A theoretical analysis of forced convection in a porous-saturated circular tube: Brinkman–Forchheimer model

Fully developed forced convection inside a circular tube filled with saturated porous medium and with uniform heat flux at the wall is investigated on the basis of a Brinkman–Forchheimer model. The matched asymptotic expansion method is applied at small Darcy numbers. For large Darcy numbers, the solution for the Brinkman–Forchheimer momentum equation is found in terms of an asymptotic expansion. Once the velocity distribution is determined, the energy equation is solved using the same asymptotic technique. The results for the two limiting cases of clear fluid and Darcy flow conditions show good agreement with those available in the literature.     

Frontogenesis in turbulent flow through porous media using non-linear Forchheimer equation

We present here both one- and two-dimensional models for turbulent flow through heterogeneous unbounded fluid saturated porous media using non-linear Forchheimer extended Darcy (DF) equation in the presence of gravitational field. The fluid is initially at rest and sets in motion due to a uniform horizontal density gradient. It is shown that a purely horizontal motion develops satisfying non-linear DF equation. Analytical solutions of this non-linear Initial Value Problem are obtained and limiting solutions valid for the Darcy regime in the case of laminar flow are derived. A measure of the stability of the flow is discussed briefly using Richardson number. The comparison between the nature of the solutions satisfying the non-linear and linear initial value problems are made. We found that even in the case of turbulent flow the vertical density gradient varies continuously both with space z and time t but the horizontal density gradient remains unchanged. The existence and uniqueness theorem of the Initial Value Problem is proved. The stability of these solutions are discussed and it is shown that the solutions are qualitatively and quantitatively different for and in the upper and lower half of the region. In particular, we have shown that the solution which is stable for infinitesimal perturbations is also stable for arbitrary perturbations both in time and space.In the case of two-dimensional motion, a piecewise initial density gradient with continuous distribution of density, stream function formulation is used and the solutions are obtained using time-series analysis. In this case solution shows crowding of the density profiles in the lower-half of the channel reflecting an increase in density gradient and incipient of frontogenesis there, because of the increase in circulation of the flow due to piecewise initial density gradient.     

On the derivation of the Forchheimer equation by means of the averaging theorem

The averaging theorem is applied to the microscopic momentum equation to obtain the macroscopic flow equation. By examining some very simple tube models of flow in porous media, it is demonstrated that the averaged microscopic inertial terms cannot lead to a meaningful representation of non-Darcian (Forchheimer) effects. These effects are shown to be due to microscopic inertial effects distorting the velocity and pressure fields, hence leading to changes in the area integrals that result from the averaging process. It is recommended that the non-Darcian flow regime be described by a Forchheimer number, not a Reynolds number, and that the Forchheimer coefficient be more closely examined as it may contain information on tortuosity.English a _i gravitational acceleration (m/s²) - A _fs interfacial area between the fluid and solid phases (m²) - Fo Forchheimer number - k permeability (m²) - k ₀ permeability at zero velocity (m²) - p thermodynamic pressure (Pa) - r _i coordinate on the microscopic scale (m) - Re Reynolds number - t time (s) - u _i ,u bulk velocity (m/s) - V volume (m³) - V _f fluid volume (m³) - w _i ,w microscopic velocity (m/s) - x _i ,x coordinate on the macroscopic scale (m) Greek the Forchheimer coefficient (1/m) - _ij extra (viscous) stress tensor (Pa) - _ij stress tensor (Pa) - Viscosity (Pa. s) - density (kg/m³) - porosity - a general variable Symbols < > phase average - < > ^f intrinsic phase average - the fluctuating part of a variable     

A theoretical equation for the flow of granular solids from a hopper

Summary Failure in shear is responsible for the inception of flow in granular solids but once the particles cease to be in contact with each other the concept of shear has little validity and there is more resemblance to the flow of a fluid. Applied to the flow of cohesion-free material from a horizontal aperture at the base of a hopper, these ideas are shown to lead to a relatively simple means of calculating a pseudo-pressure at the base of the hopper, which can be substituted into a pressure-drop relationship of theBernouilli type to give a theoretical equation for the flow rate from the aperture.The derivation depends essentially on the treatment of the moving solids as a fluid, and on the existence of an active condition in the material at the moment when movement commences. Both of these concepts are examined in detail. The theoretical flow equation is shown to fit in well with published correlations.Paper presented at a meeting of the British Society of Rheology, University of Nottingham, April 6–8, 1965.     

