Efficient solution strategy for the semi-implicit discontinuous Galerkin discretization of the Navier-Stokes equations

We deal with the numerical solution of the system of the compressible Navier–Stokes equations with the aid of the interior penalty Galerkin method. We employ a semi-implicit time discretization which leads to the solution of a sequence of linear algebraic systems. We develop an efficient strategy for the solution of these systems. It is based on a simple adaptive technique for the choice of the time step and a relatively weak stopping criterion for iterative linear algebraic solvers. The presented numerical experiments show that the proposed strategy is efficient for steady-state problems using various grids, polynomial degrees of approximations and flow regimes. Finally, we apply this strategy with a minor modification to an unsteady flow.     

Shocks and finite-time singularities in Hele-Shaw flow

Hele-Shaw flow at vanishing surface tension is ill-defined. In finite time, the flow develops cusp-like singularities. We show that this ill-defined problem admits a weak dispersive solution when singularities give rise to a graph of shock waves propagating into the viscous fluid. The graph of shocks grows and branches. Velocity and pressure have finite discontinuities across the shock. We formulate a few simple physical principles which single out the dispersive solution and interpret shocks as lines of decompressed fluid. We also formulate the dispersive weak solution in algebro-geometrical terms as an evolution of the Krichever-Boutroux complex curve. We study in detail the most generic (2, 3)-cusp singularity, which gives rise to an elementary branching event. This solution is self-similar and expressed in terms of elliptic functions.     

The application of homotopy analysis method for 2-dimensional steady slip flow in microchannels

Research on micro flow, especially on micro slip flow, is very important for designing and optimizing the micro electromechanical system (MEMS). In this Letter, similarity transformation for the Navier-Stokes equation for 2-dimensional steady slip flow in microchannels is given. We provide an analytical solution for the slip flow using a powerful, easy-to-use analytic technique for non-linear problems, that is, the homotopy analysis method (HAM). The analytical solution is presented in the form of an infinite series. The effects of the Knudsen number (Kn) is discussed on the velocity profiles. It is found that the results are in excellent agreement with the existing results in the literature for the case of laminar developed flow.     

Singularities of the Hele-Shaw flow and shock waves in dispersive media

We show that singularities developed in the Hele-Shaw problem have a structure identical to shock waves in dissipativeless dispersive media. We propose an experimental setup where the cell is permeable to a nonviscous fluid and study continuation of the flow through singularities. We show that a singular flow in this nontraditional cell is described by the Whitham equations identical to Gurevich-Pitaevski solution for a regularization of shock waves in Korteveg-de Vriez equation. This solution describes regularization of singularities through creation of disconnected bubbles.     

平面定常超音速绕流问题的新的数值解法

用特征线法解平面定常超音速绕流问题虽然有效,但当激波很弱、几乎与特征线平行时则很难处理。用有限差分法计算此问题也比较复杂。本文把作者在文章中^[1]提出的新的数值解法,发展并应用到平面定常超音速绕流的问题。仍然采用了许为厚教授提出的新拉格朗日变量^[2],这使边界条件的提法大为简化。此新的数值解法按变量指标之和,一排排地往下计算,方法简单,可以处理各种形状物体的超音速绕流。本文对向上弯曲的抛物形固壁绕流向题的实例进行了具体计算。算出了激波的形状。当激波没有形成以前,相应的普朗特一迈耶气流是有准确分析解的,把数值解与准确解进行了l比较结果是满意的。     

Conformal Symmetries of Spherical Spacetimes

We investigate the conformal geometry of spherically symmetric spacetimes in general without specifying the form of the matter distribution. The general conformal Killing symmetry is obtained subject to a number of integrability conditions. Previous results relating to static spacetimes are shown to be a special case of our solution. The general inheriting conformal symmetry vector, which maps fluid flow lines conformally onto fluid flow lines, is generated and the integrability conditions are shown to be satisfied. We show that there exists a hypersurface orthogonal conformal Killing vector in an exact solution of Einstein’s equations for a relativistic fluid which is expanding, accelerating and shearing.     

