The properties of isobars in shear concentration convective flows of a viscous incompressible fluid in a plane horizontal layer with motionless boundaries

Abstract

It is known that convection contributes to the natural mixing of melts due to the uneven distribution of their density. This mixing can be enhanced by various devices and by changing the conditions surrounding the flow region of the medium, e.g. by changing the pressure at the boundaries of the flow region of such fluid media. A pressure difference inside the region filled with a viscous fluid can induce its flow. A classic example of such a flow is the Poiseuille flow. Moreover, if the pressure distribution depends also on horizontal coordinates (i.e., there is a pressure difference not only in the vertical direction), then longitudinal pressure gradients also generate additional flows, which are superimposed on the classical flow. The paper studies the effect of pressure on the shear convective flow of a viscous incompressible binary fluid in a horizontal layer. To describe such flows, we use a system of equations of concentration convection, which includes the equation of motion of a viscous binary fluid, the equation of a concentration change, and the incompressibility equation. The solution of the system of constitutive equations is sought with the use of the class of generalized solutions, in which the velocities depend only on the vertical coordinate, the pressure and concentration being also linearly dependent on the longitudinal (horizontal) coordinates. As the boundary conditions, it is assumed that the no-slip condition is met at the lower impermeable boundary, that the upper boundary of the layer is motionless, and that the distribution of salinity and pressure is specified on it. The solution of the formulated boundary value problem is a set of polynomial functions. The highest-degree polynomial describes background pressure. The study of the properties of background pressure and the longitudinal pressure gradients is in the focus of attention. It is shown that background pressure decreases strictly monotonically with moving away from the lower boundary of the layer, regardless of the control parameters of the boundary value problem. In this case, the longitudinal pressure gradients are also described by strictly monotonic functions, but the nature of the monotonicity is determined by the values of the longitudinal concentration gradients specified at the upper boundary of the fluid layer. The relevant findings are illustrated.