A two-dimensional problem of the fluid flows with a dynamic contact angle is studied in the case of an uniformly moving contact point. Mathematical modeling of the flows is carried out with the help of the Oberbeck-Boussinesq approximation of the Navier-Stokes equations. On the thermocapillary free boundary the kinematic, dynamic conditions and the heat exchange condition of third order are fulfilled. The slip conditions (conditions of proportionality of the tangential stresses to the difference of the tangential velocities of liquid and wall) are prescribed on the solid boundaries of the channel supporting by constant temperature. The dependence of the dynamic contact angle on the contact point velocity is investigated DOI: 10.17516/1997DOI: 10.17516/ -1397DOI: 10.17516/ -2016.
IntroductionThe problems of flows of a viscous incompressible fluid in the domains with interfaces are very important for investigations. The features of the fluid flows in the domains with free boundaries and interfaces are the subject of many investigations in the last decade. Such interest is explained by need of study some phenomena in the flow structure, which arise due to the effects related with the gas phase and solid wall properties. One of the most important questions of mathematical modeling of the convective fluid flows in a domain with an interface is a correct formulation of the boundary conditions. From the mathematical point of view the non-stationary fluid flows with free boundaries remain to be very difficult for investigations because of the dynamic contact angle problem [1][2][3][4][5][6]. The problem of dynamic contact angle occurs due to the incompatibility of the conditions on the free surface of the liquid and the conditions of adhesion on a solid surface * gon@math.asu.ru c ⃝ Siberian Federal University. All rights reserved -296 - in vicinity of the moving three-phase contact line. There are various methods of statement of the problems with contact angles, describing a motion of a viscous incompressible liquid in the presence of a moving contact line (or contact point in the two-dimensional case). These problem statements assume a replacement of the no-slip conditions terms by the slip conditions on some sections of the solid walls near the contact line, the asymptotic approach, the assumption of the contact angle equality to π or to "zero" etc. (see, for example, [1,2,7,8]). For some mathematical models of fluid flows with dynamic contact angle the correctness of the initial boundary-value problems has been proved [1,2,7,[9][10][11].The problem of the fluid flows in a two dimensional domain will be studied in the case when the contact points are moving with a constant velocity. A behavior of the contact angle depends on the velocity of movement of the contact point, on the nature of the boundary thermal conditions specified on solid walls and free thermocapillary boundary, on the values of the friction coefficients and also on intensity of the gravitation field. We consider the Oberbeck-Boussinesq equations of c...