Saturated Elastic Porous Solids: Incompressible,Compressible and Hybrid Binary Models

Constitutive models for a general binary elastic-porous media are investigated by two complementary approaches. These models include both constituents treated as compressible/incompressible, a compressible solid phase with an incompressible fluid phase (hybrid model of first type), and an incompressible solid phase with a compressible fluid phase (hybrid model of second type). The macroscopic continuum mechanical approach uses evaluation of entropy inequality with the saturation condition always considered as a constraint. This constraint leads to an interface pressure acting in both constituents. Two constitutive equations for the interface pressure, one for each phase, are identified, thus closing the set of field equations. The micromechanical approach shows that the results of Didwania and de Boer can be easily extended to general binary porous media.     

Pressure-driven flow of a rate-type fluid with stress threshold in an infinite channel

In this paper we extend some of our previous works on continua with stress threshold. In particular here we propose a mathematical model for a continuum which behaves as a non-linear upper convected Maxwell fluid if the stress is above a certain threshold and as a Oldroyd-B type fluid if the stress is below such a threshold. We derive the constitutive equations for each phase exploiting the theory of natural configurations (introduced by Rajagopal and co-workers) and the criterion of the maximization of the rate of dissipation. We state the mathematical problem for a one-dimensional flow driven by a constant pressure gradient and study two peculiar cases in which the velocity of the inner part of the fluid is spatially homogeneous.     

Unsteady flow of viscoelastic fluid between two cylinders using fractional Maxwell model

The unsteady flow of an incompressible fractional Maxwell fluid between two infinite coaxial cylinders is studied by means of integral transforms.The motion of the fluid is due to the inner cylinder that applies a time dependent torsional shear to the fluid.The exact solutions for velocity and shear stress are presented in series form in terms of some generalized functions.They can easily be particularized to give similar solutions for Maxwell and Newtonian fluids.Finally,the influence of pertinent parameters on the fluid motion,as well as a comparison between models,is highlighted by graphical illustrations.     

Fully-developed conjugate heat transfer in porous media with uniform heating

We propose a computational method for approximating the heat transfer coefficient of fully-developed flow in porous media. For a representative elementary volume of the porous medium we develop a transport model subject to periodic boundary conditions that describes incompressible fluid flow through a uniformly heated porous solid. The transport model uses a pair of pore-scale energy equations to describe conjugate heat transfer. With this approach, the effect of solid and fluid material properties, such as volumetric heat capacity and thermal conductivity, on the overall heat transfer coefficient can be investigated. To cope with geometrically complex domains we develop a numerical method for solving the transport equations on a Cartesian grid. The computational method provides a means for approximating the heat transfer coefficient of porous media where the heat generated in the solid varies “slowly” with respect to the space and time scales of the developing fluid. We validate the proposed method by computing the Nusselt number for fully developed laminar flow in tubes of rectangular cross section with uniform wall heat flux. Detailed results on the variation of the Nusselt number with system parameters are presented for two structured models of porous media: an inline and a staggered arrangement of square rods. For these configurations a comparison is made with literature on fully-developed flows with isothermal walls.     

A three-dimensional non-linear constitutive law for magnetorheological fluids, with applications

In this paper we first summarize the magnetic and mechanical balance equations for magnetorheological fluids undergoing steady motion in the presence of a magnetic field. A general three-dimensional non-linear constitutive law for such a fluid is given for the case in which the magnetic induction vector is used as the independent magnetic variable. The equations are needed for the analysis of boundary-value problems involving fluids with dispersed micron-sized ferrous particles subjected to a time-independent magnetic field. For illustration, the equations are applied, in the case of an incompressible fluid, to the solution of some basic problems. We consider unidirectional flow in a region confined by two infinite parallel plates with a magnetic field applied perpendicular to the plates. Next, we examine two problems involving a circular cylindrical geometry with the fluid occupying the region between two concentric cylinders: axial flow subjected to an axial magnetic field and circumferential flow with a circumferential field. After making some simplifying assumptions on the constitutive law and choosing material parameters, numerical solutions for the velocity profiles are illustrated.     

