The no-slip boundary condition, i.e., zero fluid velocity relative to the solid at the fluid-solid interface, has been very successful in describing many macroscopic flows. A problem of principle arises when the no-slip boundary condition is used to model the hydrodynamics of immiscible-fluid displacement in the vicinity of the moving contact line, where the interface separating two immiscible fluids intersects the solid wall. Decades ago it was already known that the moving contact line is incompatible with the no-slip boundary condition, since the latter would imply infinite dissipation due to a non-integrable singularity in the stress near the contact line. In this paper we first present an introductory review of the problem. We then present a detailed review of our recent results on the contact-line motion in immiscible two-phase flow, from molecular dynamics (MD) simulations to continuum hydrodynamics calculations. Through extensive MD studies and detailed analysis, we have uncovered the slip boundary condition governing the moving contact line, denoted the generalized Navier boundary condition. We have used this discovery to formulate a continuum hydrodynamic model whose predictions are in remarkable quantitative agreement with the MD simulation results down to the molecular scale. These results serve to affirm the validity of the generalized Navier boundary condition, as well as to open up the possibility of continuum hydrodynamic calculations of immiscible flows that are physically meaningful at the molecular level.

The typical elements in a numerical simulation of fluid flow using moving meshes are a time integration scheme, a rezone method in which a new mesh is defined, and a remapping (conservative interpolation) in which a solution is transferred to the new mesh. The objective of the rezone method is to move the computational mesh to improve the robustness, accuracy and eventually efficiency of the simulation. In this paper, we consider the one-dimensional viscous Burgers' equation and describe a new rezone strategy which minimizes the L₂ norm of error and maintains mesh smoothness. The efficiency of the proposed method is demonstrated with numerical examples.

We consider a charged particle confined in a one-dimensional rectangular double-well potential, driven by an external periodic excitation at frequency Ω and with amplitude A. We find that there is the regime of the parametric resonance due to the modulation of the amplitude A at the frequency ω_prm, which results in the change in the population dynamics of the energy levels. The analysis relies on the Dirac system of Hamiltonian equations that are equivalent to the Schrödinger equation. Considering a finite dimensional approximation to the Dirac system, we construct the foliation of its phase space by subsets F_ab given by constraints a ≤ N₀ ≤ b on the occupation probabilities N₀ of the ground state, and describe the tunneling by frequencies ν_ab of the system's visiting subsets F_ab. The frequencies ν_ab determine the probability density and thus the Shannon entropy, which has the maximum value at the resonant frequency ω = ω_prm. The reconstruction of the state-space of the system's dynamics with the help of the Shaw-Takens method indicates that the quasi-periodic motion breaks down at the resonant value ω_prm.

Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source terms. In our earlier work [31–33], we designed high order well-balanced schemes to a class of hyperbolic systems with separable source terms. In this paper, we present a different approach to the same purpose: designing high order well-balanced finite volume weighted essentially non-oscillatory (WENO) schemes and RungeKutta discontinuous Galerkin (RKDG) finite element methods. We make the observation that the traditional RKDG methods are capable of maintaining certain steady states exactly, if a small modification on either the initial condition or the flux is provided. The computational cost to obtain such a well balanced RKDG method is basically the same as the traditional RKDG method. The same idea can be applied to the finite volume WENO schemes. We will first describe the algorithms and prove the well balanced property for the shallow water equations, and then show that the result can be generalized to a class of other balance laws. We perform extensive one and two dimensional simulations to verify the properties of these schemes such as the exact preservation of the balance laws for certain steady state solutions, the non-oscillatory property for general solutions with discontinuities, and the genuine high order accuracy in smooth regions.

Boundary conditions for molecular dynamics simulation of crystalline solids are considered with the objective of eliminating the reflection of phonons. A variational formalism is presented to construct boundary conditions that minimize total phonon reflection. Local boundary conditions that involve a few neighbors of the boundary atoms and limited number of time steps are found using the variational formalism. Their effects are studied and compared with other boundary conditions such as truncated exact boundary conditions or by appending border atoms where artificial damping forces are applied. In general it is found that, with the same cost or complexity, the variational boundary conditions perform much better than the truncated exact boundary conditions or by appending border atoms with empirical damping profiles. Practical issues of implementation are discussed for real crystals. Application to brittle fracture dynamics is illustrated.

This paper applies a 3-D nonuniform fast Fourier transform (NUFFT) migration method to image both free-space and buried targets from data collected by a ultra-wideband ground penetrating radar (GPR) system. The method incorporates the NUFFT algorithm into 3-D phase shift migration to evaluate the inverse Fourier transform more accurately and more efficiently than the conventional migration methods. Previously, the nonuniform nature of the wavenumber space required linear interpolation before the regular fast Fourier transform (FFT) could be applied. However, linear interpolation usually degrades the quality of reconstructed images. The NUFFT method mitigates such errors by using high-order spatial-varying kernels. The NUFFT migration method is utilized to reconstruct GPR images collected in laboratory. A plywood sheet in free space and a buried plexiglas chamber are successfully reconstructed. The results in 3-D visualization demonstrate the outstanding performance of the method to retrieve the geometry of the objects. Several buried landmines are also scanned and reconstructed using this method. Since the images resolve the features of the objects well, they can be utilized to assist the landmine discrimination.