tailieunhanh - Convection in binary mixture with free surface

Convective motion in a binary mixture without free surface have been the subject of the works. In this paper the convetion in binary mixture with free surface is studied. The existence theorem is proved. | Journal of Mechanics, NCNST of Vietnam T. XVII, 1995, No 3 (1 - 4) CONVECTION IN BINARY MIXTURE WITH FRE:E SURFACE NGO HUY CAN and TRAN THU HA lMtitute of Mechanics - Hanoi and Institute of Applied Mechanics - Ho Chi Minh City Convective motion in a binary mixture without free surface have been the subject of works [1, 2]. In this paper the convetion in binary mixture with fre~ surface is studied. The existence theorem is proved. 1. BASIC EQUATIONS For mathematical description of small convective motion in a. binary mixture with free the following equations and conditions are assumed (see [1, 2, 3]): () () () v= o, divv =0 o, C= T = (i = 1, 2), ac = at - b2v3 () o on S av•) = ata (p- 2vp ax. on r () pgv., () () Tlt=O = T(O), Where the following notations are used: v = (u 1,v2,v3) denotes the velocity, p ~the pressure, T, C - the temperature and the concentration in the mixture, p - the equilibrium state density of the mixture, g ~ the acceleration of gravity, /31, f32 - the heat and concentration coefficient, X - the coefficient of heat conductivity, a, N - the thermodiffusion and thermodynamics parameters, 1 ~ the unit vector of vertical upward axis Ox3 ii:l the cartesian coordinate system Ox 1 x2x 3 , b1 , b2 ~ the gradients of temperature and concentration in the equilibrium state of the binary mixture. 2. EXISTENCE THEOREM The following Hilbert spaces are used throughout 1 L2(0) = H2(0) x H2(0) X H2(0) with the scalar product and norm L I u;v;dO, 3 (v, u)L,(O) = •=1 0 1/2 llviiL,(O) = { L2(0) = J(O) (v,v)L,(O) } + G(O) where J(O) = { u E L2(0), divu = O, u,. = 0 on S }, G(O) = { v E L2(0), v = Vp, p = 0 on r }; H2,oo(O) = { q E H2(0), q = 0 on Sur}, H,}(O) = { q E H 2(0), Vq E H2 (0) }. W,}(O) = H,}(O) x Hi(O) x Hi(O) The scalar product in W,}(O) is defined as follows '£1 I 3 (v,w)wf(o) = Vv.;Vw.;dO+ . · •= 1 o vwdS s Hj, 0(0) = { q E H 2(0}, Vq E H 2(0), q = 0 on .