Viscous Flow Equations Research Articles

酸化物セラミックスの固相アモルファス化と放電焼結

Full density powder processing, which consists of reaction ball milling and the instrumented electric discharge consolidation method, is proposed for the development of bulky oxide ceramic materials with nanoscale grain. We find that mechanical alloying of the powder mixture, ZrO2-20 mass%Al2O3 can produce solid state amorphization, when the conditions for planetary reaction ball milling yield the optima. When electric discharge consolidation is combined with high-rate heating of the mechanically alloyed amorphous ceramic powder, we obtain fully dense ZrO2-20 mol%Al2O3 at 1347 K when a pressure of 100 MPa is applied. The densification in the supercooled liquid of ZrO2-20 mol%Al2O3 is fairly well expressed by an Arrhenius type equation of viscous flow: η=η0exp(H⁄kT). The process viscosity (η) is below 1012 Pa·s at the glass temperature and the activation energy (H) decreases from 337 to 184 kJ·mol−1 with increasing current applied to the graphite die. For cubic ZrO2-20 mol%Al2O3 with a monoclinic phase, synthesized via crystallization at 1366 K, the average grain size is estimated at 19.6 nm from X-ray line broadening. Vickers’ pyramidal hardness is 1051 DPN when the initiation of cracks at the corners of indentation is avoided.

On existence of global solutions to the Navier-Stokes equations for compressible and viscous flows on the surface of a sphere

Following tecniques proposed by A. V. Khazikhov and V. A. Weigant in 1995, we prove the global, with respect to time, existence and uniqueness of the solution to the Navier-Stokes equations for a compressible, viscous and barotropic fluid which moves on the surface of a sphere. In obtaining the main estimates we make use of the Hodge decomposition and the generalized potential theory due to K. Kodaira.

On the effective equations of a viscous incompressible fluid flow through a filter of finite thickness

On the effective equations of a viscous incompressible fluid flow through a filter of finite thickness

Discretized integral hydrodynamics

Using an interpolant form for the gradient of a function of position, we write an integral version of the conservation equations for a fluid. In the appropriate limit, these become the usual conservation laws of mass, momentum and energy. We also discuss the special cases of the Navier-Stokes equations for viscous flow and the Fourier law for thermal conduction in the presence of hydrodynamic fluctuations. By means of a discretization procedure, we show how these equations can give rise to the so-called "particle dynamics" of Smoothed Particle Hydrodynamics and Dissipative Particle Dynamics.

Arbitrary Motions of a Viscous Incompressible Liquid in a Finite Region

By using an entirely constructive analysis it is shown that there exist solutions of the viscous flow equations in which two of the Cartesian components of velocity are arbitrary and the third component can be constructed uniquely by the application of integrability conditions. The solutions are in general defined in a finite region of the fluid and can be matched onto a well‐defined deterministic solution of the Navier–Stokes equations such that all of the relevant physical quantities are continuous at the interface.

Steady plane flow of ice sheets over large-slope topography

The reduced model for a large ice sheet flow is uniformly valid when the bed topography is flat or has slopes relative to the horizontal no greater than order ϵ, where ϵ2 is a very small dimensionless viscosity based on the geometry and flow parameters. Real beds will have much greater slopes, of order unity in places. Large-scale numerical simulations of ice sheet evolution are currently based on the reduced model, and their validity when significant basal topography is present, the common situation, is not certain. Numerical solution of the full slow viscous flow equations on the unknown sheet domain, where the surface is stress-free and subject to a kinematic accumulation/melt condition, is much more difficult. No established algorithms exist at present. Here attention is focused on the most simple configuration of steady plane flow of an incompressible linearly viscous fluid, assuming isothermal conditions so that the temperature dependent rate factor is constant. It allows the application of complex potential representations for a solution of the full equations superposed on a flat bed reduced solution in order to satisfy non-slip conditions on a humped bed. A particular class of conformal mappings of a humped bed contour onto a flat bed result in boundary conditions which can be solved exactly by Cauchy integral methods to yield explicit integral representations for the potentials and related physical variables. Evaluation is necessarily numerical, but accurate. This method is applied to various humped contours with amplitude increasing from zero to determine when the reduced solution, ignoring the bed topography, starts to lose accuracy, and how far from the humped section this error persists. It is demonstrated that bed humps do influence the velocity field. © 1998 John Wiley & Sons, Ltd.

