两相流体非线性渗流模型及其应用

基于三参数非线性渗流运动定律、质量守恒定律及椭圆渗流的概念,建立了低渗透介质中两相流体椭圆非线性渗流数学模型,运用有限差分法与外推法求得了其解,导出了两相流体椭圆非线性渗流条件下油井见水前后开发指标的计算公式,进行了实例分析。结果表明:非线性渗流对含水饱和度分布影响较大;非线性渗流使得水驱油推进速度比线性渗流的快,使油井见水时间提前,使得石油开发指标变差;非线性渗流使得同一时刻的压差比线性渗流的大,使石油开发难度加大。这为低渗油藏垂直裂缝井开发工程提供了科学依据。     

The mechanisms of ground surface subsidence above compacting multiphase reservoirs and their analysis by the finite element method

An analysis of multiphase flow through a compacting porous medium is investigated in this paper. The nonlinear partial differential equations governing the flow regimes are derived and a method of calculating the effective stress gradients is given. This method involves a local averaging technique to model the interaction of two immiscible fluid phases on a small volume of the deforming porous medium. The combined stress and flow problem is then solved using the finite element method where the flow is confined to a submesh of the whole region. A simulation is made of a typical vertical cross-section of the Forties Field.     

Global nonlinear stability of convection in a heat generating fluid filled channel with a moving boundary

The energy method is employed to investigate the stability of a steady convective flow in a heat generating fluid arising due to the combined effect of buoyancy, shear and pressure gradient. By introducing a suitable generalized energy functional and using energy inequalities sufficient conditions for the existence of such a flow are found. An analysis through the variational principles is then made to find sharper limits for nonlinear stability. Comparisons are made with linear results in the literature and it is shown that the linear theory fails to capture the physics of the onset of secondary flow.     

基于不动点方法求解非线性Falkner-Skan流动方程

Falkner-Skan流动方程描述绕楔面的流动,该方程具有很强的非线性.首先通过引入变换式,将原半无限大区域上的流动问题转化为有限区间上的两点边值问题.接着基于泛函分析中的不动点理论,采用不动点方法求解两点边值问题从而得到Falkner Skan流动方程的解.最后将不动点方法给出的结果和文献中的数值结果相比较,发现不动点方法得到的结果具有很高的精度,并且解的精度很容易通过迭代而不断得到提高.表明不动点方法是一种求解非线性微分方程行之有效的方法.     

Application of the Galerkin method to the formulation of a three-dimensional nonlinear hydrodynamic numerical sea model

A mathematical formulation is presented for solving the three-dimensional nonlinear hydrodynamic equations, using the Galerkin method with an arbitrary set of basis functions.An explicit time splitting method is used to integrate these equations through time. The time splitting method is formulated in such a way that the advective terms, which are computationally expensive to evaluate, are integrated with a longer time step than the linear terms. The length of the time step used to integrate the linear terms is determined by the propagation speed of the gravity waves. The paper demonstrates that using this time splitting method an accurate and computationally economic solution of the full three-dimensional equations is possible.Numerical results are presented for the nonlinear seiche motion in a one-dimensional basin, and for the three-dimensional wind induced flow in a closed rectangular basin, using basis sets of cosine functions, Chebyshev polynomials and Gram-Schmidt orthogonalized polynomials.     

Inertia Effects in Squeeze Flows of Viscoplastic Fluids

A squeeze flow of a viscoplastic fluid through a narrow clearance between two coaxial surfaces of revolution is considered. The problem is described by boundary-layer equations. With the use of the method of integral approaches, formulas for the pressure distribution are obtained. Generally, the flow of viscoplastic fluids given by the nonlinear Shulman model is considered. The flows of viscoplastic fluids given by the Herschel, Bulkley, Bingham, Ostwald-de Waele, and Newton models are discussed in detail. Numerical examples of pressure distributions in the clearance between parallel disks are presented.     

Bifurcations and chaos of the nonlinear viscoelastic plates subjected to subsonic flow and external loads

The subharmonic bifurcations and chaotic motions of the nonlinear viscoelastic plates subjected to subsonic flow and external loads are studied by means of Melnikov method. The critical conditions for the occurrence of chaotic motions are obtained. The chaotic features on the system parameters are discussed in detail. The conditions for subharmonic bifurcations are also obtained. For the system with no structural damping, chaotic motions can occur through infinite subharmonic bifurcations of odd orders. Furthermore, we confirm our theoretical predictions by numerical simulations. The theoretical results obtained here can help us to eliminate or suppress large nonlinear vibrations and chaotic motions of the nonlinear viscoelastic plates. Based on Melnikov method, complex dynamical behaviors of the nonlinear viscoelastic plates can be controlled by modifying the system parameters.     

饱和粘土非线性渗流规律与径向固结

以多孔介质等效渗流概念得出粘土平均孔隙微尺度数量级范围在0.01 μm至0.1μm,与测试结果一致．实验结果表明饱和粘土微尺度孔隙渗流为非线性流．理论推导表明固液界面作用与渗透率平方根或孔隙半径成反比,固液界面相互作用是导致饱和粘土非线性渗流的重要原因．提出了精确描述饱和粘土微尺度孔隙非线性渗流基本规律的数学模型,其参数量纲明确,物理意义清楚．建立了饱和粘土非线性渗流新定律,统一描述了从较低到较高水力梯度全过程渗流曲线特征,达西定律是其特例．基于新定律,建立了饱和粘土非线性渗流定流量径向固结数学模型．以粘性边界层思想与稳态依序替换法及积分方程法,导出了粘土非线性渗流平均质量守恒方程及活动边界运动方程,给出了饱和粘土非线性渗流超孔隙水压力分布公式与平均固结度计算公式,获得了粘土层压力分布规律和平均固结度随时间变化规律．结果表明:饱和粘土非线性渗流使活动边界运动速度减小．研究结果为粘土地质工程与岩土工程应用提供了新的科学依据．达西渗流径向固结计算是新的非线性渗流固结计算的特例．     

Mixed convection in the stagnation-point flow of a Maxwell fluid towards a vertical stretching surface

In the present analysis, we study the steady mixed convection boundary layer flow of an incompressible Maxwell fluid near the two-dimensional stagnation-point flow over a vertical stretching surface. It is assumed that the stretching velocity and the surface temperature vary linearly with the distance from the stagnation-point. The governing nonlinear partial differential equations have been reduced to the coupled nonlinear ordinary differential equations by the similarity transformations. Analytical and numerical solutions of the derived system of equations are developed. The homotopy analysis method (HAM) and finite difference scheme are employed in constructing the analytical and numerical solutions, respectively. Comparison between the analytical and numerical solutions is given and found to be in excellent agreement. Both cases of assisting and opposing flows are considered. The influence of the various interesting parameters on the flow and heat transfer is analyzed and discussed through graphs in detail. The values of the local Nusselt number for different physical parameters are also tabulated. Comparison of the present results with known numerical results of viscous fluid is shown and a good agreement is observed.     

Dynamic response of pulsatile flow of blood in a stenosed tapered artery

The aim of this paper is to throw some light on the rheological study of pulsatile blood flow in a stenosed tapered arterial segment. Arterial wall is considered to be rigid and flexible separately for improving the similarity to the in vivo situation. The streaming blood is considered to be Newtonian. The governing nonlinear equations of motion are sought using the well‐known stream function‐vorticity method and are solved numerically by finite difference technique. Important rheological parameters, such as axial velocity component, wall shear stress, and flow separation region are estimated in the neighborhood of the stenosis. Effects of stenosis height, vessel tapering, and wall flexibility on the blood flow are investigated properly and are explained in detail through their graphical representations.     

Effects of heat transfer and space porosity on peristaltic flow in a vertical asymmetric channel

This article discusses the effect of heat transfer on the peristaltic flow of a Newtonian fluid through a porous space in a vertical asymmetric channel. Long wavelength approximation is used to linearize the governing equations. The system of the governing nonlinear PDE is solved by using the perturbation method. The solutions are obtained for the velocity and the temperature fields. The flow is investigated in a wave frame of reference moving with velocity of the wave. Numerical calculations are carried out for the pressure rise, frictional forces, and the features of the flow and temperature characteristics are analyzed by plotting graphs and discussed in detail. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

MHD stagnation-point flow of an upper-convected Maxwell fluid over a stretching surface

The present analysis comprises the steady two-dimensional magnetohydrodynamic flow of an upper-convected Maxwell fluid near a stagnation-point over a stretching surface. The governing non-linear partial differential equation for the flow are reduced to an ordinary differential equation by using similarity transformations. The analytic solution of nonlinear system is constructed in the series form using Homotopy analysis method. Convergence of the obtained series is discussed explicitly. The effects of the sundry parameters on the velocity profile is shown through graphs. The values of skin-friction coefficient for different parameters is tabulated.     

A numerical scheme for multi-point special boundary-value problems and application to fluid flow through porous layers

In this paper we propose a numerical scheme based on finite differences for the numerical solution of nonlinear multi-point boundary-value problems over adjacent domains. In each subdomain the solution is governed by a different equation. The solutions are required to be smooth across the interface nodes. The approach is based on using finite difference approximation of the derivatives at the interface nodes. Smoothness across the interface nodes is imposed to produce an algebraic system of nonlinear equations. A modified multi-dimensional Newton’s method is proposed for solving the nonlinear system. The accuracy of the proposed scheme is validated by examples whose exact solutions are known. The proposed scheme is applied to solve for the velocity profile of fluid flow through multilayer porous media.     

Nonlinear stability characterization of thin Newtonian film flows traveling down on a vertical moving plate

The paper presents both the linear and nonlinear stability theories for the characterization of thin Newtonian film flows traveling down along a vertical moving plate. The linear model is first developed to characterize the flow behavior. After showing the inadequacy of the linear model in representing certain flow characteristics, the nonlinear kinematics model is then developed to represent the system. The long-wave perturbation method is employed to derive the generalized kinematic equations with free film surface condition. The linear model is solved by using the normal mode method for three different, namely, the quiescent, up-moving and down-moving, moving conditions. Subsequently, the elaborated nonlinear film flow model is solved by the method of multiple scales. The modeling results clearly indicate that both subcritical instability and supercritical stability conditions are possible to occur in the film flow system. The effect of the down-moving motion of the vertical plate tends to enhance the stability of the film flow.     

Non-linear peristaltic transport of a second-order fluid through a porous medium

The present paper investigates phenomena brought about into the classic peristaltic mechanism by inclusion of non-Newtonian effects through a porous space in a channel. The peristaltic motion of a second-order fluid through a porous medium was studied for the case of a planar channel with harmonically undulating extensible walls. The system of the governing nonlinear PDE is solved by using the perturbation method to second-order in dimensionless wavenumber. The analytic solution has been obtained in the form of a stream function from which the axial pressure gradient has been derived. The flow is investigated in a wave frame of reference moving with velocity of the wave. Numerical calculations are carried out for the pressure rise and frictional force. The features of the flow characteristics are analyzed by plotting graphs and discussed in detail.     

Numerical modeling of turbulence-induced interfacial instability in two-phase flow with moving interface

The present paper introduces a new interfacial marker-level set method (IMLS) which is coupled with the Reynolds averaged Navier–Stokes (RANS) equations to predict the turbulence-induced interfacial instability of two-phase flow with moving interface. The governing RANS equations for time-dependent, axisymmetric and incompressible two-phase flow are described in both phases and solved separately using the control volume approach on structured cell-centered collocated grids. The transition from one phase to another is performed through a consistent balance of kinematic and dynamic conditions on the interface separating the two phases. The topological changes of the interface are predicted by applying the level set approach. By fitting a number of interfacial markers on the intersection points of the computational grids with the interface, the interfacial stresses and consequently, the interfacial driving forces are easily estimated. Moreover, the normal interface velocity, calculated at the interfacial markers positions, can be extended to the higher dimensional level set function and used for the interface advection process. The performance of linear and non-linear two-equation k–ε turbulence models is investigated in the context of the considered two-phase flow impinging problem, where a turbulent gas jet impinging on a free liquid surface. The numerical results obtained are evaluated through the comparison with the available experimental and analytical data. The nonlinear turbulence model showed superiority in predicting the interface deformation resulting from turbulent normal stresses. However, both linear and nonlinear turbulence models showed a similar behavior in predicting the interface deformation due to turbulent tangential stresses. In general, the developed IMLS numerical method showed a remarkable capability in predicting the dynamics of the considered two-phase immiscible flow problems and therefore it can be applied to quite a number of interface stability problems.     

New explicit approximate solution of MHD viscoelastic boundary layer flow over stretching sheet

In this paper, the study the momentum and heat transfer characteristics in an incompressible electrically conducting non‐Newtonian boundary layer flow of a viscoelastic fluid over a stretching sheet. The partial differential equations governing the flow and heat transfer characteristics are converted into highly nonlinear coupled ordinary differential equations by similarity transformations. The resultant coupled highly nonlinear ordinary differential equations are solved by means of, homotopy analysis method (HAM) for constructing an approximate solution of heat transfer in magnetohydrodynamic (MHD) viscoelastic boundary layer flow over a stretching sheet with non‐uniform heat source. The proposed method is a strong and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The HAM solutions contain an auxiry parameter, which provides a convenient way of controlling the convergence region of series solutions. The results obtained here reveal that the proposed method is very effective and simple for solving nonlinear evolution equations. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in physics. Copyright © 2012 John Wiley & Sons, Ltd.     

On the use of kernel-based methods in sound synthesis by physical modeling

In this paper we discuss an approach to the modeling of acoustic systems that combines prior information, exploited through physical modeling, and nonlinear dynamics reconstruction, exploited through support vector machine regression. We demonstrate our approach on two case studies, both addressing the broad class of acoustic systems for which the sound generation is obtained through the interaction of a linear system (resonator) and a nonlinear system (excitation). The first case is a physically based impact model, where the resonator is described in terms of its normal modes and the nonlinear contact force is modeled through a simplified collision equation and kernel regression. In the second case study, a model of the voice phonation is illustrated in which the vocal folds are represented by a lumped linear mass-spring system and the nonlinear flow component is modeled through simple Bernoulli-based equations and kernel regression.     

Homogenization results for a coupled system modelling immiscible compressible two-phase flow in porous media by the concept of global pressure

A model for immiscible compressible two-phase flow in heterogeneous porous media is considered. Such models appear in gas migration through engineered and geological barriers for a deep repository for radioactive waste. The main feature of this model is the introduction of a new global pressure and it is fully equivalent to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled degenerate system which consists of a nonlinear parabolic (the global pressure) equation and a nonlinear diffusion–convection one (the saturation) equation with rapidly oscillating porosity function and absolute permeability tensor. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem and give a rigorous mathematical derivation of the upscaled model by means of two-scale convergence.     

On the uniqueness of the solution of a nonlinear filtration problem through a porous medium

The purpose of this work is to study a fluid flow through a porous medium governed by a nonlinear Darcy's law. We also impose a nonlinear semi-permeability condition on some part of the boundary of this medium. The main results are the continuity of the free boundary and the uniqueness of the solution. Received May 5, 1996 / In a revised form November 16, 1996 / Accepted December 17, 1996