Nonlinear Gradient Flow of a Vertical Vortex Fluid in a Thin Layer
Received 18 July 2019
2019, Vol. 15, no. 3, pp. 271-283
Author(s): Privalova V., Prosviryakov E. Y., Simonov M. A.
A new exact solution to the Navier – Stokes equations is obtained. This solution describes
the inhomogeneous isothermal Poiseuille flow of a viscous incompressible fluid in a horizontal
infinite layer. In this exact solution of the Navier – Stokes equations, the velocity and pressure
fields are the linear forms of two horizontal (longitudinal) coordinates with coefficients depending
on the third (transverse) coordinate. The proposed exact solution is two-dimensional in terms
of velocity and coordinates. It is shown that, by rotation transformation, it can be reduced to
a solution describing a three-dimensional flow in terms of coordinates and a two-dimensional flow
in terms of velocities. The general solution for homogeneous velocity components is polynomials
of the second and fifth degrees. Spatial acceleration is a linear function. To solve the boundaryvalue
problem, the no-slip condition is specified on the lower solid boundary of the horizontal
fluid layer, tangential stresses and constant horizontal (longitudinal) pressure gradients specified
on the upper free boundary. It is demonstrated that, for a particular exact solution, up to
three points can exist in the fluid layer at which the longitudinal velocity components change
direction. It indicates the existence of counterflow zones. The conditions for the existence of
the zero points of the velocity components both inside the fluid layer and on its surface under
nonzero tangential stresses are written. The results are illustrated by the corresponding figures
of the velocity component profiles and streamlines for different numbers of stagnation points.
The possibility of the existence of zero points of the specific kinetic energy function is shown. The
vorticity vector and tangential stresses arising during the flow of a viscous incompressible fluid
layer under given boundary conditions are analyzed. It is shown that the horizontal components
of the vorticity vector in the fluid layer can change their sign up to three times. Besides,
tangential stresses may change from tensile to compressive, and vice versa. Thus, the above
exact solution of the Navier – Stokes equations forms a new mechanism of momentum transfer
in a fluid and illustrates the occurrence of vorticity in the horizontal and vertical directions in
a nonrotating fluid. The three-component twist vector is induced by an inhomogeneous velocity
field at the boundaries of the fluid layer.
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References |
|
| [1] |
Altukhov, Yu. A. and Pyshnogray, I. G., “On Allowing Slip in Plane-Parallel Flows of Polymeric Liquids”, Mekhanika kompozitsionnykh materialov i konstruktsii, 17:3 (2011), |
| [2] |
Aristov, S. N. and Knyazev, D. V., “New Exact Solution of the Problem of Rotationally Symmetric Couette – Poiseuille Flow”, J. Appl. Mech. Tech. Phys., 48:5 (2007),     ; Prikl. Mekh. Tekh. Fiz., 48:5 (2007), |
| [3] |
Aristov, S. N., Knyazev, D. V., and 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., 43:5 (2009), ; Teoret. Osnovy Khim. Tekhnolog., 43:5 (2009), |
| [4] |
Aristov, S. N., Privalova, V. V., and Prosviryakov, E. Yu., “Stationary Nonisothermal Couette Flow. Quadratic Heating of the Upper Boundary of the Fluid Layer”, Nelin. Dinam., 12:2 (2016),   |
| [5] |
Aristov, S. N. and Prosviryakov, E. Yu., “On Laminar Flows of Planar Free Convection”, Nelin. Dinam., 9:4 (2013), |
| [6] |
Aristov, S. N. and Prosviryakov, E. Yu., “Inhomogeneous Couette Flow”, Nelin. Dinam., 10:2 (2014), |
| [7] |
Aristov, S. N. and Prosviryakov, E. Yu., “Large-Scale Flows of Viscous Incompressible Vortical Fluid”, Russ. Aeronaut., 58:4 (2015), ; Izv. Vyssh. Uchebn. Zaved. Aviats. Tekh., 2015, no. 4, |
| [8] |
