tailieunhanh - A mixed problem of active aerosol pollution

The mixed problem of active aerosol pollution is stated in the mathematical modeling of environment problems [1] and was studied by some authors (see [1], [2], [3], . ). In this work we give the solution using the theory of Schwart distributions. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. 22, 2000, No 2 (87 - 92) A MIXED· PROBLEM OF ACTIVE AEROSOL POLLUTION HOANG DINH DUNG Institute of Mathematics, NCST of Vietnam ABSTRACT. The mixed problem of active aerosol pollution is stated in the mathematical modeling of environment problems [1] and was studied by some authors (see [1], [2], [3] , . ). In this work we give the solution using the theory of Schwart distributions . 1. Introduction Let G be a cylindrical region in the space R 3 of the point x = (x 1 ,x 2 ,x 3 ) with sufficiently smooth boundary BG= 8G 0 u BGc U BGh , where BGo , BGc and BGh being t he bedding (x 3 = 0), lateral and upper (x 3 = h) surfaces, respectively. Assume that the stack of a plant emits a mixture of pollutants at the point source x 0 = (x~,x~,h 0 ) E G. Denote by cpj , i = 1,N, the concentrations of aerosols. Let these pollutants be converted from one form to another, that is, we may have the following chains: 'Pi / = 'Pj1 --+ 'Pj2 --+ 'Pj3 --+ . , i = 1, N. ' For simplicity of presentation, let us consider the conversion process for only a pollutant corres~onding to the concentration cp 1 : 'Pl --+ 'P2 --+ · • · --+ 'Pm the m-th pollutant is n ot converted to another form. Then, by [1], 'Pi, i satisfy the following equations and conditions: + a1cp1 = f, Acp2 + a2cp 2 - 8-1 'P1 () = 1, m , Acp1 = 0, + am-1'Pm-1 - D-m-2'Pm-2 = 0, Acpm + Um - l 'Pm - 1 = 0, Acpm-1 87 () cpi(x, t) = cp?(x) fort= 0, x E G, i = 1, m, anc cpi = cpi on acp · if Vn -a = 0 on anh i X3 acp· -ai X3 + (Jcpi where A a = -at + V = V · n 3) in t he chain (1) and that the components 4, 5, . are not harmful and may be neglected. Thus, we consider the system () in the following form: · Bcp1 at + ocp1 U acp1 ax + V ay +W acp1 az - µ (a 2 cp1 ax 2 2 + a2 cp1) 8y 2 a ( acp1) - az LI az . + Ui 'P l = f , 2 .a'P2 acp2 +v acp2 acp2 -µ (a a cp2) - -a. ( v ocp2) ·- - +u-- + w--'P2 + .