Confined diffusion |
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Authors: | S Morelli R Santangelo and S Vincenzi |
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Institution: | (1) Osservatorio Geofisico dell'Università, via Campi 213/A, 41100 Modena, Italia;(2) Istituto per lo Studio della Dinamica delle Grandi Masse del CNR, 1364 San Polo, 30125 Venezia, Italia |
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Abstract: | Summary An equation for the unidimensional confined diffusion is proposed. The equation coincides with the well-known homogeneous
equation except the presence of a source term. This term which has the form of a dipole distribution is located on a moving
front which sharply separates two distinct regions. In the first region (from the boundary up to the front) the confined solution
coincides with a suitable solution of the homogeneous equation; in the second region (besides the front) it vanishes. The
source term, moreover, switches off the diffusing flux at the front. The sharp confinement allows to relax the original boundary
conditions of the homogeneous equation. Precisely, to the function depending on the time at the boundary, another arbitrary
function depending on the space at the initial time is added. This new function (provided not vanishing) allows to obtain
in general an acceptable evolution of the front and does not prevent the validity of the conservation law: flux at the boundary
is equal to the time variation of the diffusing quantity contained between the boundary and the front. By a suitable choice
of this new function, so that it results to be connected to the other boundary condition (that depending on time) it is possible
to arrive at an evolution of the front such as:
, where λ,K, corresponding, respectively, to a dimensionless parameter and diffusivity, depend on the medium. Under such simplifying
assumption, it is possible to obtain an analytical expression for the confined solution. This solution, evaluated in a point
of the space, arrives asymptotically at the same value reached by the solution of the homogeneous equation. |
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Keywords: | Meteorology |
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