Differential Equations, Vol. 41, No. 4, 2005, pp. 538–547. Translated from Differentsial’nye Uravneniya, Vol. 41, No. 4, 2005, pp. 508–517. Original Russian Text Copyright c  2005 by Andreev, Ryzhkov. PARTIAL DIFFERENTIAL EQUATIONS Symmetry Classification and Exact Solutions of the Thermal Diffusion Equations V. K. Andreev and I. I. Ryzhkov Institute of Computational Modelling SB RAS, Krasnoyarsk, Russia Received May 26, 2003 1. GOVERNING EQUATIONS We consider convective motion of a binary mixture assuming that its density linearly depends on the concentration of the lighter component and temperature:  =  0 (1 − β 1 T − β 2 C ). Here  0 is the mixture density at the mean values of temperature and concentration, T and C are the temperature and concentration variations, which are assumed to be small, β 1 is the thermal expansion coefficient of the mixture, and β 2 is the coefficient of density variation with concentration (β 2 > 0, since C is the concentration of the lighter component). The equations of convective motion in the Oberbeck–Boussinesq approximation have the form u t +(u ·∇)u = − −1 0 ∇p + ν Δu + g(β 1 T + β 2 C ), (1) T t + u ·∇T = χΔT, (2) C t + u ·∇C = dΔC + αdΔT, (3) ∇· u =0, (4) where u is the velocity vector, p is the difference between the actual and hydrostatic pressure, ν is the kinematic viscosity, χ is the thermal diffusivity, d is the diffusion coefficient, and α is the thermal diffusion parameter. We suppose that all characteristics of the medium are constant and correspond to the mean values of temperature and concentration. In (1), g = (0, 0, −g), where g is the gravitational acceleration (z-direction is taken vertically downwards). If normal thermal diffusion takes place, then α< 0 and the flux of the lighter component is directed towards the hot boundary, thus increasing the buoyancy force. In the case of abnormal thermal diffusion, α> 0 and the lighter component diffuses to the cold boundary, so that the buoyancy force decreases. The mechanical equilibrium can be obtained at some value of the coefficient α. The Lie symmetry algebra admitted by system (1)–(4) with g = 0 and α = 0 was found in [1]. In the present paper, the problem of symmetry classification for these equations with respect to the constants α, β 1 ,β 2 ,χ, and d is considered. We suppose that g = 0, while α, β 1 , or β 2 can be zero. (In this case, the corresponding terms in the equations are omitted.) Note that the cases g = 0 and β 1 = β 2 = 0 are equivalent from the symmetry analysis point of view. 2. SOLUTION OF THE DETERMINING EQUATIONS We introduce the following notation: x =(x 1 ,x 2 ,x 3 ) is a point of R 3 , and u =(u 1 ,u 2 ,u 3 ) is the velocity vector. Let h(t),f (t, x) be some functions. Then their derivatives are denoted as follows: dh dt = h ′ , d 2 h dt 2 = h ′′ , ∂f ∂t = f t , ∂f ∂x i = f i , ∂ 2 f ∂t∂x i = f ti , ∂ 2 f ∂x i ∂x j = f ij (i, j =1, 2, 3, i ≤ j ). Using this notation, we can rewrite system (1)–(4) in coordinate form: u i t + u 1 u i 1 + u 2 u i 2 + u 3 u i 3 + p i / 0 − ν (u i 11 + u i 22 + u i 33 )=0, i =1, 2, (5) 0012-2661/05/4104-0538 c  2005 Pleiades Publishing, Inc.