1. Mathematica l Model 1.1. Filtratio n flow of an incompressible fluid. Consider a planar flo w of a homogeneous flui d throug h a homogeneous porou s medium. Suc h a flow i s modeled b y a time-dependent vecto r field in the plane, v = (v { , v 2 ), whic h is called the fluid velocity . Suppose that the fluid is incompressible. Thi s property is expressed by the differential equatio n dv* dv~ (u) d i v v = ^ + ^ = °- The main la w of filtration i s Darcy's law. 1 I t states that th e fluid velocit y is proportional t o the pressure gradient : (1.2) v = -/cgrad/? . Here p i s the pressure , K 0 i s a proportionality coefficient . I t i s known that K i s inverse proportional to the dynamical fluid viscosity ju: K = k//i. The coefficient k depend s solel y on the properties of the porous medium . Thus, the fluid velocity is a potential vector field: (1.3) v = gradO, & =-Kp, and its potential O i s a harmonic function : (1.4) AJ=— T +—5 - = 0. dx2 dy 2 Introduce the complex coordinate z = x + iy . Let the region of flow contain sources and sinks. Le t their coordinates and rates b e z x , ... , z n an d q x , ... , q n respectively . Thi s mean s tha t nea r th e point z . (1.5) v(z ) = J _ + smooth vector-function , 2n{z z •) or, equivalently , q. (1.6) O(z ) = 7^-log| z - z.\ - h smooth function . 2n J This la w wa s discovere d experimentall y i n 185 6 b y A . Darcy , a Frenc h engineer , whe n designing a system of public fountains . 1 http://dx.doi.org/10.1090/ulect/003/01
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