The nonlinear behavior of viscous fingering in miscible displacements is studied. A Fourier spectral method is used as the basic scheme for numerical simulation. In its simplest formulation, the problem can be reduced to two algebraic equations for flow quantities and a first-order ordinary differential equation in time for the concentration. There are two parameters, the Peclet number (Pe) and mobility ratio (M), that determine the stability characteristics. The result shows that at short times, both the growth rate and the wavelength of fingers are in good agreement with predictions from our previous linear stability theory. However, as the time goes on, the nonlinear behavior of fingers becomes important. There are always a few dominant fingers that spread and shield the growth of other fingers. The spreading and shielding effects are caused by a spanwise secondary instability, and are aided by the transverse dispersion. It is shown that once a finger becomes large enough, the concentration gradient of its front becomes steep as a result of stretching caused by the cross-flow, in turn causing the tip of the finger to become unstable and split. The splitting phenomenon in miscible displacement is studied by the authors for the first time. A study of the averaged one-dimensional axial concentration profile is also presented, which indicates that the mixing length grows linearly in time, and that effective one-dimensional models cannot describe the nonlinear fingering.

A theoretical treatment of the stability of miscible displacement in a porous medium is presented. For a rectilinear displacement process, since the base state of uniform velocity and a dispersive concentration profile is time dependent, we make the quasi-steady-state approximation that the base state evolves slowly with respect to the growth of disturbances, leading to predictions of the growth rate. Comparison of results with initial value solutions of the partial differential equations shows that, excluding short times, there is good agreement between the two theories. Comparison of the theory with several experiments in the literature indicates that the theory gives a good prediction of the most dangerous wavelength of unstable fingers. An approximate analysis for transversely anisotropic media has elucidated the role of transverse dispersion in controlling the length scale of fingers.

We study a pressureless Euler system with a non-linear density-dependent alignment term, originating in the Cucker-Smale swarming models. The alignment term is dissipative in the sense that it tends to equilibrate the velocities. Its density dependence is natural: the alignment rate increases in the areas of high density due to species discomfort. The diffusive term has the order of a fractional Laplacian (−∂ xx ) α/2 , α ∈ (0, 1). The corresponding Burgers equation with a linear dissipation of this type develops shocks in a finite time. We show that the alignment nonlinearity enhances the dissipation, and the solutions are globally regular for all α ∈ (0, 1). To the best of our knowledge, this is the first example of such regularization due to the non-local nonlinear modulation of dissipation.Here, N i (t) = {j : |x i (t) − x j (t)| ≤ r}, with some r > 0 fixed, ∆θ is a uniformly distributed random variable in [−1, 1], and η > 0 is a parameter measuring the strength of the noise.

We study the large-time behaviour of Eulerian systems augmented with non-local alignment. Such systems arise as hydrodynamic descriptions of agent-based models for self-organized dynamics, e.g. Cucker & Smale (2007 IEEE Trans. Autom. Control 52 , 852–862. (doi: 10.1109/TAC.2007.895842 )) and Motsch & Tadmor (2011 J. Stat. Phys. 144 , 923–947. (doi: 10.1007/s10955-011-0285-9 )) models. We prove that, in analogy with the agent-based models, the presence of non-local alignment enforces strong solutions to self-organize into a macroscopic flock. This then raises the question of existence of such strong solutions. We address this question in one- and two-dimensional set-ups, proving global regularity for subcritical initial data. Indeed, we show that there exist critical thresholds in the phase space of the initial configuration which dictate the global regularity versus a finite-time blow-up. In particular, we explore the regularity of non-local alignment in the presence of vacuum.

A linear stability analysis of miscible displacement for a radial source flow in porous media is presented. Since there is no characteristic time or length scale for the system, it is shown that solutions to the stability equations depend only upon a similarity variable, with disturbances growing algebraically in time. Two parameters, the mobility ratio and a Peclet number based upon the source strength, determine the stability. Results for the growth constant as a function of mobility ratio and Peclet number are given. It is shown that there is a critical Peclet number Pec above which displacement becomes unstable. For Pe>Pec, there is always a cutoff scale attributable to dispersion, and a most dangerous mode, with the two corresponding wavenumbers increasing with Pe. The growth constant increases with Pe as well. The effect of mobility ratio is also studied. The result indicates that, as expected, increasing mobility ratio destabilizes the displacement. Asymptotic results for the growth rate, cutoff, and preferred scales that hold as Pe→∞ are given, and are found to be in good agreement with the numerical results.

We study the critical thresholds for the compressible pressureless Euler equations with pairwise attractive or repulsive interaction forces and non-local alignment forces in velocity in one dimension. We provide a complete description for the critical threshold to the system without interaction forces leading to a sharp dichotomy condition between global in time existence or finite-time blow-up of strong solutions. When the interaction forces are considered, we also give a classification of the critical thresholds according to the different type of interaction forces. We also remark on global in time existence when the repulsion is modeled by the isothermal pressure law.subject to initial density and velocity (ρ(·, t), u(·, t))| t=0 = (ρ 0 , u 0 ).Since the total mass is conserved in time, we may assume, without loss of generality, that ρ is a probability density function, i.e., ρ(·, t) L 1 = 1.

Viscous fingering in miscible displacements in the presence of permeability heterogeneities is studied using two-dimensional simulations. The heterogeneities are modeled as stationary random functions of space with finite correlation scale. Both the variance and scale of the heterogeneities are varied over modest ranges. It is found that the fingered zone grows linearly in time in a fashion analogous to that found in homogeneous media by Tan and Homsy [Phys. Fluids 31, 1330 (1988)], indicating a close coupling between viscous fingering on the one hand and flow through preferentially more permeable paths on the other. The growth rate of the mixing zone increases monotonically with the variance of the heterogeneity, as expected, but shows a maximum as the correlation scale is varied. The latter is explained as a ‘‘resonance’’ between the natural scale of fingers in homogeneous media and the correlation scale.

The Euler-Poisson-Alignment (EPA) system appears in mathematical biology and is used to model, in a hydrodynamic limit, a set agents interacting through mutual attraction/repulsion as well as alignment forces. We consider one-dimensional EPA system with a class of singular alignment terms as well as natural attraction/repulsion terms. The singularity of the alignment kernel produces an interesting effect regularizing the solutions of the equation and leading to global regularity for wide range of initial data. This was recently observed in [5]. Our goal in this paper is to generalize the result and to incorporate the attractive/repulsive potential. We prove that global regularity persists for these more general models.