| Filtration is a kind of common phenomenon in nature, which indicates the movement of liquid in porous media. For example, the water flowing among the soil is a kind of filtration. The research of nitration is very important to the exploitation of ground water resources and the discovery of petroleum or gas, especially to the agriculture. At the same time, when we investigate the problem about the saline-alkali soil and melioration, the using fertilizer intelligently, the industrial waste water disposal and the protection of the ground water resource, which are involved the solute movement and the heat transfer, we must consider the dynamics of the solute in the fiitration and the heat transportation.The experimental research of filtration phenomenon originated from the famous experiment of H. Dary's [1] in 1956. In many years after that, a lot of mathematical models were established, and researches on numerical computation and the theoretical qualitative analysis have been achieved a great deal progress. In this paper,we are interested in a flow with a homogeneous, isotropic and rigid porous medium filled with a fluid. Firstly, by the continuity equa-tion, we have39+ divu = 0, (1)otwhere v denotes the macroscopic velocity of the fluid, 9 the volumetric moistrue content. The Darcy's law yields (2)where k(9] denotes the hydraulic conductivity and the total potential. If we ignore the absorption and chemical, osmotic and thermal effects could be expressed as (3)where the first term is the hydrostatic potential due to capillary suction and z the gravitational potential. Here z is a variable which direction accords with gravitation.Combining (1), (2), (3), we obtainFor many medium, could be a function of 9, i.e.\I = {9}. Then we have the following equation of the forman. (5)otAnd the experimentation yields that the hydraulic conductivityis not negative, i.e. A(s] is a non-decreasing function. This is a kindof typical filtration equation.On the other hand, if 9 depends on , i.e. 0 = ), the equation (4) is induced to.In one dimensional case, we could get the following equation by some proper transformdC(u) _d_ dB(u] If the effect of gravitation is ignored (e.g. the direction of x is horizontal), the equation (6) has the form ofdC(u)dt dxrwhere C(u) is in general a non-decreasing function. This equation is applied to the research of filtration with saturated and unsaturated region.For the equations (5), (6). (7) of above, we are interested in the degenerate case. Generally, equation (5) is the typical parabolic-4hyperbolic mixed equation, which degenerates when A'(s] = 0. While equations (6) and (7) are elliptic-parabolic mixed types, which degenerates when C'(s] = 0.The theory on the solutions of the degenerate equations (5) could ascend to 1958. In this year. Oleinik, Kalashinkov and Zhou Yulin [2] studied the Cauchy problem of the equations with thefollowing formdu d(f)(x,t,u} ' Where they required (, x,u) is defined for u 0 and with thefollowing properties> 0, 0u(t,x-,u) > 0, when u > 0, ,0 = 0.Due to the degeneracy, classical solutions may not exist. They put forward the definitions of the generalized solutions of the first boundary value condition and the second boundary value condition. Using the method of parabolic regularitzation, they proved the existence of generalized solutions. And they also proved the uniqueness of solutions and obtained the conditions for solutions to have the properties of finite propagation of disturbances.After that. Gilding and Peletier [3] considered the Cauchy problem for the equationdu dt dx2 dx and proved that it admits at most one generatlized solution whenever n |(m + 1), and it admits a generalized solution if isnongenative, bounded and continuous with u[" lying Lipschitz continuous. Soon...
|
PDF Full Text Request