Non-helical exact solutions to the Euler equations for swirling axisymmetric fluid flows

Swirling axisymmetric stationary flows of an ideal incompressible fluid are considered within the framework of the Euler equations. A number of new exact solutions to the Euler equations are presented, where, as distinct from the known Gromeka-Beltrami solutions, vorticity is noncollinear with velocity. One of the obtained solutions corresponds to the flow inside a closed volume, with the nonpermeability condition fulfilled at its boundary, the vector lines of vorticity being coiled on revolution surfaces homeomorphic to a torus. © 2019 Samara State Technical University. All rights reserved.Funding. This work was partially supported by the Complex Program of UB RAS, project no. 18–1–1–5

Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation

This article discusses the solvability of an overdetermined system of heat convection equations in the Boussinesq approximation. The Oberbeck-Boussinesq system of equations, supplemented by an incompressibility equation, is overdetermined. The number of equations exceeds the number of unknown functions, since non-uniform layered flows of a viscous incompressible fluid are studied (one of the components of the velocity vector is identically zero). The solvability of the non-linear system of Oberbeck-Boussinesq equations is investigated. The solvability of the overdetermined system of non-linear Oberbeck-Boussinesq equations in partial derivatives is studied by constructing several particular exact solutions. A new class of exact solutions for describing three-dimensional non-linear layered flows of a vertical swirling viscous incompressible fluid is presented. The vertical component of vorticity in a non-rotating fluid is generated by a non-uniform velocity field at the lower boundary of an infinite horizontal fluid layer. Convection in a viscous incompressible fluid is induced by linear heat sources. The main attention is paid to the study of the properties of the flow velocity field. The dependence of the structure of this field on the magnitude of vertical twist is investigated. It is shown that, with nonzero vertical twist, one of the components of the velocity vector allows stratification into five zones through the thickness of the layer under study (four stagnant points). The analysis of the velocity field has shown that the kinetic energy of the fluid can twice take the zero value through the layer thickness. © 2019 Samara State Technical University. All rights reserved.12281GU/2017Competing interests. We declare that we have no conflicts of interest in the authorship or publication of this contribution. Authors’ contributions and responsibilities. We are fully responsible for submitting the final manuscript in print. Each of us has approved the final version of the manuscript. Funding. This work was supported by the Foundation for Assistance to Small Innovative Enterprises in Science and Technology (the UMNIK program, agreement 12281GU/2017)

Investigation of a velocity field for the Marangoni shear convection of a vertically swirling viscous incompressible fluid

The article considers a new exact solution for describing large-scale flows of a vertically swirling fluid initiated by thermocapillary forces acting on a free surface. The behavior of the velocity field is analyzed. It is shown that the topology of this field depends on the values of the given parameters. It is also shown that the components of the velocity field can have several stagnant points, as a result of which the specific kinetic energy has a substantially nonmonotonic behavior. © 2018 Author(s)

Exact solution of the convective flow of a viscous fluid layer with a heated lower boundary

A new exact solution of the layered convection problem is considered. The obtained solution describes the flow layer of a viscous incompressible fluid with nonzero gradients of temperature and pressure. The horizontal velocity components depend only on the vertical transverse coordinate of the fluid layer. At the lower layer boundary, nonzero temperature gradients and the Navier slip condition are specified, tangential stresses and longitudinal pressure gradients being specified at the upper boundary. The possibility of the occurrence of counterflow areas and the corresponding changes in the tangential stresses and the vorticity vector are shown for the obtained particular exact solution. © 2019 Author(s)

Complex large-scale convection of a viscous incompressible fluid with heat exchange according to Newton's law

The paper considers the construction of analytical solutions to the Oberbeck-Boussinesq system. This system describes layered Bénard-Marangoni convective flows of an incompressible viscous fluid. The third-kind boundary condition, i. e. Newton's heat transfer law, is used on the boundaries of a fluid layer. The obtained solution is analyzed. It is demonstrated that there is a fluid layer thickness with tangential stresses vanishing simultaneously, this being equivalent to the existence of tensile and compressive stresses. © 2017 Author(s)

Exact solution for the layered convection of a viscous incompressible fluid at specified temperature gradients and tangential forces on the free boundary

A new exact analytical solution of a system of thermal convection equations in the Boussinesq approximation describing layered flows in an incompressible viscous fluid is obtained. A fluid flow in an infinite layer is considered. Convection in the fluid is induced by tangential stresses specified on the upper non-deformable boundary. At the fixed lower boundary, the no-slip condition is satisfied. Temperature corrections are given on the both boundaries of the fluid layer. The possibility of physical field stratification is investigated. © 2017 Author(s).This work was supported by the Foundation for Assistance to Technology (the UMNIK program, agreement 12281GU/2017)

Unidirectional thermocapillary flows of a viscous incompressible fluid with the Navier boundary condition

A new exact solution for Marangoni shear convection in a flat horizontal layer, induced by taking into account the Navier condition at the lower boundary, is obtained. The study of the behavior of hydrodynamic fields has shown that they have a complex topology and allow the appearance of several zones with reverse flow. © 2019 Author(s)

Unidirectional Marangoni-Poiseuille flows of a viscous incompressible fluid with the Navier boundary condition

The paper obtains an exact solution describing the shear convection of a swirling viscous incompressible fluid in a horizontal layer taking into account the Marangoni condition, the Navier condition and the non-uniform pressure distribution on one of the layer boundaries. It is shown that this exact solution is able to describe the appearance of the stratification of a velocity field and thermal force fields. © 2019 Author(s)

Layered convective flows of vertically swirling incompressible fluid affected by tangential stresses

Using the construction of several particular exact solutions, the article investigates the overdetermined system of equations of thermal convection of a vertically swirling viscous fluid in the Boussinesq approximation. The vertical component of vorticity in a non-rotating fluid is generated by an inhomogeneous velocity field at the lower boundary of the infinite horizontal fluid layer. The main causes of convection in the problem under consideration are linear heat sources and the field of shear stresses. The main attention is paid to the study of the properties of the flow velocity field and the dependence of the structure of this field on the magnitude of the vertical swirl. © 2019 Author(s)

Convective Couette-type flows under condition of slip and heating at the lower boundary

A Couette type boundary value problem is considered for a new exact solution of the layered convection problem. The obtained solution describes the flow of a viscous incompressible fluid layer with nonzero temperature and pressure gradients along the longitudinal (horizontal) coordinates. The horizontal velocity components depend only on the vertical (transverse) coordinate of the fluid layer. The Navier slip condition and nonzero temperature gradients are specified on the lower absolutely solid boundary of the layer. The tangential stresses and constant (atmospheric) pressure are specified at the upper boundary. The possibility the occurrence of countercurrent regions and the corresponding changes in the tangential stresses and the vorticity vector are shown for the obtained particular exact solution. © 2019 Author(s)