Bénard-Marangoni convection with a deformable surface
The author studies the system of equations \[ (u_t+ u\nabla u)/\text{Pr}+ \nabla p= \Delta u-\rho(T)\nabla y,\quad \nabla u= 0,\quad T_t+ u\nabla T= \Delta T, \] which describes the thermal convection of an incompressible fluid. Here \(T\) is the temperature, \(u= (u_1,u_2)\) -- the fluid velocity, Pr -- the Prandtl number, and \(\rho\) -- the fluid density considered as linearly dependent on \(T\). The two-dimensional motion is considered in the infinite strip \[ \Omega(t)= \{(x,y): -\infty< x<\infty,\;-1< y<\eta(x,t)\}, \] where \(\eta(x,t)\) is the unknown free surface, variable in time. On \(\partial\Omega(t)\), the author imposes natural boundary conditions: no-slip on the solid boundary, and stress-free motion on the free boundary. The above system is written in the weak form, generalized to the \(n\)-dimensional space, and is studied with functional-analytical methods in the linearized and nonlinearized form. In both these cases, the author proves the existence and uniqueness of solutions, and establishes some solution properties.
scientific article; zbMATH DE number 1251224