Diffusion Poiseuille flow of a viscous incompressible binary fluid in a horizontal layer with motionless boundaries

摘要

A layered steady-state convective flow of a viscous incompressible fluid in an infinite horizontal layer induced by an inhomogeneous pressure distribution at one of the layer boundaries and by the presence of an impurity (salinity) in the fluid is considered. In addition to the equation of motion of a viscous fluid and to the law of conservation of mass for an incompressible fluid, the determining system of relations also includes an equation describing the distribution of the volume fraction of the impurity (salinity) over the entire region of the flow of the fluid under consideration. The solution of the determining system of equations is sought with the use of the class of generalized solutions, in which the velocities depend only on the vertical (transverse) coordinate, and the impurity concentration and pressure are linearly distributed along the horizontal (longitudinal) coordinates. A general solution for the determining system of equations within the chosen class is presented, and the corresponding number of boundary conditions necessary to find the values of the integration constants that appear in this general solution is formulated. A complete solution for the boundary value problem is also given. The features of the velocity field, the concentration field, and the pressure field are analyzed. The dependences of the properties of these fields on the values of parameters determining the distribution of the pressure field and the concentration field at the upper boundary of the layer are studied. It is shown that the constructed exact solution is able to describe multiple stratifications of the above-mentioned hydrodynamic. All the results obtained during the study are illustrated.