A theoretical analysis of viscous fingering instability for a reactive system ??+??????? 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, |??????????|, resulting from a chemical reaction for a given endpoint viscosity contrast ????, leads to a more unstable system. However, there exist some reactions when ????>?????, the onset delays than the equivalent non-reactive case, ????=?????. It suggests that the stability of the flow is primarily influenced when instability develops downstream within the flow. Furthermore, ???? 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.

This study examines the impact of sinusoidal time-dependent injection velocities on miscible thermo-viscous fingering instabilities observed in enhanced oil recovery. Linear stability analysis (LSA) and nonlinear simulations (NLS) are used to investigate fingering dynamics, considering parameters such as thermal mobility ratio (R?), solutal mobility ratio (Rc), Lewis number (Le), and thermal-lag coefficient (?). The LSA employs a quasi-steady state approximation in a transformed self-similar coordinate system, while NLS uses a finite element solver. Two injection scenarios are explored: injection-extraction (?=2) and extraction-injection (?=?2), with fixed periodicity (T=100). Results show that for unstable solutal and thermal fronts (Rc>0,R?>0), increasing Le with fixed ??1 leads to more prominent mixing and interfacial length for ?=2 compared to constant injection and ?=?2. While for unstable solutal fronts (Rc>0) and stable thermal fronts (R?<0), increasing Le results in more prominent mixing and interfacial length for ?=?2, except during early diffusion. Thus, when porous media are swept using cold fluid, increasing the Lewis number intensifies the level of flow instability for ?=?2; whereas when hot fluid is used, the instability enhances for ?=2. Furthermore, it is observed that the high thermal diffusion (Le?1) and enhanced thermal redistribution between solid and fluid phases (??1) effectively mitigate destabilizing effects associated with positive R?, reducing overall instability. Overall, in extraction-injection scenarios, the phenomenon of tip-splitting and coalescence is attenuated, and the channeling regime is observed

This paper addresses how the ocean currents can influence the class II Bragg resonance of surface waves with non-zero surface tension interacting with bottom topography comprising of two different wavenumbers under the assumption of a small-amplitude wave theory. A multi-scale analysis technique is applied up to the third-order to obtain the analytical expressions for reflection and transmission coefficients by solving the coupled evolution equations involving the amplitudes of both reflected and transmitted waves. In the absence of current and surface tension, the findings are validated with the existing literature results. The Bragg peak decreases with increasing U and reaches closely to zero for U lying between 2.75 and 2.81cm/s and further peak started to increase with increasing U. Phase shifting in the Bragg peak is observed in a non-monotonic manner with an increase in U as there is upshift when U changes from 0 to 2.75cm/s, with sudden downshift at U=2.77 cm/s and followed again by upshift for U=2.81 cm/s. There is asymmetrical behavior in the amplitude of left and right subharmonic peaks that are observed for U lying between 2.75 and 2.81cm/s and also, in the same range of U, the resonance bandwidth B is too short. These different qualitative behavior in the Bragg resonance is contingent upon the role played by the both positive and negative group velocities. The role of surface tension parameter T is also not trivial as the Bragg peak increases when T increases up to U=2.81 cm/s but beyond this range, the effect of T is opposite which means the Bragg peak decreases when T increases. In addition to this, this study predicted that the choice of number of ripples M to achieve full reflection is dependent upon the value of U, and it is possible to accurately capture the effect of reflected energy for large values of M. Moreover, the subcritical detuning is observed in proximity to the Bragg resonance. This study has potential to build an understanding about the role of currents on higher-order Bragg resonance on the coastal bathymetries and extremely important for coastal rehabilitation.

