implemented for the description of convection in magnetic fluids [37-40] and nanofluids with various properties [37-40] by numerical integration of the Oberbeck-Boussinesq equations. To eliminate the deficiency of exact solutions for the Oberbeck-Boussinesq 533 P r i v a l o v a V. V., P r o s v i r y a k o v E. Yu. system, an exact solution for the stationary creeping convective flow of a horizontal infinite layer of a viscous incompressible fluid is obtained in this paper. Tangential stresses are given on the upper thermally insulated permeable layer boundary, constant pressure and the Newton-Richman law of heat exchange being taken into account at the upper boundary. The no-slip condition is satisfied and the heat source is given at the lower boundary. The obtained exact solution is studied in detail for the diagnostics of the stratification of hydrodynamic fields describing the modification of the Couette-Hiemenz flow [15-17]. 1. Problem statement. The plane motion equations of a viscous incompressible fluid in an infinite layer with plane boundaries (Fig. 1), which describe the effect of temperature on the distribution of hydrodynamic fields, are considered in the Boussinesq approximation [1, 15-17] as (︁# About the authors

### Valentina V Privalova

Institute of Engineering Science, Urals Branch, Russian Academy of Sciences
Email: valentprival@gmail.com

34, Komsomolskaya st., Ekaterinburg, 620049, Russian Federation Cand. Phys. & Math. Sci.; Researcher; Sector of Nonlinear Vortex Hydrodynamics

### Evgeny Yu Prosviryakov

Institute of Engineering Science, Urals Branch, Russian Academy of Sciences; Ural Federal University named after the First President of Russia B. N. Yeltsin
Email: evgen_pros@mail.ru

34, Komsomolskaya st., Ekaterinburg, 620049, Russian Federation; 19, Mira st., Ekaterinburg, 620002, Russian Federation Dr. Phys. & Math. Sci., Professor; Head of Sector; Sector of Nonlinear Vortex Hydrodynamics

