• Abstract

We present numerical simulations of a new system of micro-pump based on the thermal creep effect described by the kinetic theory of gases. This device is made of a simple smooth and curved channel with a periodic temperature distribution. Using the Boltzmann-BGK model as the governing equation for the gas flow, we develop a numerical method based on a deterministic finite volume scheme, implicit in time, with an implicit treatment of the boundary conditions. This method is comparatively less sensitive to the slow flow velocity than the usual Direct Simulation Monte Carlo method in case of long devices, and turns out to be accurate enough to compute macroscopic quantities like the pressure field in the channel. Our simulations show the ability of the device to produce a one-way flow that has a pumping effect.

• Abstract

This paper introduces a three dielectric layer hybrid solvation model for treating electrostatic interactions of biomolecules in solvents using the Poisson-Boltzmann equation. In this model, an interior spherical cavity will contain the solute and some explicit solvent molecules, and an intermediate buffer layer and an exterior layer contain the bulk solvent. A special dielectric permittivity profile is used to achieve a continuous dielectric transition from the interior cavity to the exterior layer. The selection of this special profile using a harmonic interpolation allows an analytical solution of the model by generalizing the classical Kirkwood series expansion. Discrete image charges are used to speed up calculations for the electrostatic potential within the interior and buffer layer regions. Semi-analytical and least squares methods are used to construct an accurate discrete image approximation for the reaction field due to solvent with or without salt effects. In particular, the image charges obtained by the least squares method provide accurate approximations to the reaction field independent of the ionic concentration of the solvent. Numerical results are presented to validate the accuracy and effectiveness of the image charge methods.

• Abstract

An efficient implicit lower-upper symmetric Gauss-Seidel (LU-SGS) solution approach has been applied to a high order spectral volume (SV) method for unstructured tetrahedral grids. The LU-SGS solver is preconditioned by the block element matrix, and the system of equations is then solved with a LU decomposition. The compact feature of SV reconstruction facilitates the efficient solution algorithm even for high order discretizations. The developed implicit solver has shown more than an order of magnitude of speed-up relative to the Runge-Kutta explicit scheme for typical inviscid and viscous problems. A convergence to a high order solution for high Reynolds number transonic flow over a 3D wing with a one equation turbulence model is also indicated.

• Abstract

An indirect-forcing immersed boundary method for solving the incompressible Navier-Stokes equations involving the interfaces and irregular domains is developed. The rigid boundaries and interfaces are represented by a number of Lagrangian control points. Stationary rigid boundaries are embedded in the Cartesian grid and singular forces at the rigid boundaries are applied to impose the prescribed velocity conditions. The singular forces at the interfaces and the rigid boundaries are then distributed to the nearby Cartesian grid points using the immersed boundary method. In the present work, the singular forces at the rigid boundaries are computed implicitly by solving a small system of equations at each time step to ensure that the prescribed velocity condition at the rigid boundary is satisfied exactly. For deformable interfaces, the forces that the interface exerts on the fluid are computed from the configuration of the elastic interface and are applied to the fluid. The Navier-Stokes equations are discretized using finite difference method on a staggered uniform Cartesian grid by a second order accurate projection method. The ability of the method to simulate viscous flows with interfaces on irregular domains is demonstrated by applying to the rotational flow problem, the relaxation of an elastic membrane and flow in a constriction with an immersed elastic membrane.

• Abstract

From a spectral method combined with a predictor-corrector scheme, we numerically study the behavior in time of solutions of the three-dimensional generalized Kadomtsev-Petviashvili equations. In a systematic way, the dispersion, the blowup in finite time, the solitonic behavior and the transverse instabilities are numerically inspected.

• Abstract

Inspiral of binary black holes occurs over a time-scale of many orbits, far longer than the dynamical time-scale of the individual black holes. Explicit evolutions of a binary system therefore require excessively many time-steps to capture interesting dynamics. We present a strategy to overcome the Courant-Friedrichs-Lewy condition in such evolutions, one relying on modern implicit-explicit ODE solvers and multidomain spectral methods for elliptic equations. Our analysis considers the model problem of a forced scalar field propagating on a generic curved background. Nevertheless, we encounter and address a number of issues pertinent to the binary black hole problem in full general relativity. Specializing to the Schwarzschild geometry in Kerr-Schild coordinates, we document the results of several numerical experiments testing our strategy.

• Abstract

In this paper, we study the Marangoni effects in the mixture of two Newtonian fluids due to the thermo-induced surface tension heterogeneity on the interface. We employ an energetic variational phase field model to describe its physical phenomena, and obtain the corresponding governing equations defined by a modified Navier-Stokes equations coupled with phase field and energy transport. A mixed Taylor-Hood finite element discretization together with full Newton's method are applied to this strongly nonlinear phase field model on a specific domain. Under different boundary conditions of temperature, the resulting numerical solutions illustrate that the thermal energy plays a fundamental role in the interfacial dynamics of two-phase flows. In particular, it gives rise to a dynamic interfacial tension that depends on the direction of temperature gradient, determining the movement of the interface along a sine/cosine-like curve.

• Abstract

This article is devoted to the construction of a numerical scheme to solve the equations of radiative hydrodynamics. We use this numerical procedure to compute shock profiles and illustrate some earlier theoretical results about their smoothness and monotonicity properties. We first consider a scalar toy model, then we extend our analysis to a more realistic system for the radiative hydrodynamics that couples the Euler equations and an elliptic equation.

• Abstract

In this study, a stable and robust interface-capturing method is developed to resolve inviscid, compressible two-fluid flows with general equation of state (EOS). The governing equations consist of mass conservation equation for each fluid, momentum and energy equations for mixture and an advection equation for volume fraction of one fluid component. Assumption of pressure equilibrium across an interface is used to close the model system. MUSCL-Hancock scheme is extended to construct input states for Riemann problems, whose solutions are calculated using generalized HLLC approximate Riemann solver. Adaptive mesh refinement (AMR) capability is built into hydrodynamic code. The resulting method has some advantages. First, it is very stable and robust, as the advection equation is handled properly. Second, general equation of state can model more materials than simple EOSs such as ideal and stiffened gas EOSs for example. In addition, AMR enables us to properly resolve flow features at disparate scales. Finally, this method is quite simple, time-efficient and easy to implement.

• Abstract