Aristov, S. N. and Prosviryakov, E. Yu., “Unsteady Layered Vortical Fluid Flows”, Fluid Dynam., 51:2 (2016), ; Izv. Ross. Akad. Nauk. Mekh. Zidk. Gaza, 2016, no. 2, |
| [9] |
Aristov, S. N. and Prosviryakov, E. Yu., “Nonuniform Convective Couette Flow”, Fluid Dynam., 51:5 (2016), ; Izv. Ross. Akad. Nauk. Mekh. Zidk. Gaza, 2016, no. 5, |
| [10] |
Aristov, S. N. and Skulskiy, O. I., “Viscoelastic Effects of Blood Flow in Nondeformable Capillary”, Ross. Zh. Biomekh., 3:4 (1999), |
| [11] |
Asmolov, E. S. and Manuilovich, S. V., “Instability of a Horizontal Plane-Channel Flow of a Dilute Suspension”, Fluid Dynam., 44:1 (2009), ; Izv. Ross. Akad. Nauk. Mekh. Zidk. Gaza, 2009, no. 1, |
| [12] |
Birikh, R. V., Pukhnachev, V. V., and Frolovskaya, O. A., “Convective Flow in a Horizontal Channel with Non-Newtonian Surface Rheology under Time-Dependent Longitudinal Temperature Gradient”, Fluid Dynam., 50:1 (2015), ; Izv. Ross. Akad. Nauk. Mekh. Zidk. Gaza, 2015, no. 1, |
| [13] |
Blokhin, A. M. and Tkachev, D. L., “Linear Asymptotic Instability of a Stationary Flow of a Polymeric Medium in a Plane Channel in the Case of Periodic Perturbations”, J. Appl. Ind. Math., 8:4 (2014), ; Sibirsk. Zh. Industr. Matem., 17:3 (2014), |
| [14] |
Blokhin, A. M. and Semenko, R. E., “Stationary Magnetohydrodynamical Flows of Non-Isothermal Polymeric Liquid in the Flat Channel”, Vestn. Yuzhno-Ural. Gos. Univ. Ser. Matem. model. i progr., 11:4 (2018), |
| [15] |
Calderer, M. C. and Mukherjee, B., “On Poiseuille Flow of Liquid Crystals”, Liq. Cryst., 22:2 (1997), |
| [16] |
Davey, A. and Drazin, P. G., “The Stability of Poiseuille Flow in a Pipe”, J. Fluid Mech., 36:2 (1969), |
| [17] |
Drazin, P. G. and Riley, N., The Navier – Stokes Equations: A Classification of Flows and Exact Solutions, London Math. Soc. Lecture Note Ser., 334, Cambridge Univ. Press, Cambridge, 2006, 196 pp. |
| [18] |
Gavrilenko, S. L., Vasin, R. A., and Shilko, S. V., “A Method for Determining Flow and Rheological Constants of Viscoplastic Biomaterials: P. 1”, Ross. Zh. Biomekh., 6:3 (2002), |
| [19] |
Gol'dshtik, M. A., Shtern, V. N., and Yavorskiy, N. I., Viscous Flows with Paradoxical Properties, Nauka, Novosibirsk, 1989, 336 pp. (Russian) |
| [20] | Hartmann, J., Hg-Dynamics: 1. Theory of the Laminar Flow of an Electrically Conductive Liquid in a Homogeneous Magnetic Field, v. 15, Det Kgl. Danske Videnskabernes Selskkab. Math.-fys. Medd., Levin & Munksgaard, København, 1937, 28 pp. |
| [21] |
Knyazev, D. V. and Kolpakov, I. Yu., “Exact Solutions of the Problem of the flow of a Viscous Fluid in a Cylindrical Region with a Varying Radius”, Nelin. Dinam., 11:1 (2015), |
| [22] | Korotaev, G. K., Mikhailova, E. N., and Shapiro, N. B., Theory for Equatorial Countercurrents in the World Ocean, Naukova Dumka, Kiev, 1986, 208 pp. (Russian) |
| [23] | Kulikovskii, A. G. and Lyubimov, G. A., Magnetic Hydrodynamics, Logos, Mocsow, 2005, 328 pp. (Russian) |
| [24] |
Kuznetsova, Yu. L. and Skul'skiy, O. I., “Shear Banding of the Fluid with a Nonmonotonic Dependence of Flow Stress upon Strain Rate”, Vychisl. Mekh. Sploshn. Sred, 11:1 (2018), |
| [25] |
Kuznetsova, Ju. L., Skul'skiy, O. I., and Pyshnograi, G. V., “Pressure Driven Flow of a Nonlinear Viscoelastic Fluid in a Plane Channel”, Vychisl. Mekh. Sploshn. Sred, 3:2 (2010), |
| [26] |
Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow, Gordon & Breach, New York, 1969, 184 pp. |
| [27] |
Landau, L. D. and Lifshitz, E. M., Course of Theoretical Physics: In 10 Vols., v. 6, Fluid Mechanics, 2nd ed., Butterworth-Heinemann, Oxford, 2003, 552 pp. |
| [28] |
Medvedev, A. E., “Three-Dimensional Motion of a Viscous Incompressible Fluid in a Narrow Tube”, J. Appl. Mech. Tech. Phys., 50:4 (2009), ; Prikl. Mekh. Tekhn. Fiz., 50:4(296) (2009), |
| [29] |
Medvedev, A. E., “Unsteady Motion of a Viscous Incompressible Fluid in a Tube with a Deformable Wall”, J. Appl. Mech. Tech. Phys., 54:4 (2013), ; Prikl. Mekh. Tekhn. Fiz., 54:4(320) (2013), |
