An inhomogeneous Couette-type flow with a perfect slip condition at the lower boundary of an infinite fluid layer

In this paper, we investigate an exact solution to the three-dimensional problem of an isobaric flow of a viscous incompressible fluid layer. The solution is a linear function of the longitudinal coordinates. The solution type under study describes a vertical twist in the fluid, which arises due to the inclusion of inertial forces and inhomogeneous velocity distribution at the free boundary of the fluid layer. The solution allows to us describe the counterflow of an incompressible fluid in a thin layer. The exact solution is obtained for the perfect slip condition at one of the boundaries of the fluid layer. Conditions for the existence of points inside the fluid layer at which the velocity vanishes are defined. © 2019 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)

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)

Nonlinear gradient flow of a vertical vortex fluid in a thin layer

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 boundary-value 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. © 2019 Institute of Computer Science Izhevsk. All rights reserved.19-19-00571The work was supported by the Russian Scientific Foundation (project 19-19-00571)

An exact solution for the description of the gradient flow of a vortex fluid

An isothermal nonlinear gradient flow of a horizontal layer of a vertically vortex fluid is considered. The Navier-Stokes equation uses a solution describing the velocity and pressure fields. This solution is a linear function of the longitudinal (horizontal) coordinates with coefficients depending on the transverse (vertical) coordinate. For the obtained general exact solution, the boundary-value problem is solved with the no-slip conditions, nonzero tangential stresses, and constant pressure gradients set at the boundaries of the infinite fluid layer. It is shown that, for the considered boundary conditions, up to three stagnation points can arise in the fluid layer. The velocity or its components change their direction to the opposite at these stagnation points. © 2019 Author(s)

The elemental composition of Patrinia scabiosifolia

Aim. To study the mineral composition of the aerial organs of Patrinia scabiosifolia Fisch. ex Link. Materials and methods. Objects of research - leaves and flowers of Patrinia scabiosifolia Fisch. The multi-element analysis was performed on an 1CP-MS quadrupole Agilent 7500ce mass spectrometer using a micro-flow nebulizer and a sample introduction system. As external standards, a sample of Baikal water and multicell standard solutions 1CP-MS-68A-A and 1CP-MS-68A-B were used. The analysis was made of the results of quantitative content of mineral components of at least 100 pg/kg and a relative error of determination of not more than 5 %. Results. As a result of the study, the content of 8 macro- and 64 micro- and ultra-microelements was determined in the samples of leaves and flowers of Patrinia scabiosifolia. Of these, 13 elements are classified as essential and conditionally essential - Cu, Fe, 1, Co, C.R., M.N., Mo, S.E., Zn, A.L., B., V., Co, Si. The magnesium content in the test samples is close to the content of this element in Valeriana officinalis. Conclusion. The composition and content of the chemical elements in the flowers and leaves of Patrinia scabiosifolia, which grows on the territory of Eastern Siberia, has been studied. The similarity of the mineral composition of the samples studied and the row of Valeriana officinalis is similar