# References

- Gershuni G. Z., Zhukhovitskii E. M. Convective Stability of Incompressible Fluids. Jerusalem, Keter Publishing House, 1976, vi+330 pp., https://ntrl.ntis.gov/NTRL/dashboard/searchResults/titleDetail/PB255645.xhtml.
- Drazin T. Introduction to Hydrodynamic Stability. Cambridge, Cambridge University Press, 2002, xviii+258 pp. doi: 10.1017/cbo9780511809064.
- Andreev V. K., Gaponenko Ya. A., Goncharova O. N., Pukhnachev V. V. Mathematical Models of Convection, De Gruyter Studies in Mathematical Physics, vol. 5. Berlin, Boston, De Gryuter Publ., 2012, xvi+420 pp. doi: 10.1515/9783110258592.
- Nepomnyashchy A., Simanovskii I., Legros J. C. Interfacial Convection in Multilayer System. New York, Springer-Verlag, 2012, xiii+498 pp. doi: 10.1007/978-0-387-87714-3.
- Shtern V. Counterflows. Paradoxical Fluid Mechanics Phenomena. Cambridge, Cambridge University Press, 2012, xiv+470 pp. doi: 10.1017/CBO9781139226516.
- Aristov S. N., Shvarts K. G. Vortical Flows of the Advective Nature in a Rotating Fluid Layer. Perm, Perm State Univ., 2006, 155 pp. (In Russian)
- Aristov S. N., Shvarts K. G. Vortical Flows in Thin Fluid Layers. Kirov, Vyatka State Univ., 2011, 207 pp. (In Russian)
- Aristov S. N., Shvarts K. G. Convective heat transfer in a locally heated plane incompressible fluid layer, Fluid Dyn., 2013, vol. 48, no. 3, pp. 330-335. doi: 10.1134/s001546281303006x.
- Aristov S. N., Knyazev D. V. Localized convective flows in a nonuniformly heated liquid layer, Fluid Dyn., 2014, vol. 49, no. 5, pp. 565-575. doi: 10.1134/S0015462814050020.
- Ryzhkov I. I. Thermodiffusion in Mixtures: Equations, Symmetries, Solutions and their Stability. Novosibirsk, SB RAS Publ., 2013, 199 pp. (In Russian)
- Sidorov A. F. Two classes of solutions of the fluid and gas mechanics equations and their connection to traveling wave theory, J. Appl. Mech. Tech. Phys., 1989, vol. 30, no. 2, pp. 197-203. doi: 10.1007/bf00852164.
- Andreev V. K., Cheremnykh E. N. The joint creeping motion of three viscid liquids in a plane layer: A priori estimates and convergence to steady flow, J. Appl. Ind. Math., 2016, vol. 10, no. 1, pp. 7-20. doi: 10.1134/S1990478916010026.
- Vlasova S. S., Prosviryakov E. Yu. Two-dimensional convection of an incompressible viscous fluid with the heat exchange on the free border, Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2016, vol. 20, no. 3, pp. 567-577. doi: 10.14498/vsgtu1483.
- Aristov S. N., Prosviryakov E. Yu. On one class of analytic solutions of the stationary axisymmetric convection Bénard-Maragoni viscous incompreeible fluid, Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2013, no. 3(32), pp. 110-118 (In Russian). doi: 10.14498/vsgtu1205.
- Aristov S. N., Privalova V. V., Prosviryakov E. Yu. Stationary nonisothermal Couette flow. Quadratic heating of the upper boundary of the fluid layer, Rus. J. Nonlin. Dyn., 2016, vol. 12, no. 2, pp. 167-178 (In Russian). doi: 10.20537/nd1602001.
- 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, no. 2, pp. 92-109 (In Russian). doi: 10.17804/2410-9908.2018.2.092-109.
- Privalova V. V., Prosviryakov E. Yu. Steady convective Coutte flow for quadratic heating of the lower boundary fluid layer, Rus. J. Nonlin. Dyn., 2018, vol. 14, no. 1, pp. 69-79 (In Russian). doi: 10.20537/nd1801007.
- Aristov S. N., Prosviryakov E. Yu. Nonuniform convective Couette flow, Fluid Dyn., 2016, vol. 51, no. 5, pp. 581-587. doi: 10.1134/S001546281605001X.
- Deryabina M. S., Martynov S. I. Periodic flow of a viscous fluid with a predetermined pressure and temperature gradient, Rus. J. Nonlin. Dyn., 2018, vol. 14, no. 1, pp. 81-97 (In Russian). doi: 10.20537/nd1801008.
- Ostroumov G. A. Free convection under the condition of the internal problem, NASA Technical Memorandum 1407. Washington, National Advisory Committee for Aeronautics, 1958.
- Birikh R. V. Thermocapillary convection in a horizontal layer of liquid, J. Appl. Mech. Tech. Phys., 1966, no. 7, pp. 43-44. doi: 10.1007/bf00914697.
- Shliomis M. I., Yakushin V. I. The Convection in a Two-layer Binary System with Evaporation, Hydrodynamics, 1972, no. 4, pp. 129-140 (In Russian).