| [30] |
Meleshko, S. V., Petrova, A. G., and Pukhnachev, V. V., “Characteristic Properties of the System of Equations for an Incompressible Viscoelastic Maxwell Medium”, J. Appl. Mech. Tech. Phys., 58:5 (2017), ; Prikl. Mekh. Tekhn. Fiz., 58:5(345) (2017), |
| [31] | Pedley, T. J., The Fluid Mechanics of Large Blood Vessel, Cambridge Univ. Press, Cambridge, 1980, 464 pp. |
| [32] |
Poddar, A., Mandal, Sh., Bandopadhyay, A., and Chakraborty, S., “Electrical Switching of a Surfactant Coated Drop in Poiseuille Flow”, J. Fluid Mech., 870 (2019), |
| [33] |
Poiseuille, J.-L.-M., “Recherches expérimenteles sur le mouvement des liquides dans les tubes de très petits diamètres”, Comptes rendus hebdomadaires des séances de l'Académie des sciences, 11 (1840), |
| [34] |
Poiseuille, J.-L.-M., “Recherches expérimenteles sur le mouvement des liquides dans les tubes de très petits diamètres(suite)”, Comptes rendus hebdomadaires des séances de l'Académie des sciences, 12 (1841), |
| [35] |
Privalova, V. V. and Prosviryakov, E. Yu., “Exact Solutions for Three-Dimensional Nonlinear Flows of a Viscous Incompressible Fluid”, AIP Conf. Proc., 2053:1 (2018), 040077, 5 pp. |
| [36] |
Privalova, V. V. and Prosviryakov, E. Yu., “Vortex Flows of a Viscous Incompressible Fluid at Constant Vertical Velocity under Perfect Slip Conditions”, Diagnostics, Resource and Mechanics of Materials and Structures, 2019, no. 2, |
| [37] |
Proskurin, A. V. and Sagalakov, A. M., “A New Branch of Instability of the Magnetohydrodynamic Poiseuille Flow in a Longitudinal Magnetic Field”, Tech. Phys. Lett., 34:3 (2008), ; Pis'ma Zh. Tekh. Fiz., 34:5 (2008), |
| [38] |
Proskurin, A. V. and Sagalakov, A. M., “Stability of the Poiseuille Flow in a Longitudinal Magnetic Field”, Tech. Phys., 57:5 (2012), ; Zh. Tekh. Fiz., 82:5 (2012), |
| [39] |
Proskurin, A. V. and Sagalakov, A. M., “The Numerical Investigation of the Stability of the Localized Perturbation in Poiseuille Flow”, J. Comput. Technolog., 18:3 (2013), |
| [40] |
Pukhnachev, V. V., “Symmetries in the Navier – Stokes Equations”, Uspekhi Mekh., 4:1 (2006), |
| [41] |
Regirer, S. A., “On the Movement of Fluid in a Tube with a Deforming Wall”, Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza, 3:4 (1968), |
| [42] |
Regirer, S. A., “Quasi-One-Dimensional Theory of Peristaltic Flows”, Fluid Dynam., 19:5 (1984), ; Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza, 1984, no. 5, |
| [43] |
Rykov, V. A., Titarev, V. A., and Shakhov, E. M., “Rarefied Poiseuille Flow in Elliptical and Rectangular Tubes”, Fluid Dynam., 46:3 (2011), ; Izv. Ross. Akad. Nauk. Mekh. Zidk. Gaza, 2011, no. 3, |
| [44] |
Shil'ko, S. V., Gavrilenko, S. L., Khizhenok, V. F., Stakan, I. N., and Salivonchik, S. P., “A Method for Defining Flow and Rheological Constants of Viscoplastic Biomaterials: P. 2”, Ross. Zh. Biomekh., 7:2 (2003), |
| [45] |
Sadovnichii, V. A., Dubrovskii, V. V., Kadchenko, S. I., and Kravchenko, V. F., “Computation of the First Eigenvalues of a Boundary Value Problem on the Hydrodynamic Stability of a Poiseuille Flow in a Circular Tube”, Differ. Equ., 34:1 (1998), ; Differ. Uravn., 34:1 (1998), |
| [46] | Skul'skii, O. I. and Aristov, S. N., Mechanics of Anomalously Viscous Fluids, R&C Dynamics, Institute of Computer Science, Izhevsk, 2003, 156 pp. (Russian) |
| [47] |
Takagi, D. and Balmforth, N. J., “Peristaltic Pumping of Rigid Objects in an Elastic Tube”, J. Fluid Mech., 672 (2011), |
| [48] |
Tverier, V. M. and Gladysheva, O. S., “A Biomechanical Model of the Mammary Gland”, Master's J., 2013, no. 2, |
| [49] |
Yin, F. and Fung, Y. C., “Peristaltic Waves in Circular Cylindrical Tubes”, ASME J. Appl. Mech., 36:3 (1969), |
| [50] |
Zikanov, O. Yu., “On the Instability of Pipe Poiseuille Flow”, Phys. Fluids, 8:11 (1996), |
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