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 ( Rθ), solutal mobility ratio ( Rc), Lewis number (Le), and thermal-lag coefficient ( λ ). 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 ( Γ = 2 ) and extraction-injection ( Γ = − 2 ), with fixed periodicity ( T = 100 ). Results show that for unstable solutal and thermal fronts ( Rc > 0 , Rθ > 0 ), increasing Le with fixed λ ≠ 1 leads to more prominent mixing and interfacial length for Γ = 2 compared to constant injection and Γ = − 2 . While for unstable solutal fronts ( Rc > 0 ) and stable thermal fronts ( Rθ < 0 ), increasing Le results in more prominent mixing and interfacial length for Γ = − 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 Γ = − 2 ; whereas when hot fluid is used, the instability enhances for Γ = 2 . Furthermore, it is observed that the high thermal diffusion ( Le ≫ 1 ) and enhanced thermal redistribution between solid and fluid phases ( λ ≪ 1 ) effectively mitigate destabilizing effects associated with positive Rθ, reducing overall instability. Overall, in extraction-injection scenarios, the phenomenon of tip-splitting and coalescence is attenuated, and the channeling regime is observed.

A theoretical analysis of viscous fingering instability for a reactive system A+B→C 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, |Rb-Rc|, resulting from a chemical reaction for a given endpoint viscosity contrast Rb, leads to a more unstable system. However, there exist some reactions when Rc>Rb, the onset delays than the equivalent non-reactive case, Rc=Rb. It suggests that the stability of the flow is primarily influenced when instability develops downstream within the flow. Furthermore, Rb 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 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 2.81 cm/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 2.75 cm/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 2.81 cm/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 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 (Γ) 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 Γ < 0, the fingering can be suppressed whereas Γ > 0, yields in vigorous fingers as compared to when Γ = 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.

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 Γ 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 (Γ>1) and extraction-injection (Γ<−1). The onset of fingers and vigorous mixing is observed for Γ>1, whereas for Γ<−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 χ(t) is more pronounced for injection-extraction scenario and least for extraction-injection R<0,Γ=−2. The average convective forces are more dominant for Γ>1,R=2 till the deformed rare interface interact with diffusive frontal interface. The average diffusive forces are significant for Γ<−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 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 L2-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.

A non-modal linear stability analysis (NMA) of the miscible viscous fingering in a porous medium is studied for a toy model of non-monotonic viscosity variation. The onset of instability and its physical mechanism are captured in terms of the singular values of the propagator matrix corresponding to the non-autonomous linear equations. We discuss two types of non-monotonic viscosity profiles, namely, with unfavourable (when a less viscous fluid displaces a high viscous fluid) and with favourable (when a more viscous fluid displaces a less viscous fluid) endpoint viscosities. A linear stability analysis yields instabilities for such viscosity variations. Using the optimal perturbation structure, we are able to show that an initially unconditional stable state becomes unstable corresponding to the most unstable initial disturbance. In addition, we also show that to understand the spatio-temporal evolution of the perturbations it is necessary to analyse the viscosity gradient with respect to the concentration and the location of the maximum concentration. For the favourable endpoint viscosities, a weak transient instability is observed when the viscosity maximum moves close to the pure invading or defending fluid. This instability is attributed to an interplay between the sharp viscosity gradient and the favourable endpoint viscosity contrast. Further, the usefulness of the non-modal analysis demonstrating the physical mechanism of the quadruple structure of the perturbations from the optimal concentration disturbances is discussed. We demonstrate the dissimilarity between the quasi-steady-state approach and NMA in finding the correct perturbation structure and the onset, for both the favourable and unfavourable viscosity profiles. The correctness of the linear perturbation structure obtained from the non-modal stability analysis is validated through nonlinear simulations. We have found that the nonlinear simulations and NMA results are in good agreement. In summary, a non-monotonic variation of the viscosity of a miscible fluid pair is seen to have a larger influence on the onset of fingering instabilities than the corresponding Arrhenius type relationship.

We analyse the linear growth of the viscous fingering instability for miscible, non-reactive, neutral buoyant fluids using the non-modal analysis (NMA). The onset of instability is obscured due to the continually changing base state, and the normal mode analysis is not applicable to the non-autonomous linearized perturbed equations. Commonly used techniques such as frozen time method or amplification theory approach with random initial condition using transient amplifications yield substantially different results for the threshold of instability. We present the classical non-modal methods in the short-time limit using singular value decomposition of the propagator matrix. Using the non-modal approach we characterize the existence of a transition region between a domain exhibiting strong convection and a domain where initial perturbations are damped due to diffusion. Further, at the early times the algebraic growth of perturbations is possible which suggest that NMA could play an important role in describing the onset of instability in the physical phenomenon involving VF.