- Aristov S. N., Prosviryakov E. Yu. A new class of exact solutions for three-dimensional thermal diffusion equations, Theor. Found. of Chem. Engin., 2016, vol. 50, no. 3, pp. 286-293. doi: 10.1134/S0040579516030027.
- Pukhnachev V. V. Non-stationary analogues of the Birikh solution, Izv. Alt. Gos. Univ., 2011, no. 1-2, pp. 62-69 (In Russian).
- Pukhnachev V. V. Exact solutions of the hydrodynamic equations derived from partially invariant solutions, J. Appl. Mech. Tech. Phys., 2003, vol. 44, no. 3, pp. 317-323. doi: 10.1023/A:1023472921305.
- Aristov S. N., Prosviryakov E. Yu., Spevak L. F. Nonstationary laminar thermal and solutal Marangoni convection of a viscous fluid, Comp. Cont. Mech., 2015, vol. 8, no. 4, pp. 445-456 (In Russian). doi: 10.7242/1999-6691/2015.8.4.38.
- Goncharova O. N. Exact solution of linearized equations of convection of a weakly compressible fluid, J. Appl. Mech. Tech. Phys, 2005, vol. 46, no. 2, pp. 191-201. doi: 10.1007/s10808-005-0032-6.
- Bratsun D. A., Vyatkin V. A., Mukhamatullin A. R. On exact nonstationary solutions of equations of vibrational convection, Comp. Cont. Mech., 2017, vol. 10, no. 4, pp. 433-444 (In Russian). doi: 10.7242/1999-6691/2017.10.4.35.
- 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. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2017, vol. 21, no. 1, pp. 180-196 (In Russian). doi: 10.14498/vsgtu1527.
- Burmasheva N. V., Prosviryakov E. Yu. A large-scale layered stationary convection of a incompressible viscous fluid under the action of shear stresses at the upper boundary. Temperature and presure field investigation, Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2017, vol. 21, no. 4, pp. 736-751 (In Russian). doi: 10.14498/vsgtu1568.
- Lin C. C. Note on a class of exact solutions in magneto-hydrodynamics, Arch. Rational Mech. Anal., 1957, vol. 1, no. 1, pp. 391-395. doi: 10.1007/bf00298016;. doi: 10.1142/9789814415651_0022.
- 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, Theor. Found. Chem. Eng., 2009, vol. 43, no. 5, pp. 642-662. doi: 10.1134/s0040579509050066.
- Hiemenz K. Die Grenzschicht an einem in den gleichförmigen Flüssigkeit-sstrom eingetauchten geraden Kreiszylinder, Dingler’s Politech. J., 1911, vol. 326, pp. 321-324.
- Pukhnachev V. V. Group properties of the Navier-Stokes equations in the plane case, J. Appl. Math. Tech. Phys., 1960, no. 1, pp. 83-90.
- Aristov S. N., Knyazev D. V. Viscous fluid flow between moving parallel plates, Fluid Dyn., 2012, vol. 47, no. 4, pp. 476-61. doi: 10.1134/s0015462812040060.
- Petrov A. G. Exact solution of the Navier-Stokes equations in a fluid layer between the moving parallel plates, J. Appl. Math. Tech. Phys., 2012, vol. 53, no. 5, pp. 642-646. doi: 10.1134/S0021894412050021.
- Tsai R., Huang J. S. Heat and mass transfer for Soret and Dufour’s effects on Hiemenz flow through porous medium onto a stretching surface, Int. J. Heat Mass Trans., 2009, vol. 52, no. 9-10, pp. 2399-2406. doi: 10.1016/j.ijheatmasstransfer.2008.10.017.
- Afify A. A. Similarity solution in MHD: effects of thermal diffusion and diffusion thermo effects on free convective heat and mass transfer over a stretching surface considering suction or injection, Commun. Nonlinear Sci. Numer. Simul., 2009, vol. 14, no. 5, pp. 2202-2214. doi: 10.1016/j.cnsns.2008.07.001.
- Beg O. A., Bakier A. Y., Prasad V. R. Numerical study of free convection magnetohydrodynamic heat and mass transfer from a stretching surface to a saturated porous medium with Soret and Dufour effects, Comput. Mater. Sci., 2009, vol. 46, no. 1, pp. 57-65. doi: 10.1016/j.commatsci.2009.02.004.
- Osalusi E., Side J., Harris R. Thermal-diffusion and diffusion-thermo effects on combined heat and mass transfer of a steady MHD convective and slip flow due to a rotating disk with viscous dissipation and Ohmic heating, Int. Commun. Heat Mass Trans., 2008, vol. 35, no. 8, pp. 908-915. doi: 10.1016/j.icheatmasstransfer.2008.04.011.
- Prasolov V. V. Polynomials, Algorithms and Computation in Mathematics, vol. 11. Berlin, Springer-Verlag, 2004, xiv+301 pp. doi: 10.1007/978-3-642-03980-5.