| In analogy to 3 types of the heat diffusion, super- and sub-diffusion besides normal diffusion, as discussed by Zeldovich and Kompaneets, 3 types of different pressure distribution in a porous medium may appear. The combination of external forces with surface tension (which is caused by cohesion within the liquid) and adhesive forces between the liquid and container is essential for flow of liquid. One of its manifestations is Laplace' pressure, it is the pressure difference between the inside and the outside of a bubble or droplet. If the intermolecular forces within the liquid are weaker than those between the solid wall of pore canal and the liquid, the wetting phenomenon hinders the flow through porous medium. If the intermolecular forces exceed those between the solid wall and the liquid, the wetting facilitates the fluid flow in the capillary canal, and adhesion forces between the fluid and the solid inner wall pulls the liquid along the wall. Hence, in analogy to heat diffusion two opposite cases can be considered, the sub-seepage in which the fluid distribution is strictly limited and super-seepage which is developed with far reaching tails. Consider a liquid in isothermal conditions, with the density $\rho$ in which the pressure $p$ prevails. Let the liquid be described by a linear state equation \begin \rho = \rho_0 + \alpha p \end where $\rho_0$ and $\alpha$ are the constants. Let the equation describing the flow of the liquid be of Darcy's type \begin {\bf j} = - K \nabla p \end where $K$ is the permeability of porous medium. The continuity relation links temporal variation of the density of particles $f(x,t)$ with the spatial variation of the particle stream $j$ \begin %5 \frac {\partial \rho}{\partial t} = - \frac {\partial j_i(x, t)}{\partial x_i} \end If this is combined with Darcy's phenomenological law of seepage, the pressure equation is obtained. A sub-seepage occurs, when the permeability coefficient takes the form \begin K=a(t) p^n \end where $ n $ is a positive constant, and $p=p(x,t)$ is a solution to the nonlinear equation of pressure distribution (4). A super-seepage occurs, when \begin K=a(t)(1/p)^n \end where $ n $ is a positive constant. Such processes can be called the Zeldovich--Kompaneets nonlinear seepages, in analogy to a nonlinear heat flow analysed by those authors in 1950. Wigner semicircle distribution of pressure is an example of sub-seepage with the coefficient $K = a p^2$, and $p(x,t) = \frac{\pi x_0^2} \sqrt{x_0^2 - x^2}$. The value of $x_0$ is varying with time $t$, as $x_0^2 = (4/\pi) A^{1/2}$, $A = A(t) = \int_0^t a(\tau) d \tau$. References \begin \bibitem %1 Y. B. Zeldovich and A. S. Kompaneets, The theory of heat propagation in the case where conductivity depends on temperature, in: {\it Sbornik posviaschenny 70-letiyu akademika A. F. Ioffe}, {\it Collection of papers celebrating the seventieth birthday of Academician A. F. Ioffe}, ed. P. I. Lukirskii, Izdat. Akad. Nauk SSSR, Moskva 1950, pp. 61-71 (in Russian). \bibitem %2 L. D. Landau and E. M. Lifshitz, {\it Fluid mechanics}, Pergamon Press, Oxford 1959. \end |
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