Đang chuẩn bị liên kết để tải về tài liệu:
Sabatier Agrawal Machado Advances in Fractional Calculus Episode 6

Tham khảo tài liệu 'sabatier agrawal machado advances in fractional calculus episode 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | SOLUTE SPREADING IN HETEROGENEOUS POROUS 187 Fig. 1. The porous medium on the small scale left and an elementary volume of porous medium containing many spherical aggregates right . A similar situation is observed in the cooling shell of an atomic reactor when the lead melt contains grains of iron or different iron compounds and atoms of some other substance 15 . The model also works for particles of solute dissolved in a fluid at rest filling the voids of a porous matrix made of aggregates whose largest diameter is less than a0. The volume of grains dVgr in the elementary volume dV is dVr dV K x where K x is the volume of grains per unit volume near point x. Then a grain-free volume element dV near volume dV satisfies dV dV - dVgr dV 1 - K x s x dV. 1 Here - x dV 1 - K x is the porosity of the medium at point x. A grain-free area of the surface element dS of the elementary volume dV satisfies dS s x dS . Indeed let us consider a parallelepiped whose basis perpendicular to OZ has area dS while the height is l Fig. 1 . Due to Eq. 2 the volume filled with grains is dVgr K x l dS. Consider a plane perpendicular to OZ and intersecting the elementary parallelepiped at level z . Let dSgr z be the area common to the plane and to the grains. The volume of the grains cutting the plane is dVgr dSgr z dz and the grain-free area in the plane is dS z dS - dSgr z . We focus on scales greater than the largest grain size a0 i.e. l a0. Nevertheless only scale-averaged area values make sense. If we average dS z over 0 l then we get dS 1 d z dz 1 dS - dSgr dz dS - dV- 2 which implies 188 Logvinova and Néel dS dS - K x dS s x dS. 3 2.2 An equation for the solute concentration on the small scale In any volume V limited by a closed surface S the total mass balance is Õ udV -j jdS. 4 In Eq. 4 j represents the density of particles flux in V the right-hand side is the flux through S and Eqs. 2 3 and 4 imply 8tus div jff 0 5 with j D.Vu if Fick s law holds locally. Here D is the diffusion .