tailieunhanh - The bifurcation theorem on the problem of thermal convection and contaminant transport in underground water

In the paper the bifurcation theorem on the problem of thermal convection and contaminant transport in underground water is proven. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. 22, 2000, No 2 (65 - 70) THE BIFURCATION THEOREM ON THE PROBLEM OF THERMAL CONVECTION AND CONTAMINANT TRANSPORT IN UNDERGROUND WATER NGO HUY CAN * , TRAN THU HA ** * Institute of Mechanics, NCST of Vietnam * * Institute of Applied Mechanics, NCST of Vietnam ABSTRACT. In the paper the bifurcation theorem on the problem of thermal conve,ction and contaminant transport in underground water is proven. 1. Formulation of the problem Equations of the problem on thermal convection and contaminant transport in underground water assume the dimensionless form [1 ] -a\lp + RTT"f - RcC"f v= aT () at + v · "VT = , ac c- ot 1 +v· \JC = - divv = 0, () Le () ' (} inn, with the boundary conditions: () and initial conditions To = T(xo, Yo, zo), Co= C(xo, Yo, zo) +, where the following notations are used: temperature, C - concentration a = v Hv () denotes the velocity, p - pressure, T k - coefficient of permeability, H - a length scale, v - coefficient of kinematical viscosity, c . at t =to, = p_, (}' cf> - porosity, u - heat capacity of porous media, "{ - unit vector of the vertical upward axis Ox3 in the 65 Cartesian coordinate system Ox1x2 xa, Le - .Lewis number, RT - thermal Rayleigh number, Re - concentration Rayleigh number. In [2] it is proved that there exists a mechanical equilibrium in the fluid: v=O, () Co= -Ag'xa +Bf, To = -A6 xa + Bf , Ag, IJf , A6, B'{ the constants . In [2] the existence theorem and the spectrum theorem of the linear problem are proved. In t his paper we prove the bifurcation theorem of the problem v = -a"Vp- RcC1 + RTT1, v · VT - v · 'Y = . . DT, 1 v · "VC-v·;;v= -DC ' T = = 0, Vn () () Le ' o, c = 0 () on n. () 2. The theorem From () - (), () we can obtain v = -a"Vp .!. 8T at - v . 'Y = ac eat RcC'Y + RTT1, () DT, () 1 () -v·'Y= Le4c, divv .