Computing the flow around a moving profile by numerically integrating the Navier-Stokes equations

A vorticity velocity formulation is proposed for the solution of the equations for viscous flow around a moving profile. A non-inertial reference frame is used and the velocities are computed from a Poincaré integral formula. The studies are directed towards the need to understand helicopter blade aerodynamics. Worked examples are given which validate the method and programme for laminar flows, at least for low Reynolds numbers. © 1998 John Wiley & Sons, Ltd.

Quantitative analysis on boundary sliding and its accommodation mode during superplastic deformation of two-phase Ti-6Al-4V alloy

A study has been made to investigate boundary sliding and its accommodation mode with respect to the variation of grain size and α/β volume fraction during superplastic deformation of a two-phase Ti-6Al-4V alloy. A load relaxation test has been performed at 600 °C and 800 °C to obtain the flow stress curves and to analyze the deformation characteristics by the theory of inelastic deformation. The results show that grain matrix deformation (GMD) is found to be dominant at 600 °C and is well described by the plastic state equation. Whereas, at 800 °C, phase/grain boundary sliding (P/GBS) becomes dominant and is fitted well with the viscous flow equation. The accommodation mode for fine-grained microstructures (3 µm) well agrees with the isostress model, while that for large-grained structures (11 µm) is a mixed mode of the isostress and isostrain-rate models. The sliding resistance analyzed for the different boundaries is lowest in the α/β boundary, and increases on the order of α/β≪α/α≈β/β, which plays an important role in controlling the superplasticity of the alloys with various α/β phase ratios.

Analogy between heat and momentum transfer

Analogy between heat and momentum transfer—the concepts as developed by O. Reynolds and L. Prandtl—are based on certain hypotheses about the mechanism of turbulent transport phenomena. Therefore, heat transfer equations based on these concepts are semi-empirical. In addition, a new heat transfer equation is presented which applies to non-developed turbulent flow e.g. in plate heat exchangers. This equation can be derived from the classical differential equations for viscous flow and heat conduction which are not bound to any modeling concept of turbulent transport phenomena whatsoever, and may be taken as rigorous.

The thin viscous flow equation in higher space dimensions

We prove local integral (entropy) estimates for nonnegative solutions of the fourth-order degenerate parabolic equation $$ u_t+ \div (u^n\nabla\Delta u)=0 $$ in space dimensions two and three. These estimates enable us to show that solutions have finite speed of propagation if $n\in(\frac 18,2)$ and that the support cannot shrink if the growth exponent $n$ is larger than $3/2$. In addition, we prove decay estimates for solutions of the Cauchy problem and a growth estimate for their support.

Operator splitting and upwinding for the Navier-Stokes equations

We present an operator splitting scheme for the unsteady Navier-Stokes equations for incompressible viscous fluid flow. Like other operator splitting methods applied to these equations, the difficulties associated with the nonlinearity and the incompressibility condition are decoupled. At each time step we obtain two subproblems of Stokes type and a linear one of elliptic type. The linear problem gives us uncoupled scalar problems of transport type; then, we may take advantage of well known upwind techniques for such kind of problems in order to handle large Reynolds numbers flow with coarse meshes. To show the efficiency of the scheme we report numerical results up to Reynolds numbers Re=4000 obtained with very coarse meshes.

A wave equation approach to the numerical solution of the Navier-Stokes equations for incompressible viscous flow

In this Note we describe a novel method for the solution of the Navier-Stokes equations for incompressible viscous fluids. This method, which can be viewed as an alternative to the methods of characteristics, takes advantage of a time discretization by operator splitting to decouple incompressibility-diffusion from advection. The incompressibility-diffusion steps can be treated by classical Stokes solvers. Concerning the advection steps, thanks to the incompressibility of the advecting field, we can replace the corresponding transport equations by second order in time wave equations, which are much easier to solve numerically despite the fact that they are associated to degenerate elliptic operators. Numerical experiments confirm the good computational properties of the new method.

DISCUSSION OF NUMERICAL DEFICIENCY OF APPLYING A PARTIALLY WEIGHTED UPWIND FINITE-ELEMENT MODEL TO INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

The streamline upwind technique is extended to quadratic dements to analyze incompressible and viscous flow equations cast in the steady state. The biased part of the weighting Junctions is devised to achieve a nodally exact discretized one-dimensional equation, with an emphasis on grid nonuniformity. Two classes of upwinding finite-element models are considered. Our primary goal is to address the deficiency of the partially weighted finite-element model. Assessment is made of the stability and accuracy of the schemes devised. The integrity of the weighting functions chosen and the finite-element models considered is demonstrated analytically, and their performance is assessed systematically.

Numerical study of three‐dimensional incompressible thermal flows in complex geometries

In part I of this study, a three‐dimensional finite difference iterative solver capable of handling the coupled Navier‐Stokes and energy equations for incompressible viscous flows was described and validated with two‐ and three‐dimensional benchmarks. Part II describes the results of the computational study of two distinct complex geometries: 1) two‐dimensional and three‐dimensional natural convection in cavity whose surface is cooled while two internal blocks are heated; 2) two‐dimensional and three‐dimensional natural convection in the region defined by two interconnected cavities of different sizes which are differentially heated. All computations have been performed for a Prandtl number of 1.0, and different values of the Rayleigh number ranging between 103 and 106 depending on the problem. In the first problem, three‐dimensional effects in the top region of the cavity trap fluid in vortices near the top of the heated blocks adversely affecting heat transfer in the region while enhancing it in the region between the two heated blocks. In the second problem, the sudden expansion of fluid as it leaves the top cavity and enters the bottom one generates three‐dimensional wakes in the bottom cavity that enhance the convective heat transfer across the system walls near them. These studies tend to suggest that three‐dimensional effects play a very important role in the enhancement of convective heat transfer in complex geometries, especially at higher Rayleigh numbers.

Numerical study of three‐dimensional incompressible thermal flows in complex geometries

In part I uses an iterative point successive over‐relaxation (PSOR) finite difference scheme to solve the coupled unsteady Navier‐Stokes and energy equations for incompressible, viscous and laminar flows in their primitive variable form. Presents the details concerning the derivation of the solution scheme, as well as details on its computer implementation. For validation purposes, includes the results of the two‐dimensional and three‐dimensional benchmark problem of natural convection in a cavity with differentially heated vertical walls. Benchmark computations have been performed for a Prandtl number of 0.71, and different values of the Rayleigh number ranging between 103 and 106 depending on the problem. By comparison with other approaches in the literature, the scheme has been found to be accurate even for large Rayleigh numbers.

Unsteady Aerodynamic Flow Phenomena in a Transonic Compressor Stage

A three-dimensional unsteady, viscous aerodynamic analysis has been developed for the flow inside a transonic, high-through-flow, single stage compressor. The compressor stage is comprised of a low-aspect-ratio rotor and a closely coupled stator. The analysis is based on a numerical method for solving the three-dimensional Navier-Stokes equation for unsteady viscous flow through multiple turbomachinery blade rows. The method solves the fully three-dimensional Navier-Stokes equation with an implicit scheme. A two-equation turbulence model with a low-Reynolds-number modification is applied for the turbulence closure. A third-order accurate upwinding scheme is used to approximate convection terms while a second-order accurate central difference scheme is used for the discretization of the viscous terms. A second-order accurate scheme is employed for the temporal discretization. The numerical method is applied to study the unsteady flow field inside a transonic, high-through-flow, axial compressor stage. The numerical results are compared with available experimental data.

Finite element methods in the simulation and analysis of intracranial blood flow

This paper presents an introduction to the use of finite element methods in the simulation and analysis of intracranial blood flow and lays the foundation for more detailed clinically oriented studies. An overview of finite element theory is provided and includes the formulation of both the continuous and discrete equations of viscous fluid flow. A discussion of appropriate assumptions and boundary conditions governing arterial blood flow is presented. Two-dimensional, rigid-walled models are developed for flow in a straight artery, a 90 curved artery and a bifurcated artery. For each model, a description of the finite element mesh, numerical solution and computational results are presented. This paper is the first in a series which will detail computational analysis of the relationships between pressure, velocity, development of intracranial aneurysms and therapeutic approaches to aneurysm management. The goals of this research are to investigate the fluid dynamics that arise as a result of pulsatile flow in the arteries of the circle of Willis, relate these hemodynamics to the formation of aneurysms, develop a computational understanding of the effects of various therapies on blood flow related to aneurysms, and to develop and utilize patient specific computer simulations for treatment planning. [Neural Res 1997; 19: 174–186]

Powder processing and mechanical characteristics of fully dense nano-intermetallics

This article describes the basic idea that process control of electric discharge consolidation makes it possible to obtain fully dense high-strength nano-intermetallics. This powder processing provides sensors of pulse discharging and densification for making a process model and a control methodology. The discharging of rectangular pulses, monitored by voltage wave, is characterized by spark and arc, resulting in the neck formation between spherical amorphous particles. Then, densification of the amorphous Ti 50Al 50 powder under an electrical field is expressed by an Arrhenius-type equation of viscous flow: η p=η p0(I)exp{H(I)/kT}, where η p is process viscosity. A mechanism of bulk transport via enhanced viscous flow through artificial multiple necks accounts for rapid pore-free consolidation. For ductile nano-intermetallic Ti 50Al 50 synthesized via crystallization, the highest specific strength is 790 MPa Mg m −3 at the γ grain size of 17 nm; and the softening temperature is comparable to those of cast TiAl and Ni-base superalloy.

Anomalous behavior during leveling of thin coating layers with surfactant

Our recently-published linear analysis [Schwartz et al., Langmuir 11, 3690 (1995)] demonstrated that an initially rippled thin layer of Newtonian liquid with uniformly distributed surfactant may level in unexpected ways. While the presence of surfactant will, in general, slow the rate of leveling compared to that of a perfectly clean system, there was shown to exist a realistic parameter range where increasing, rather than reducing, the amount of surfactant present will hasten leveling. Here, for the two-dimensional problem, we investigate the importance of nonlinearity though numerical solution of (i) the unsteady lubrication form of the evolution equations with surfactant, and (ii) finite-element solution of the exact governing equations for slow viscous flow. Confirmation of the linear results is demonstrated and quantitative discrepancy only appears for large-amplitude and short-wavelength ripples. Surface tension gradient driven flow explains the anomalies; for moderate surfactants, the surface quickly ‘hardens,’ leading to a decay rate of one-quarter of the clean-surface rate, while for weak surfactants, leveling proceeds to a plateau level which decays much slower than the hard-surface result.

Time-dependent viscous deformation of a drop in a rapidly rotating denser fluid

Viscous stretching of a cigar-shaped drop due to the centrifugal pressure field in a surrounding rapidly rotating denser fluid is analysed. Scaling arguments are used to examine the various contributions to the viscous stresses resisting deformation, and a number of asymptotic regimes are identified which are delineated by the relative magnitudes of the aspect ratio, the viscosity ratio and unity. These asymptotic regimes may usefully be described as the bubble, pipe, sliding-rod and toffee-strand limits. Detailed analysis based upon a slenderness assumption combined with an integral representation of Stokes equations is used to derive evolution equations for the shape of the drop as a function of time in the different regimes. In the limit that interfacial-tension effects are negligible, similarity solutions are developed in which the length of the drop is found to increase ast2/5,t1/4, (tlnt)1/4andt. The analytical results are in good agreement with numerical simulations based upon a boundary-integral solution to the full viscous flow equations.