Inhomogeneous Kaluza-Klein model with heat flow

We present a cosmological model in Kaluza-Klein spacetime with a matter field obeying an equation of state and a radial heat flow. The (3+1)space resembles that of a Robertson-Walker spacetime, but unlike the corresponding four-dimensional case our five-dimensional spacetime admits a radial flow of heat even when the three-space curvature remains a constant. Further our solution exhibits the desirable feature of dimensional reduction.     

A representation of Kerr-like nonlinearity for the analytical TM surface polariton solution

The TM-polarized waves propagating along the interface between a nonlinear Kerr-like material and linear cladding are investigated. We analyse the nonlinear dielectric permittivity as a function of the electromagnetic field. It is shown that an exact analytical solution of Maxwell's equations corresponding to the TM surface polariton in the form described by sech function do exist in a Kerr-like nonlinear medium with the permittivity profile given by a hypergeometric function. We compare our analytical solution and analogous exact numerical solution in a Kerr medium. The power flow down the interface between two media is also discussed.     

Kinetic theory of hydrodynamic flows. I. The generalized normal solution method and its application to the drag on a sphere

We consider the flow of a dilute gas around a macroscopic heavy object. The state of the gas is described by an extended Boltzmann equation where the interactions between the gas molecules and the object are taken into account in computing the rate of change of the distribution function of the gas. We then show that the extended Boltzmann is equivalent to the usual Boltzmann equation, supplemented by boundary conditions imposed on the distribution function at the surface of the object. The remainder of the paper is devoted to a study of the solution of the extended Boltzmann equation in the case that the mean free path of a gas molecule is small compared to some characteristic dimension of the macroscopic object. We show that the Chapman-Enskog normal solution of the ordinary Boltzmann equation is not in general a solution of the extended equation near the surface of the object and must be supplemented by a boundary layer term. We then introduce a projection operator method which allows us to decompose the solution of the extended equation into a normal solution part and a boundary layer part when the gas flow is sufficiently slow. As a specific example of the method we consider the flow around a sphere, and derive the Stokes-Boussinesq form for the frequency-dependent force on the sphere for arbitrary slip coefficient. This derivation is the first one that starts from the Boltzmann equation for a general dilute gas and incorporates the effect of the boundary layer on the drag force.Work supported by the National Science Foundation.     

Mathematical modeling of the thermal and optical parameters of the zone of laser influence for determining the feedback parameters for a system of adaptive precision laser material processing

We develop a mathematical model of channel formation in the ablation regime at the initial non-stationary stage of growth that establishes the relationship between channel parameters such as temperature, optical parameters of the irradiated surface and erosion flow, and the parameters of laser action. The solution to a two-dimensional heat equation, coordinated with the solution to the problem of propagation of an ablation front and the formation of flows of evaporated material and plasma, is found in a grid with variable boundaries.     

Global Shock Waves¶for the Supersonic Flow Past a Perturbed Cone

We prove the global existence of a shock wave for the stationary supersonic gas flow past an infinite curved and symmetric cone. The flow is governed by the potential equation, as well as the boundary conditions on the shock and the surface of the body. It is shown that the solution to this problem exists globally in the whole space with a pointed shock attached at the tip of the cone and tends to a self-similar solution under some suitable conditions. Our analysis is based on a global uniform weighted energy estimate for the linearized problem. Combining this with the local existence result of Chen–Li [1] we establish the global existence and decay rate of the solution to the nonlinear problem. Received: 1 August 2001 / Accepted: 14 January 2002     

Entropic lattice Boltzmann method for turbulent flow simulations: Boundary conditions

We introduce a new concept of boundary conditions for realization of the lattice Boltzmann simulations of turbulent flows. The key innovation is the use of a universal distribution function for particles, analogous to the Tamm–Mott-Smith solution for the shock wave in the classical Boltzmann kinetic equation. Turbulent channel flow simulations demonstrate that the new boundary enables accurate results even with severely under-resolved grids. Generalization to complex boundary is illustrated with an example of turbulent flow past a circular cylinder.     

