Konstantin Shvarts

In this paper a new exact solution of the Navier – Stokes equations in the Boussinesq approximation describing advective flow in a horizontal liquid layer with free boundaries, where the vertical velocity component is a constant value, is obtained. The temperature is linear along the boundaries of the layer. Solutions of this kind are used to close three-dimensional equations averaged across the layer in the derivation of two-dimensional models of nonisothermal large-scale flows in a thin layer of liquid or incompressible gas. The properties of advective flow at different values of Reynolds number and Prandtl number are investigated.

This paper presents a derivation of new exact solutions to the Navier – Stokes equations in Boussinesq approximation describing two advective flows in a rotating thin horizontal fluid layer with no-slip or free boundaries in a vibrational field. The layer rotates at a constant angular velocity; the axis of rotation is aligned with the vertical axis of coordinates. The temperature is linear along the boundaries of the layer. The case of longitudinal vibration is considered. The resulting solutions are similar to those describing the advective flows in a rotating fluid layer with solid or free boundaries without vibration. In both cases, the velocity profile is antisymmetric. Thus, in particular, in the absence of rotation, the longitudinal vibration in the presence of advection can be considered as a kind of “one-dimensional” rotation. The presence of rotation initiates the vortex motion of the fluid in the layer. Longitudinal vibration has a stronger effect on the xth component of the velocity than on the yth component. At large values of the Taylor number and (or) the vibration analogue of the Rayleigh number thin boundary layers of velocity, temperature and amplitude of the pulsating velocity component arise, the thickness of which is proportional to the root of the fourth degree from the sum of these numbers.