caa a22 4500 463162935 CHVBK 20180406164753.0 cr unu---uuuuu 170326e20070901xx s 000 0 eng 10.1007/s11242-006-9072-5 doi (NATIONALLICENCE)springer-10.1007/s11242-006-9072-5 Sieniutycz Stanisław Faculty of Chemical Engineering, Warsaw University of Technology, 1 Waryńskiego Street, 00-645, Warsaw, Poland aut Frictional passage of fluid through inhomogeneous porous system: a variational principle [Elektronische Daten] [Stanisław Sieniutycz] A Fermat-like principle of minimum time is formulated for nonlinear steady paths of fluid flow in inhomogeneous isotropic porous media where fluid streamlines are curved by a location dependent hydraulic conductivity. The principle describes an optimal nature of nonlinear paths in steady Darcy's flows of fluids. An expression for the total path resistance leads to a basic analytical formula for an optimal shape of a steady trajectory. In the physical space an optimal curved path ensures the maximum flux or shortest transition time of the fluid through the porous medium. A sort of "law of bending” holds for the frictional fluid flux in Lagrange coordinates. This law shows that—by minimizing the total resistance—a ray spanned between two given points takes the shape assuring that a relatively large part of it resides in the region of lower flow resistance (a ‘rarer' region of the medium). Springer Science+Business Media, Inc., 2007 Variational principles nationallicence Frictional flows nationallicence Inhomogeneous media nationallicence Darcy's law nationallicence A : Variable area perpendicular to fluid flow nationallicence A 0 : Constant transfer area projected on axis y nationallicence c : Bending constant for a frictional ray nationallicence g : Gravity accelleration nationallicence H : Hamiltonian function nationallicence h : Hydraulic head $$P(\rho_{\rm m}\hbox{g})^{-1}+z$$ nationallicence I : Total fluid flux passing through the system nationallicence k : Hydraulic conductivity, related to gradient of hydraulic head nationallicence L : Vertical distance in Fig. 1 nationallicence l : Length coordinate nationallicence n : Refraction coefficient nationallicence p : Momentum type integral or ∂R/∂y nationallicence Q : Mass of transferred fluid nationallicence R 1,2 : Total or cumulative resistance for frictional path between points 1 and 2 nationallicence R ( x , y ) : Minimum resistance potential nationallicence t : Physical time nationallicence u : Controlled direction dy/dx nationallicence x : Coordinate perpendicular to the resistance gradient nationallicence y : Coordinate tangent to the resistance gradient nationallicence z : Vertical coordinate nationallicence α : Angle nationallicence β : Resistance ratio, ρ 2/ρ 1 nationallicence γ : Coefficient of exponential change nationallicence Δ : Deviation angle nationallicence δ : First variation nationallicence β : Resistance ratio nationallicence ρ : Specific frictional resistance or reciprocal of conductivity k nationallicence ρ m : Mass density of the fluid nationallicence Transport in Porous Media Kluwer Academic Publishers 69/2(2007-09-01), 239-257 0169-3913 69:2<239 2007 69 11242 https://doi.org/10.1007/s11242-006-9072-5 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11242-006-9072-5 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Sieniutycz Stanisław Faculty of Chemical Engineering, Warsaw University of Technology, 1 Waryńskiego Street, 00-645, Warsaw, Poland aut NATIONALLICENCE 773 0- Transport in Porous Media Kluwer Academic Publishers 69/2(2007-09-01), 239-257 0169-3913 69:2<239 2007 69 11242 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer