A NUMERICAL STUDY OF THE FLOW OF GRANULAR MATERIALS BETWEEN TWO VERTICAL FLAT PLATES WHICH ARE AT DIFFERENT TEMPERATURES

The present study considers an assembly of spherical particles, densely packed, between two vertical flat plates which are at different temperatures. A continuum model is used and the flow due to such a temperature difference is investigated. For a fully developed flow of these materials, the governing equations reduce to a system of coupled, non-linear ordinary differential equations. The equations are integrated numerically. In so doing, the equations are discretized using the conventional central finite difference approximation technique and the unknown conditions are assumed to reduce the problem to an initial value problem. After the trial solutions the known boundary conditions at the edge of the integration interval are compared with the correspondent values provided by the trial solutions. If the solutions at this point do not agree with the known boundary conditions the Newton-Raphson method is used to correct the initial guesses and the iteration is repeated. This procedure continues until the solutions converge to the given values. The technique is found to be powerful for this type of application.     

Use of optimization methods to solve heat convection problems with unknown wall boundary conditions

In convection heat transfer problems, it is convenient to solve the governing differential equations by using straightforward marching techniques for numerical integration. However, the boundary layer equations in these heat transfer problems are not of the initial value type even though they are parabolic equations. Consequently marching techniques are not successful until correct initial values are known. A means of determining these initial values is presented herein. Selected optimization methods are utilized in conjunction with marching integration techniques to solve third- and fourth-order ordinary differential equations which have three unknown initial boundary conditions. Two optimization methods, one deterministic and the other non-deterministic in nature, are used both independently and in combination as determined by the particular circumstances. Two applications are used to demonstrate the technique and comparison is made with existing solutions.     

Finite difference analysis for laminar flow heat transfer in concentric annuli with simultaneously developing hydrodynamic and thermal boundary layers

The results of a finite difference analysis are presented for the problem of incompressible laminar flow heat transfer in concentric annuli with simultaneously developing hydrodynamic and thermal boundary layers, the boundary conditions of one wall being isothermal and the other wall adiabatic. This corresponds to the fundamental solution of the third kind according to the four fundamental solutions classified by Reynolds, Lundberg and McCuen¹. Firstly, the hydrodynamic entry length problem, based on the boundary layer simplifications of the Navier–Stokes equations, was solved by means of an extension of the linearized finite difference scheme used previously by Bodia and Osterle² to solve a similar problem between parallel plates. The energy equation is then solved, using the velocity profiles previously obtained, by means of an implicit finite difference technique. The accuracy of the numerical solution was checked by comparing results for the annulus of radius ratio 0.25 with the avaiable solution of Shumway and McEligot³.     

A Galerkin-reproducing kernel method: Application to the 2D nonlinear coupled Burgers' equations

The paper introduces a Galerkin method in the reproducing kernel Hilbert space. It is implemented as a meshless method based on spatial trial spaces spanned by the Newton basis functions in the “native” Hilbert space of the reproducing kernel. For the time-dependent PDEs it leads to a system of ordinary differential equations. The method is used for solving the 2D nonlinear coupled Burgers' equations having Dirichlet and mixed boundary conditions. The numerical solutions for different values of Reynolds number (Re) are compared with analytical solutions as well as other numerical methods. It is shown that the proposed method is efficient, accurate and stable for flow with reasonably high Re in the case of Dirichlet boundary conditions.     

STOKES FLOW THROUGH A SYSTEM OF PARALLEL INFINITE CYLINDERS WITH AXES ORIENTED AT AN ANGLE TO THE DIRECTION Of MEAN FLOW

Abstract

Theoretical models based on Stokes flow of air through a fibrous filter predict a significantly higher pressure drop than experimentally measured values. This discrepancy persists even when the interaction of the flow between) neighboring fibers is accounted for. Various authors have attributed this discrepancy to the inhomogeneity of the fiber distribution within the filter and to the possibility that some fibers are partially orientation in the directon of mean flow. It has been shown that fiber density inhomogeneity does indeed contribute to this discrepancy

In this paper, the effect on the flow and subsequent pressure drop when the fibers are oriented at an angle to the directon of mean flow is studied. The solution of the three dimensional equation for creeping, incompressible flow in a doubly periodic, infinite lattice of infinite circular cylinders when there is a constant mean flow whose direction makes an acute angle with the axes of the cylinders is given. If the volume fraction of fibers is small, the periodic boundary conditions can be replaced by requiring zero vorticity at the outer boundary of an imagined cylindrical cell of fluid surrounding one of the cylinders. The resulting parallel and transverse problems have known solutions and give an approximate solution to the flow through the periodic lattice. The resulting drag is used to compute the dimensionless pressure drop across a filter for several values of the volume fraction of fiber and is compared to the experimentally determined formula of Davies. It is shown that the average drag over a uniform distribution of fiber orientations yields a pressure drop which is significantly closer to the experimental values of Davies than that resulting from strictly transverse flow.     

Wavelet–Galerkin solutions for one-dimensional partial differential equations

In this paper we describe how wavelets may be used to solve partial differential equations. These problems are currently solved by techniques such as finite differences, finite elements and multi-grid. The wavelet method, however, offers several advantages over traditional methods. Wavelets have the ability to represent functions at different levels of resolution, thereby providing a logical means of developing a hierarchy of solutions. Furthermore, compactly supported wavelets (such as those due to Daubechies¹) are localized in space, which means that the solution can be refined in regions of high gradient, e.g. stress concentrations, without having to regenerate the mesh for the entire problem. In order to demonstrate the wavelet technique, we consider the one-dimensional counterpart of Helmholtz's equation. By comparison with a simple finite difference solution to this problem with periodic boundary conditions, we show how a wavelet technique may be efficiently developed. Dirichlet boundary conditions are then imposed, using the capacitance matrix method described by Proskurowski and Widlund² and others. The convergence rates of the wavelet solutions are examined and they are found to compare extremely favourably to the finite difference solutions. Preliminary investigations also indicate that the wavelet technique is a strong contender to the finite element method, at least for problems with simple geometries.     

Two dimensional combined convection in vertical parallel plate ducts,including situations of flow reversal

Numerical solutions are presented for the problem of steady laminar combined convection flows in vertical parallel plate ducts. Axial diffusion is neglected in the analysis and the resulting governing equations, which are of a parabolic nature, are expressed in an implicit finite difference scheme using a vorticity-stream function formulation and are solved using a marching technique. A constant wall temperature boundary condition is used and investigations are restricted to the case Pr = 0.72. A large range of values of the ratio Gr/Re is considered, ?300 ≤ Gr/Re ≤ 70, and comparisons are made with the case of pure forced convection. For large values of the ratio |Gr/Re| reverse flow occurs in the duct. A modification to the standard marching technique is introduced and complete solutions are achieved for these situations for the first time. Results are presented in terms of velocity profiles, Nusselt numbers, friction factors and temperature distributions.     

On extra boundary condition in the stagnation point flow of a second grade fluid

The two dimensional stagnation point flow of a second grade fluid is considered. The flow is governed by a boundary value problem in which the order of differential equations is one more than the number of available boundary conditions. It is shown that without augmenting the boundary conditions at infinity it is possible to obtain a numerical solution of the problem for all values of K, where K is the dimensionless viscoelastic fluid parameter. The numerical results using the algorithm foreshadow an asymptotic behavior for large K. The asymptotic solution is derived up to terms of O(K⁻¹). Perturbation solutions are also obtained up to the terms of O(K²). Finally an approximate solution is developed, based on stretching of the independent variable and minimizing the residual of the differential equation in the least square sense. All these solutions are compared with the exact numerical solution and the appropriate conclusions are drawn.     

A note on a plane creeping corner flow

Abstract

A simple method of similarity transformation is formulated to analyze a two‐dimensional creeping corner flow. By this peculiar transformation, governing equations for the plane velocity are reduced to a pair of ordinary differential equations. With a particular selection of appropriate boundary conditions, the field variables of velocity, pressure, vorticity, and stream function are obtained analytically. A special case with constant velocity at one boundary is explored. The salient characteristics of this example are used to compare with previous investigations. The present study shows that both approaches provide exactly the same solutions. A very interesting feature is that the velocity components in the coordinate system are independent of the radial direction.     

An interface constitutive modeling problem for compressible flows with mass transfer

The complete solution of a two-phase flow problem requires the solution of appropriate partial differential equations (PDEs) of mass, momentum, and energy in the region occupied by the vapor and in the region occupied by the liquid. The moving interface between each phasic region requires the specification of additional interface conditions. These additional conditions are jump conditions imposed by the mass, momentum, and energy balances at the interface and additional interface ‘constitutive’ equations. These additional interface constitutive equations can also be thought of as internal interface boundary conditions that must be imposed on the PDEs on each side of the interface to secure a well posed problem.It is well known that the characteristic equations in any hyperbolic system give a complete picture of the required boundary conditions or interface shock conditions. In this paper, the limiting form of a set of characteristic equations will be used to determine the nature of the phasic interface conditions that are required in a compressible, viscous, conducting fluid at a mass transfer interface. The analysis will show that the traditional interface modeling constitutive equations are insufficient in number and hence lead to multiple solutions and therefore to an ill-posed problem. The source of the insufficient number of interface conditions will be discussed.     

Solution of magnetohydrodynamic flow problems using the boundary element method

A boundary element solution is implemented for magnetohydrodynamic (MHD) flow problem in ducts with several geometrical cross-section with insulating walls when a uniform magnetic field is imposed perpendicular to the flow direction. The coupled velocity and induced magnetic field equations are first transformed into uncoupled inhomogeneous convection–diffusion type equations. After introducing particular solutions, only the homogeneous equations are solved by using boundary element method (BEM). The fundamental solutions of the uncoupled equations themselves (convection–diffusion type) involve the Hartmann number (M) through exponential and modified Bessel functions. Thus, it is possible to obtain results for large values of M (M≤300) using only the simplest constant boundary elements. It is found that as the Hartmann number increases, boundary layer formation starts near the walls and there is a flattening tendency for both the velocity and the induced magnetic field. Also, velocity becomes uniform at the center of the duct. These are the well-known behaviours of MHD flow. The velocity and the induced magnetic field contours are graphically visualized for several values of M and for different geometries of the duct cross-section.     

A numerical study of steady plane granular chute flows using the Jenkins–Savage model and its extension

The granular flow model proposed by Jenkins and Savage and extended by us is used here to construct numerical solutions of steady chute flows thought to be typical of granular flow behaviour. We present the governing differential equations and discuss the boundary conditions for two flow cases: (i) a fully fluidized layer of granules moving steadily under rapid shear and (ii) a fluidized bottom-near bed covered by a rigid slab of gravel in steady motion under its own weight. The boundary value problem is transformed into a dimensionless form and the emerging system of non-linear ordinary differential equations is numerically integrated. Singularities at the free surface and (in one case) also at an unknown point inside the solution interval make the problem unusual. Since the non-dimensionalization is performed with the maximum particle concentration and the maximum velocity, which are both unknown, these two parameters also enter the formulation of the problem through algebraic equations. The two-point boundary value problem is solved with the aid of the shooting method by satisfying the boundary conditions at the end of the soluton interval and these normalizing conditions by means of a minimization procedure. We outline the numerical scheme and report selective numerical results. The computations are the first performed with the exact equations of the Jenkins–Savage model; they permit delineation of the conditions of applicability of the model and thus prove to be a useful tool for the granular flow modeller.     

ON THE INADMISSIBILITY OF SEPARATION OF VARIABLES SOLUTIONS IN LINEAR ANISOTROPIC VISCOELASTICITY

SUMMARY

The field and constitutive equations as well as the boundary conditions of linear anisotropic viscoelasticity are examined for possible solutions which are separable into products of time and spatial functions. It is shown that under no circumstances other than for constant Poisson's ratios are such separation of variables solutions admissible, even though under these or some additional restrictive conditions they can exist in isotropic viscoelasticity.     

Application of the differential transformation method to the solutions of Falkner-Skan wedge flow

Summary. This paper adopts the differential transformation method to investigate the velocity and shear stress fields associated with the Flakner-Skan boundary-layer problem. A group of transformations are used to reduce the boundary value problem into a pair of initial value problems, which are then solved by means of the differential transformation method. The proposed method yields closed series solutions of the boundary layer equations, which can then be calculated numerically. Numerical results for the dimensionless velocity and the shear stress profiles of the wedge flow are presented graphically for different values of . It is found that the current results are in good agreement with those provided by other numerical methods. Therefore, the proposed method is proven to be an effective scheme for the solution of nonlinear boundary-layer problems.     

基于动态近似边界条件的气动弹性数值模拟

发展了一种利用欧拉方程计算非定常气动力的数值方法,通过在固定物面边界上满足动态近似边界条件计算出非定常气动力,避免了在每个时间步重新生成网格或需用动网格技术进行网格变形处理过程,提高了计算效率。运用这种方法计算了一系列非定常气动力算例,并与非结构动网格准确边界条件下的欧拉方程解和实验数据进行了比较,进一步分析了翼型俯仰角和马赫数对非定常气动力相对误差的影响。将气动力解算器与结构方程耦合进行气动弹性数值模拟,计算了跨音速具有S型颤振边界的二元气动弹性标准算例-Isogaiwing。算例结果表明,利用动态近似边界条件的欧拉方程具有简便、高效的特点,并能在小振幅情况下得到与精确边界条件精度相当的非定常流场解,还可以用于气动弹性分析。     

Flow of a third grade fluid through a porous flat channel

The steady, laminar flow of a third grade fluid through a porous flat channel is considered, when the rate of injection of the fluid at one boundary is equal to the rate of suction at the other boundary. The flow is governed by a non-linear boundary value problem (BVP) in which the order of the differential equation is three, but only two boundary conditions are available. Two numerical schemes are developed to obtain the appropriate solution of the BVP. In the first scheme the dilemma is resolved by assuming that the solution is analytical in the neighborhood of K=0, where K is the non-dimensional viscoelastic fluid parameter. This scheme is practical to use only up to certain values of T, the third grade fluid parameter. The second scheme allows arbitrary values of T, but is restricted to small values of K and R, the cross-flow Reynolds number.A perturbation solution valid for small values of T is also derived. Finally two approximate solutions, based on Collatz’ iterative scheme, but with different starting trial solutions are obtained. A comparison is made of the results computed by using various methods and appropriate conclusions are drawn.     

The indirect design of fluid channels using a global variational method with trial functions not satisfying the prescribed boundary conditions

The indirect design problem for compressible fluid flow in a channel determines the physical shape of the channel from the prescribed flow on the channel boundary. In this paper, a global variational method for the solution of this problem is described. The non-linear equations are solved by iteration; at each step of this iteration the solution to the equations (which are now linear) is found by a global variational method. The variational method differs from conventional methods in that the trial function does not satisfy the prescribed boundary conditions, but the method reproduces the conditions on the boundary. Results are presented for various physical quantities associated with the channel. Some comments are made on the extensions of these methods to supersonic flow and to the realistic cascade design problem.     

Nodal velocity derivatives of finite element solutions: The FiND method for incompressible Navier–Stokes equations

This paper presents the development and application of the finite node displacement (FiND) method to the incompressible Navier–Stokes equations. The method computes high‐accuracy nodal derivatives of the finite element solutions. The approach imposes a small displacement to individual mesh nodes and solves a very small problem on the patch of elements surrounding the node. The only unknown is the value of the solution ( u , p) at the displaced node. A finite difference between the original and the perturbed values provides the directional derivative. Verification by grid refinement studies is shown for two‐dimensional problems possessing a closed‐form solution: a Poiseuille flow and a flow mimicking a boundary layer. For internal nodes, the method yields accuracy slightly superior to that of the superconvergent patch recovery (SPR) technique of Zienkiewicz and Zhu (ZZ). We also present a variant of the method to treat boundary nodes. The local discretization is enriched by inserting an additional mesh point very close to the boundary node of interest. Computations show that the resulting nodal derivatives are much more accurate than those obtained by the ZZ SPR technique. Copyright © 2008 John Wiley & Sons, Ltd.     

两类非线性拟抛物方程解的有限时间爆破和长时间行为

本文研究两类非线性拟抛物方程的初边值问题,它们包括了GBBM方程,Sobolev—Galpern方程,多维粘性扩散方程及半线性拟抛物方程作为特殊情形。对这两类方程我们分别采用积分估计方法和特征函数法证明了,当方程的非线性项满足某些条件时,问题的解按时间t的指数形式衰减为零,而当方程的非线性项满足另外某些条件时,问题的解在有限时间内爆破。本文从实质上改进和推广了已有结果。