配置点谱方法-人工压缩法(SCM-ACM)求解同心圆筒内流体流动

发展了配置点谱方法SCM(Spectral collocation method)和人工压缩法ACM(Artificial compressibility method)相结合的SCM-ACM数值方法,计算了柱坐标系下稳态不可压缩流动N-S方程组。选取典型的同心圆筒间旋转流动Taylor-Couette流作为测试对象,首先,采用人工压缩法获得人工压缩格式的非稳态可压缩流动控制方程;再将控制方程中的空间偏微分项用配置点谱方法进行离散,得到矩阵形式的代数方程;编写了SCM-ACM求解不可压缩流动问题的程序;最后,通过与公开发表的Taylor-Couette流的计算结果对比,验证了求解程序的有效性。结果证明,本文发展的SCM-ACM数值方法能够用于求解圆筒内不可压缩流体流动问题,该方法既保留了谱方法指数收敛的特性,也具有ACM形式简单和易于实施的特点。本文发展的SCM-ACM数值方法为求解柱坐标下不可压缩流体流动问题提供了一种新的选择。     

On thermodynamics of incompressible viscoelastic rate type fluids with temperature dependent material coefficients

We derive a class of thermodynamically consistent variants of Maxwell/Oldroyd-B type models for incompressible viscoelastic fluids. In particular, we study the models that allow one to consider temperature dependent material coefficients. This naturally calls for the formulation of a temperature evolution equation that would accompany the evolution equations for the mechanical quantities. The evolution equation for the temperature is explicitly formulated, and it is shown to be consistent with the laws of thermodynamics and the evolution equations for the mechanical quantities. The temperature evolution equation contains terms that are ignored or even not thought of in most of the practically oriented (computational) works dealing with this class of fluids. The impact of the additional terms in the temperature evolution equation on the flow dynamics is documented by the solution of simple initial/boundary value problems.     

Gravitational forces in dual-porosity systems: I. Model derivation by homogenization

We consider the problem of modeling flow through naturally fractured porous media. In this type of media, various physical phenomena occur on disparate length scales, so it is difficult to properly average their effects. In particular, gravitational forces pose special problems. In this paper we develop a general understanding of how to incorporate gravitational forces into the dual-porosity concept. We accomplish this through the mathematical technique of formal two-scale homogenization. This technique enables us to average the single-porosity, Darcy equations that govern the flow on the finest (fracture thickness) scale. The resulting homogenized equations are of dual-porosity type. We consider three flow situations, the flow of a single component in a single phase, the flow of two fluid components in two completely immiscible phases, and the completely miscible flow of two components.This work was supported in part by the National Science Foundation and by the State of Texas Governor's Energy Office.     

MHD flow of upper-convected Maxwell fluid over porous stretching sheet using successive Taylor series linearization method

This paper investigates the magnetohydrodynamic(MHD) boundary layer flow of an incompressible upper-convected Maxwell(UCM) fluid over a porous stretching surface.Similarity transformations are used to reduce the governing partial differential equations into a kind of nonlinear ordinary differential equations.The nonlinear problem is solved by using the successive Taylor series linearization method(STSLM).The computations for velocity components are carried out for the emerging parameters.The numerical values of the skin friction coefficient are presented and analyzed for various parameters of interest in the problem.     

Simulation of Single-Phase Multicomponent Flow Problems in Gas Reservoirs by Eulerian–Lagrangian Techniques

