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The front of filtration combustion and the possibility of its structuring are studied. Cellular structures of the front are demonstrated to form under conditions of instability of a plane combustion front, large filtration coefficients, and insufficient mass of the active gaseous reagent inside the pores. These cellular structures are similar to cellular flames in the case of combustion of gas mixtures. In such systems, the concentration fields of condensed products of filtration combustion form frontal structures that were called fingering instabilities.

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