Reactive Flow and Free Boundary Problems
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Understanding the behavior of reactive fluids flowing through porous media is essential in many geochemical situations such as determining the genesis and integrity of compartments containing mineral deposits. One problem of particular interest is the effect of a layered porous medium on the dynamics of a reaction front. Our contribution to this field involved a novel combination of homogenization methods with free boundary problems to analyze the stability of reactive fronts in layered porous media. |
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Abstract: A model of reactive flow in a layered porous medium is considered in which the layering is represented by small-scale periodic structure. A novel form of homogenization a nalysis is presented, combining geometric optics and multiple scales expansions together with matched asymptotics to derive an effective free boundary problem for the motion of the reactive interface. Applications of the effective free boundary equations arc given in which travelling wave solutions and the stability of shape perturbations arc considered. |
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Abstract:
The linearized shape stability of melting and solidifying fronts with surface
tension is discussed in this paper by using asymptotic analysis. We show
that the melting problem is always linearly stable regardless of the presence
of surface tension, and that the solidification problem is linearly unstable
without surface tension, but with surface tension it is linearly stable
for those modes whose wave numbers lie outside a certain finite interval
determined by the parameters of the problem. We also show that if the perturbed
initial data is zero in the vicinity of the front, but otherwise quite general,
it does not affect the stability. The present results complement those in
Chadam & Ortoleva which are only valid asymptotically for large time
or equivalently for slow-moving interfaces. The theoretical results are
verified numerically. |
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Abstract:
The shape stability of the reaction interface for reactive flow in a layered
porous medium is studied. This is done using a complete linearized stability
analysis in the setting of a free boundary model of this phenomenon. The
spectrum of the linearized problem is obtained in the general case, and
it is compared with that obtained for homogeneous media in two typical cases. |
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Abstract: Unsteady axial mixing due to addition of a batch of sodium chloride solution at the top of a water-filled tube (2.63 cm i.d.) has been studied by measuring the developing concentration profile and the advancing front with dye added to the brine. Data have also been obtained with added baffle plates, with the use of a viscous aqueous solution, and in smaller diameter(1.48, 1.91 cm) tubes. Results can be approximately by correlated by means of a model based on unsteady one-dimensional turbulent dispersion. Laminar flow affacts the behavior of the advancing front at which the salt concentration is lowest. |
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Abstract:
The infiltration flow of a reactive fluid in a porous medium is investigated.
The reaction causes porosity/permeability changes in the porous medium as
well as viscosity changes in the fluid. The coupling of the associated reaction-infiltration
and Saffman-Taylor instabilities are considered. A mathematical model for
this phenomenon is given in the form of a moving free-boundary problem.
The morphological instability of a planar dissolution front is demonstrated
using a linear stability analysis. An unexpected simplification occurs in
that the resulting fourth-order equation can be solved explicitly. |