On the Global Existence and Stability of a Multi-Dimensional Supersonic Conic Shock Wave

We establish the global existence and stability of a three-dimensional supersonic conic shock wave for a compactly perturbed steady supersonic flow past an infinitely long circular cone with a sharp angle. The flow is described by a 3-D steady potential equation, which is multi-dimensional, quasilinear, and hyperbolic with respect to the supersonic direction. Making use of the geometric properties of the pointed shock surface together with the Rankine–Hugoniot conditions on the conic shock surface and the boundary condition on the surface of the cone, we obtain a global uniform weighted energy estimate for the nonlinear problem by finding an appropriate multiplier and establishing a new Hardy-type inequality on the shock surface. Based on this, we prove that a multi-dimensional conic shock attached to the vertex of the cone exists globally when the Mach number of the incoming supersonic flow is sufficiently large. Moreover, the asymptotic behavior of the 3-D supersonic conic shock solution, which is shown to approach the corresponding background shock solution in the downstream domain for the uniform supersonic constant flow past the sharp cone, is also explicitly given.     

有激波的一维不定常气流的新的数值解法

解有激波的气体力学问题的数值解法,主要有特征线法和有限差分法两类。特征线法一般能给出高的精度,但当激波很弱、与特征线几乎平行时就需特殊处理^[1]。有限差分法的研究和发展更广,有人工粘性法^[2]、激波捕捉法^[8]、分离奇性法^[1]等处理激波的方案,它们又各有其特殊的技巧和问题。     

Exact solutions to one-dimensional acoustic fields with temperature gradient and mean flow

An exact solution for one-dimensional acoustic fields in ducts in the presence of an axial mean temperature gradient and mean flow is presented in this paper. The analysis is valid for mean Mach numbers such that the square of the mean Mach number is much less than one. The one-dimensional wave equation for ducts with axial mean temperature gradient and mean flow is derived. By appropriate transformations, the wave equation is reduced to an analytically solvable hypergeometric differential equation for the case of a linear mean temperature profile. The developed solution is applied to investigate the dependence of sound propagation in a duct on factors such as temperature gradient and mean flow. The results obtained using the analytical solution compare very well with the numerical results. The developed solution is also compared with an existing analytical solution.     

Weak solution for the Hele-Shaw problem: Viscous shocks and singularities

In Hele-Shaw flows, a boundary of a viscous fluid develops unstable fingering patterns. At vanishing surface tension, fingers evolve to cusp-like singularities preventing a smooth flow. We show that the Hele-Shaw problem admits a weak solution where a singularity triggers viscous shocks. Shocks form a growing, branching tree of a line distribution of vorticity where pressure has a finite discontinuity. A condition that the flow remains curl-free at a macroscale uniquely determines the shock graph structure. We present a self-similar solution describing shocks emerging from a generic (2, 3)-cusp singularity—an elementary branching event of a branching shock graph.     

Time dependent compressible flow about a spherically symmetric polymer in solution

We obtain the general solution of the linear Navier-Stokes equation for time dependent compressible viscous flow about a spherically symmetric polymer molecule. The solution is presented in a covariant form valid in a general cartesian coordinate frame. In the course of deriving the solution we obtain a general decomposition of the unperturbed flow in the absence of the polymer. Our solution generalizes the earlier solution derived by Schmitz and Felderhof for the case of creeping flow.     

Vacillating breathing and tumbling of vesicles under shear flow

The dynamics of vesicles under a shear flow are analyzed analytically in the small deformation regime. We derive two coupled nonlinear equations which describe the vesicle orientation in the flow and its shape evolution. A new type of motion is found, namely, a "vacillating-breathing" mode: the vesicle orientation undergoes an oscillation around the flow direction, while the shape executes breathing dynamics. This solution coexists with tumbling. Moreover, we provide an explicit expression for the tumbling threshold. A rheological law for a dilute vesicle suspension is outlined.     

A micro/nanocasting method for replicating soft micro/nanosurface structures using a dynamic electrical field

We propose a new electric field-induced micro/nanocasting method to replicate soft patterns using micro/nanocasting techniques without pressure. The process uses an alternating current (AC) electrical field and rotation of one electrode, generating a dynamic electrical field that induces electrokinetic flow motion in a dielectric solution (polydimethylsilane, PDMS). We used a lotus leaf as a replication template and characterised the PDMS flow motion to observe the effects of various process parameters (e.g., electrical field strength, rotation speed of an electrode, and electrode shape). The unstable flow motion was significantly dependent on the processing parameters, especially the rotation speed of the electrode. Using the optimised processing conditions, the replication efficiency was about 88%. We believe that this method has potential for fabricating soft micro/nanosized structures.