In the present study, the heat transfer characteristics of thermally developing magnetohydrodynamic flow of nanofluid through microchannel are delineated by following a semi-analytical approach. The combined influences of pressure-driven flow, electroosmotic transport and magnetic field is taken into account for the analysis of the complex microscale thermal transport processes. Solutions for the normalized temperature distributions and the Nusselt number variations, considering the simultaneous interplay of electrokinetic effects (electroosmosis), magnetic effects, Joule heating and viscous dissipation are obtained, for constant wall temperature condition. Particular attention is paid to assess the role of nanofluids in altering the transport phenomena, through variations in the effective nanoparticle volume fractions, as well as the aggregate structure of the particulate phases. It is observed that magnetohydrodynamic effect reduces advective transport of the liquid resulting in gradual reduction of heat transfer. Increase in nanoparticle volume fraction shows decrease in heat transfer. Similar effects are observed with increase in aggregate sizes of the nanoparticles. The effect of the nanofluids on system irreversibility is also studied through entropy generation analysis due to flow and heat transfer in the microchannel. Total entropy generation is found to be dominant at the thermally developing region of the microchannel, whereas it drops sharply at the thermally developed region. Presence of nanoparticles in the base fluid reduces the total entropy generation in the microchannel, thereby indicating decrease in thermodynamic irreversibility with increasing nanoparticle volume fraction.

The effect of a linear adsorption isotherm on the onset of fingering instability in a miscible displacement in the application of liquid chromatography, pollutant contamination in aquifers, etc., is investigated. Such fingering instability on the solute dynamics arise due to the miscible viscus fingering (VF) between the displacing fluid and sample solvent. We use a Fourier pseudo-spectral method to solve the initial value problem that appears in the linear stability analysis. The present linear stability analysis is of generic type and it captures the early-time-diffusion-dominated region which was never expressible through the quasi-steady-state analysis (QSSA). In addition, it measures the onset of instability more accurately than the QSSA methods. It is shown that the onset time depends nonmonotonically on the retention parameter of the solute adsorption. This qualitative influence of the retention parameter on the onset of instability resemblances with the results obtained from direct numerical simulations of the nonlinear equations. Moreover, the present linear stability method helps for an appropriate characterization of the linear and nonlinear regimes of miscible VF instability and also can be useful for the fluid flow problems with the unsteady base state.

The nonmodal linear stability of miscible viscous fingering in a two-dimensional homogeneous porous medium has been investigated. The linearized perturbed equations for Darcy's law coupled with a convection-diffusion equation is discretized using a finite difference method. The resultant initial value problem is solved by a fourth-order Runge-Kutta method, followed by a singular value decomposition of the propagator matrix. Particular attention is given to the transient behavior rather than the long-time behavior of eigenmodes predicted by the traditional modal analysis. The transient behaviors of the response to external excitations and the response to initial conditions are studied by examining the ε-pseudospectra structures and the largest energy growth function, respectively. With the help of nonmodal stability analysis we demonstrate that at early times the displacement flow is dominated by diffusion and the perturbations decay. At later times, when convection dominates diffusion, perturbations grow. Furthermore, we show that the dominant perturbation that experiences the maximum amplification within the linear regime lead to the transient growth. These two important features were previously unattainable in the existing linear stability methods for miscible viscous fingering. To explore the relevance of the optimal perturbation obtained from nonmodal analysis, we performed direct numerical simulations using a highly accurate pseudospectral method. Furthermore, a comparison of the present stability analysis with existing modal and initial value approach is also presented. It is shown that the nonmodal stability results are in better agreement than the other existing stability analyses, with those obtained from direct numerical simulations.

The influence of viscosity contrast on buoyantly unstable miscible fluids in a porous medium is investigated through a linear stability analysis (LSA) as well as direct numerical simulations (DNS). The linear stability method implemented in this paper is based on an initial value approach, which helps to capture the onset of instability more accurately than the quasi-steady-state analysis. In the absence of displacement, we show that viscosity contrast delays the onset of instability in buoyantly unstable miscible fluids. Further, it is observed that by suitably choosing the viscosity contrast and injection velocity a gravitationally unstable miscible interface can be stabilized completely. Through LSA we draw a phase diagram, which shows three distinct stability regions in a parameter space spanned by the displacement velocity and the viscosity contrast. DNS are performed corresponding to parameters from each regime and the results obtained are in accordance with the linear stability results. Moreover, the conversion from one dimensionless formulation to another and its importance when comparing the different type of flow problem associated with each dimensionless formulation are discussed. We also calculate ε-pseudo-spectra of the time dependent linearized operator to investigate the response to external excitation.