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.