Abstract:
We consider initial-boundary value problems describing
the in-situ leaching of rare metals (uranium, nickel and so on)
with an acid solution. Assuming that the solid skeleton of the ground
is an absolutely rigid body, we describe the physical process
in the pore space at the microscopic level (with characteristic size
about 5–20 microns) by the Stokes equations for
an incompressible fluid coupled with
diffusion–convection equations for
the concentrations of the acid and the chemical
reaction products in the pore space. Since
the solid skeleton changes its geometry during dissolution, the boundary
‘pore space–solid skeleton’ is unknown (free).
Using the homogenization method for media with a special periodic structure,
we rigorously derive a macroscopic mathematical model (with characteristic size
of several meters or tens of meters) of incompressible fluid corresponding to
the original microscopic model of the physical process
and prove the global-in-time existence and uniqueness theorems
for classical solutions of the resulting macroscopic mathematical model.
Keywords:free boundary problems, two-scale convergence, homogenization of periodic structures, fixed point theorem.
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