The effect of non-uniform, oscillating bottom profiles on a two-layer stable density stratification model has been examined using the method of weakly non-linear analysis. The study of bottom profiles in the context of two-layered stratified fluids has focused on three specific types: (a) profiles that exhibit a monotonically decreasing pattern, (b) profiles that decay exponentially, and (c) profiles that display Gaussian oscillations. The analysis of the second-order reflection and transmission coefficients for the nonlinear boundary value problem was conducted using a combination of the regular perturbation method and the Fourier transforms technique. The numerical findings pertaining to various physical parameters have been presented, demonstrating the impact of the Class I Bragg resonance in all three profiles and the elevation of the tails in the monotonically decreasing oscillatory profile. Specifically, the presence of high reflections due to the tail-lifting phenomenon is observed in a profile that exhibits a monotonically decreasing pattern, in contrast to the other profiles. The findings of the study indicate that interface modes demonstrate pronounced reflections when the density ratios are low, but divergent results are observed when the density ratios are high. As the density ratio R increases, there is a greater migration of wave energy from the interface mode to the surface mode, resulting in increased levels of reflection. As the value of R approaches 1, it is observed that lower frequencies exhibit significantly more pronounced internal mode reflections compared to surface mode. Several contrasting aspects can be observed in three oscillatory profiles when compared to periodic profiles. These aspects include the disappearance of zero reflection, also known as complete transmission, as well as the absence of oscillations. The findings of this study demonstrate that a monotonically oscillating decreasing profile can be considered as an efficacious Bragg breakwater. Furthermore, this study investigates the energy transfer that occurs during the movement of surface and internal waves across non-periodic oscillatory profiles. As a result, an energy balance relationship is derived, which specifically applies to surface and interface modes. This work has hydrodynamical relevance to wave propagation in coastal regions and to the hydrodynamics of tsunamis in the open ocean, both of which are affected by changes in the bathymetry of the fluid region.

The effect of a sinusoidal injection on the fingering instability in a miscible displacement in the application of liquid chromatography, pollutant contamination in aquifers, etc., is investigated. The injection velocity, U(t) is characterized by its amplitude of ? and time-period of T. The solute transport, flow in porous media, and mass conservation in a two-dimensional porous media is modeled by the convection-diffusion equation, Darcy's equation, and the continuity equation, respectively. The numerical simulation is performed in COMSOL Multiphysics utilizing a finite-element based approach. The fingering dynamics for various time-period have been studied for two scenarios namely, injection-extraction (?>1) and extraction-injection ( ?<?1 ). The onset of fingers and vigorous mixing is observed for ?>1, whereas for ?<?1, the onset gets delayed. The viscosity contrast between the sample and the surrounding fluid is characterized by the log-mobility ratio R. When R>0 the rare interface becomes unstable, while for R<0 the frontal interface deformed. In the case of R<0, the extraction-injection process attenuates the fingering dynamics, which is beneficial in chromatographic separations or pollutant dispersion in underground aquifers. The injection-extraction process is observed to have a longer mixing length, indicating early interaction between both interfaces. The degree of mixing ?(t) is more pronounced for injection-extraction scenario and least for extraction-injection R<0,?=?2. The average convective forces are more dominant for ?>1,R=2 till the deformed rare interface interact with diffusive frontal interface. The average diffusive forces are significant for ?<?1,R=?2 which can be helpful in separation of chemicals in chromatography. This study therefore provided new insights into the role of alternate injection-extraction injections in altering the fingering dynamics of the miscible sample.

The influence of time-injection velocity on the miscible displacement in porous media is studied numerically. We examined the scenario when a more viscous fluid of finite length is confined within a less viscous one in a Hele-Shaw cell. The injection velocity is assumed to be in form of a sinusoidal form characterized by its amplitude (?) and time-period (T). The physical mechanism is analysed by solving three coupled equations, namely, Darcy’s equation, continuity equation and convection-advection equation. The non-linear simulations for the rectilinear flow have been carried out using COMSOL multi-physics (version 5.3a). The obtained results suggest that for ? < 0, the fingering can be suppressed whereas ? > 0, yields in vigorous fingers as compared to when ? = 0 (the constant injection strategy). It can be concluded that the timedependent strategy may help in analysing and controlling the spread of contaminants and chemical separations.

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The influence of dispersion or equivalently of the Péclet number (Pe) on miscible viscous fingering in a homogeneous porous medium is examined. The linear optimal perturbations maximizing finite-time energy gain is demonstrated with the help of the propagator matrix approach based non-modal analysis (NMA). We show that onset of instability is a monotonically decreasing function of Pe and the onset time determined by NMA emulates the non-linear simulations. Our investigations suggest that perturbations will grow algebraically at early times, contrary to the well-known exponential growth determined from the quasi-steady eigenvalues. One of the over-arching objective of the present work is to determine whether there are alternative mechanisms which can describe the mathematical understanding of the spectrum of the time-dependent stability matrix. Good agreement between the NMA and non-linear simulations is observed. It is shown that within the framework of L-norm, the non-normal stability matrix can be symmetrizable by a similarity transformation and thereby we show that the non-normality of the linearized operator is norm dependent. A framework is thus presented to analyze the exchange of stability which can be determined from the eigenmodes.

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