Over the past two decades most discussions of the simulation of miscible displacement in porous media were related to incompressible flow problems; recently, however, attention has shifted to compressible problems. The first goal of this paper is the derivation of the governing equations (mathematical models) for a hierarchy of miscible isothermal displacements in porous media, starting from a very general single-phase, multicomponent, compressible flow problem; these models are then compared with previously proposed models. Next, we formulate an extension of the modified method of characteristics with adjusted advection to treat the transport and dispersion of the components of the miscible fluid; the fluid displacement must be coupled in a two-stage operator-splitting procedure with a pressure equation to define the Darcy velocity field required for transport and dispersion, with the outer stage incorporating an implicit solution of the nonlinear parabolic pressure equation and an inner stage for transport and diffussion in which the mass fraction equations are solved sequentially by first applying a globally conservative Eulerian–Lagrangian scheme to solve for transport, followed by a standard implicit procedure for including the diffusive effects. The third objective is a careful investigation of the underlying physics in compressible displacements in porous media through several high resolution numerical experiments. We consider real binary gas mixtures, with realistic thermodynamic correlations, in homogeneous and heterogeneous formations.     

Development of a three-dimensional solver based on the local domain-free discretization and immersed boundary method and its application for incompressible flow problems

Recently, the author and two other coauthors have proposed a two-dimensional hybrid local domain-free discretization and immersed boundary method (LDFD-IBM), which can be used to solve the flow problem with complex geometries. In this paper, the LDFD-IBM is extended to solve a three-dimensional unsteady incompressible flow with the complex computational domain. The technical issues related to the implementation of the LDFD-IBM in three-dimensional problems are discussed in detail, particularly for the discretization of Navier-Stokes equations, mesh strategies for a three-dimensional flow, and the fast algorithm on the identification of the status of mesh nodes (ie, to identify if the mesh node is located in the solid domain, in the fluid domain, or near the immersed boundary). Numerical tests show that the LDFD-IBM can accurately solve three-dimensional incompressible problems with ease.     

An effective model for the lubrication with micropolar fluid

In this paper, using asymptotic analysis, we study the lubrication process with incompressible micropolar fluid. Starting from 3D micropolar equations, we derive the higher-order asymptotic model explicitly acknowledging the microstructure effects. The effective equations are similar to the Brinkman model for porous medium flow.     

A new method for obtaining the exact solutions of the Navier-Stokes (NS) equations and its applications

In this paper we propose a new method for obtaining the exact solutions of the Mavier-Stokes (NS) equations for incompressible viscous fluid in the light of the theory of simplified Navier-Stokes (SNS) equations developed by the first author^[1,2], Using the present method we can find some new exact solutions as well as the well-known exact solutions of the NS equations. In illustration of its applications, we give a variety of exact solutions of incompressible viscous fluid flows for which NS equations of fluid motion are written in Cartesian coordinates, or in cylindrical polar coordinates, or in spherical coordinates. The project supported by National Natural Science Foundation of China.     

Trochoidal Solutions to the Incompressible Two-Dimensional Euler Equations

Within the Lagrangian framework we present an approach yielding some explicit solutions to the incompressible two-dimensional Euler equations, generalizing the celebrated Gerstner flow. The solutions so obtained, for which explicit formulas of each particle trajectory are provided, represent either flows in domains with a rigid boundary or free-surface flows for a fluid of infinite depth. For some of these solutions the trajectories are epitrochoids or hypotrochoids. Possibilities for obtaining further flows of this type are indicated.     

Flow on oscillating rectangular duct for Maxwell fluid

This paper presents an analysis for the unsteady flow of an incompressible Maxwell fluid in an oscillating rectangular cross section.By using the Fourier and Laplace transforms as mathematical tools,the solutions are presented as a sum of the steady-state and transient solutions.For large time,when the transients disappear,the solution is represented by the steady-state solution.The solutions for the Newtonian fluids appear as limiting cases of the solutions obtained here.In the absence of the frequency of oscillations,we obtain the problem for the flow of the Maxwell fluid in a duct of a rectangular cross-section moving parallel to its length.Finally,the required time to reach the steady-state for sine oscillations of the rectangular duct is obtained by graphical illustrations for different parameters.Moreover,the graphs are sketched for the velocity.     

