Non-modal linear stability analysis of reactive front A+B→C for infinitely fast chemical reactions

A theoretical analysis of viscous fingering instability for a reactive system <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>+</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo stretchy="false">→</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> with an infinitely fast reaction in a porous medium for a rectilinear flow is presented. By contrast to the traditional quasi-steady-state analysis (QSSA), a non-modal analysis (NMA) based on the fundamental matrix formulation is employed to study the reactive displacement, considering reactants and products with mismatched viscosities. This study investigates the transient growth of perturbations by analysing the singular values and singular vectors to address the optimal energy amplification. We illustrate that an increase in the viscosity contrast, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>|</mml:mi> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo>−</mml:mo> </mml:mrow> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mi>|</mml:mi> </mml:mrow> </mml:math> , resulting from a chemical reaction for a given endpoint viscosity contrast <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:math> , leads to a more unstable system. However, there exist some reactions when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>&gt;</mml:mo> </mml:mrow> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:mrow> </mml:math> , the onset delays than the equivalent non-reactive case, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>=</mml:mo> </mml:mrow> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:mrow> </mml:math> . It suggests that the stability of the flow is primarily influenced when instability develops downstream within the flow. Furthermore, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:math> is found to significantly affect the spatio-temporal evolution of perturbations and the underlying physical mechanism. It is demonstrated that the QSSA is inadequate to address the transient growth, and NMA is the most suitable approach to studying the underlying physical mechanism of instability. Furthermore, NMA results align more consistently with non-linear simulations compared with QSSA.