The effect of geometry is observable on the onset of instability, where we obtain significant differences from existing results in the rectilinear geometry.

The influence of fluid dispersion on the Saffman-Taylor instability in miscible fluids has been investigated in both the linear and the nonlinear regimes. The convective characteristic scales are used for the dimensionless formulation that incorporates the Péclet number (Pe) into the governing equations as a measure for the fluid dispersion. A linear stability analysis (LSA) is performed in a similarity transformation domain using the quasi-steady-state approximation. LSA results confirm that a flow with a large Pe has a higher growth rate than a flow with a small Pe. The critical Péclet number (Pec) for the onset of instability for all possible wave numbers and also a power-law relation of the onset time and most unstable wave number with Pe are observed. Unlike the radial source flow, Pec is found to vary with t0. A Fourier spectral method is used for direct numerical simulations (DNS) of the fully nonlinear system. The power-law dependence of the onset of instability ton on Pe is obtained from the DNS and found to be inversely proportional to Pe and it is in good agreement with that obtained from the LSA. The influence of the anisotropic dispersion is analyzed in both the linear and the nonlinear regimes. The results obtained confirm that for a weak transverse dispersion merging happens slowly and hence the small wave perturbations become unstable. We also observ that the onset of instability sets in early when the transverse dispersion is weak and varies with the anisotropic dispersion coefficient, ε, as ∼√[ε], in compliance with the LSA. The combined effect of the Korteweg stress and Pe in the linear regime is pursued. It is observed that, depending on various flow parameters, a fluid system with a larger Pe exhibits a lower instantaneous growth rate than a system with a smaller Pe, which is contrary to the results when such stresses are absent.

Viscous fingering (VF) is an interfacial hydrodynamic instability phenomenon observed when a fluid of lower viscosity displaces a higher viscous one in a porous media. In miscible viscous fingering, the concentration gradient of the undergoing fluids is an important factor, as the viscosity of the fluids are driven by concentration. Diffusion takes place when two miscible fluids are brought in contact with each other. However, if the diffusion rate is slow enough, the concentration gradient of the two fluids remains very large during some time. Such steep concentration gradient, which mimics a surface tension type force, called the effective interfacial tension, appears in various cases such as aqua-organic, polymer-monomer miscible systems, etc. Such interfacial tension effects on miscible VF is modeled using a stress term called Korteweg stress in the Darcy's equation by coupling with the convection-diffusion equation of the concentration. The effect of the Korteweg stresses at the onset of the instability has been analyzed through a linear stability analysis using a self-similar Quasi-steady-state-approximation (SS-QSSA) in which a self-similar diffusive base state profile is considered. The quasi-steady-state analyses available in literature are compared with the present SS-QSSA method and found that the latter captures appropriately the unconditional stability criterion at an earlier diffusive time as well as in long wave approximation. The effects of various governing parameters such as log-mobility ratio, Korteweg parameters, disturbances' wave number, etc., on the onset of the instability are discussed for, (i) the two semi-infinite miscible fluid zones and (ii) VF of the miscible slice cases. The stabilizing property of the Korteweg stresses effect is observed for both of the above mentioned cases. Critical miscible slice lengths are computed to have the onset of the instability for different governing parameters with or without Korteweg stresses. These stabilizing properties of the Korteweg stresses captured in this present study are in agreement with the numerical simulations of fully nonlinear problem and the experimental observations reported in the literature.