Modelling of two‐dimensional laminar flow using finite element method

In this paper, a Galerkin weighted residual finite element numerical solution method, with velocity material time derivative discretisation, is applied to solve for a classical fluid mechanics system of partial differential equations modelling two‐dimensional stationary incompressible Newtonian fluid flow. Classical examples of driven cavity laminar flow and laminar flow past a cylinder are presented. Numerical results are compared with data found in the literature. Copyright © 1999 John Wiley & Sons, Ltd.     

A high-order compact scheme for solving the 2D steady incompressible Navier-Stokes equations in general curvilinear coordinates

In this paper, a high-order compact finite difference algorithm is established for the stream function-velocity formulation of the two-dimensional steady incompressible Navier-Stokes equations in general curvilinear coordinates. Different from the previous work, not only the stream function and its first-order partial derivatives but also the second-order mixed partial derivative is treated as unknown variable in this work. Numerical examples, including a test problem with an analytical solution, three types of lid-driven cavity flow problems with unusual shapes and steady flow past a circular cylinder as well as an elliptic cylinder with angle of attack, are solved numerically by the newly proposed scheme. For two types of the lid-driven trapezoidal cavity flow, we provide the detailed data using the fine grid sizes, which can be considered the benchmark solutions. The results obtained prove that the present numerical method has the ability to solve the incompressible flow for complex geometry in engineering applications, especially by using a nonorthogonal coordinate transformation, with high accuracy.     

Flows through Porous Media Containing a Shielded Spherical Inclusion

Compact formulas are obtained for constructing flow potentials in media containing a spherical inclusion shielded by a high- or low-permeability film (fracture or barrier) using the known potentials of steady-state incompressible fluid flows through homogeneous porous media. The formulas obtained can be extended both to inhomogeneous porous media in which the permeability functions have different constant factors inside and outside the inclusion and to bounded zones. The type of singular points of the potentials (sources, sinks, etc.) and the boundary conditions assigned in media without an inclusion are conserved for media containing a shielded inclusion. As an example, translational flow past a spherical shielded polluted zone is studied. This is of interest in connection with environmental problems.     

Nonisothermal two‐ and three‐dimensional flow simulations of inelastic and viscoelastic fluids by a finite‐volume method

This paper applies the finite‐volume method to computations of steady flows of viscous and viscoelastic incompressible fluids in complex two and three‐dimensional geometries. The materials adopted in the study obey different constitutive laws: Newtonian, purely viscous Carreau–Yasuda as also Upper‐Convected Maxwell and Phan‐Thien/Tanner differential models, with a Williams–Landel–Ferry (WLF) equation for temperature dependence. Specific analyses are made depending on the rheological model. A staggered grid is used for discretizing the equations and unknowns. Stockage possibilities allow us to solve problems involving a great number of degrees of freedom, up to 1 500 000 unknowns with a desk computer. In relation to the fluid properties, our numerical simulations provide flow characteristics for various 2D and 3D configurations and demonstrate the possibilities of the code to solve problems involving complex nonlinear constitutive equations with thermal effects. Copyright © 2009 John Wiley & Sons, Ltd.     

Calculation of three-dimensional fluid flow in a radial-axial turbine

Several studies have been made concerning the calculation of three-dimensional fluid flow in turbomachines [1–9, 11].The results of [13] are based on the idea proposed in [9, 12] of the possibility of representing the streamline with the aid of two stream surfaces. In this method the problem for the equations of motion (second order) reduces to a variational problem. The author has used the method of [9] to calculate the flow in the interblade passage of a radial-axial water turbine wheel.A curvilinear nonorthogonal coordinate system is introduced in place of the cylindrical system. As the first family of coordinate surfaces we take surfaces of revolution that are similar in form to the turbine housing, and as the second family we take cylindrical stream surfaces that have directrices in the plane perpendicular to the turbine axes which are logarithmic spirals. The introduction of the curvilinear nonorthogonal coordinate system complicates the form of the equations describing the fluid flow and increases the volume of the computational work, but it does give the possibility of calculating the fluid flow in a turbomachine with radial-axial flow.Results are presented of the calculation of the vortical flow of an incompressible inviscid fluid in a turbine with a total pressure gradient at the channel inlet.