Exact Solutions for Stationary and Unsteady Layered Convection of a Viscous Incompressible Fluid with the Specified Velocities at the Bottom

The layered convective flow of a viscous incompressible fluid is considered with the specified velocities at the bottom of an infinite layer. A new exact stationary and nonstationary solution of the Oberbeck-Boussinesq system is presented. The account of fluid velocity at the bottom is characterized by the presence of two stagnant points, this being indicative of the nonmonotonic kinetic energy profile with two local extrema.

Substituting the hydrodynamic fields (2), we obtain a loosely coupled system of equations. This system consists of heat-type evolution equations and stationary gradient relationships, System ( We formulate the initial-boundary value problem for finding a particular solution of system (3). Consider that, at the initial time 0 t = , the velocities and the reference temperature are zero, At the lower fluid layer boundary described by the plane equation 0 z = , the following boundary conditions are specified for 0 t ≥ : The boundary conditions (5) determine the perturbation of the velocity field at the lower boundary. This condition can be interpreted as the non-satisfaction of the no-slip conditions, as well as the specification of the law of motion of the lower boundary in experimental hydromechanical studies. On the free boundary ( z h = ), the boundary conditions of the form found in [8,10] are valid, Here, σ is the temperature surface tension coefficient, η is the dynamic (shear) viscosity coefficient.
The curvature of the free boundary is negligible when large-scale processes are considered [8,10].

Steady-state layered thermocapillary convection
To solve the initial-boundary value problems of layered thermocapillary convection, we reduce equation (3) and conditions (4)- (6) to the non-dimensional form. We select h , l , System (3) describing steady convective flows of a viscous incompressible fluid has the following nondimensional form: The boundary conditions (5) and (6) The boundary value problem (7)-(9) in the dimensionless form has the exact stationary solution Re cos 24 4 3 We Gr Re sin 24 6 4 6 Gr Winter The study of the temperature and pressure gradients is trivial; therefore, we will analyze the velocity field. The velocity x V assumes one zero value (figure 1) when the inequality ( ) ( ) is satisfied.
For the velocity y V , the condition for the existence of a critical value has the form Re 1 0 sin Gr 24 Counterflows at Re 0 = (no slip at the bottom) were studied for thermal and concentration convection in [1,2,4,16]. When the no-slip condition at the bottom is satisfied, the stagnant point is formed only with respect to the velocity y V . The velocity x V is of a constant sign. If there is a slip on the bottom of the fluid layer, the velocity x V can have one stagnant point or two stagnant points (see figures 1, 2).

Nonstationary layered thermocapillary convection
On a specified time interval, the parabolic equations of system (3) are sequentially solved in time steps with the corresponding boundary conditions (4) and (5) by the boundary element method, similarly to the way it was done in [14].

Conclusion
The convective flow of a viscous incompressible fluid has been studied. The velocity is set at the lower boundary of the fluid layer. An exact solution of the overdetermined Oberbeck-Boussinesq equation system has been obtained, and this solution can be used with any dimensionless Reynolds and Grashof numbers. This exact solution describes incompressible fluid flows with a nonmonotonic kinetic energy profile. In this case, the velocity field can take a zero value at two points.