用VLW 状态方程计算水的冲击Hugoniot曲线

采用L-J(Lennard-Jones12-6)势函数,结合VLW 状态方程,计算了水的冲击压缩Hugoniot曲线。 将理论计算结果与实验结果进行了比较,当势函数参数/K=120、b0=30.42时,计算值与实验符合较好。与 采用BKW 状态方程计算结果相比,VLW 状态方程能够更准确地描述水的冲击Hugoniot曲线。     

A theoretical Taylor–Galerkin model for first‐order hyperbolic equation

This paper presents a class of Taylor–Galerkin (TG) finite‐element models for solving the first‐order hyperbolic equation which admits discontinuities. Five parameters are introduced for purposes of controlling stability, monotonicity and accuracy. In this paper, the total variation diminishing concept and the theory of M‐matrix are applied to construct a monotonic TG model for capturing discontinuities. To avoid making the scheme overly diffusive, we apply a flux‐corrected transport (FCT) technique of Boris and Book to overcome the difficulty with anti‐diffusive flux. In smooth flow regions, our strategyof developing the temporal and spatial high‐order TG finite‐element model is based on modified equation analysis. In regions where discontinuity is encountered, we resort to two dispersively more accurate models to make the prediction accuracy as high as that obtained in smooth cases. These models are developed using the entropy‐increasing principle and the theory of group velocity. Guided by this theory, a slower group velocity should be used ahead of the shock. To avoid a train of post‐shocks, free parameters should be chosen properly to obtain a group velocity which takes on a larger value than the exact phase velocity. In this paper, we also apply the entropy‐increasing principle to determine free parameters introduced in the finite‐element model. Under the entropy‐increasing requirement, it is mandatory that coefficients of the even and odd derivative terms shown in the modified equation should change signs alternatively in order to avoid non‐physical wiggles. Several benchmark problems have been investigated to confirm the integrity of these proposed characteristic models. Copyright © 2003 John Wiley & Sons, Ltd.     

The damping coefficient in welded bodies: A theoretical study

Summary The purpose of the present paper is to obtain formulae for damping coefficient increase in welded bodies. Weld area is assumed to be homogeneous, with different properties from the remaining part of the body. Attention is confined to uniform stress and compression, torsion and flexion vibrations, for simple form bodies and simple bounds.

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Sommario Scopo del presente lavoro è ricavare formule relative all'aumento del coefficiente di smorzamento in corpi saldati. Si assume che la zona interessata dalla saldatura sia omogenea, pur con proprietà diverse dal resto del corpo. Vengono considerati, per corpi di forma semplice e semplicemente vincolati, i casi di sforzo uniforme e di vibrazione in compressione, torsione e flessione.

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The Onset of Darcy–Forchheimer Convection in Inclined Porous Layers Heated from Below

It is well known that the onset of convection in an inclined porous layer heated from below takes the form of longitudinal vortices when Darcy’s law is valid. In this paper we consider briefly how the onset criterion alters when form drag, as modelled by the Forchheimer terms, is significant. In general, the critical Rayleigh number increases substantially as form drag effects strengthen, but the wavenumber rises by only a small amount. This numerical study is supplemented by a brief asymptotic analysis of the case when the Forchheimer terms dominate and it is shown that the critical Rayleigh number increases in direct proportion with the form drag parameter.     

The non-homogeneous biharmonic plate equation: fundamental solutions

Real materials and structural components are often non-homogeneous, either by design or because of the physical composition and imperfections in the underlying material. Thus, analytical solutions for non-homogeneous materials under mechanical loads are of considerable interest to engineers and have widespread applications, given the prevalence of these materials in fields as diverse as aerospace, construction, electronics, etc. More precisely, those are essentially composites with carefully manufactured properties that yield desirable mechanical characteristics and properties, such as optimal arrangement of the material, minimum weight, etc. To this end, the displacement fundamental solution (or Green’s function) corresponding to a point force for the non-homogenous biharmonic equation in two dimensions are derived in this work by employing a conformal mapping technique in conjunction with the Radon transformation. These functions, besides being useful in their own right, can also be used within the context of integral equation formulations for the solution of boundary-value problems. Finally, a series of numerical examples that deal with the non-homogeneous plate on elastic foundation problem serve to illustrate the present method.     

Magnetic field assisted fluidization： A modified Richardson-Zaki equation

Magnetic particles can be uniformly fluidized by coupling the gas flow with an externally imposed magnetic field. Interparticle forces generated by the magnetic field cause aggregation of the particles in chain-like structures preferentially oriented along the magnetic field lines. In the present paper, we study the implications of the formation of these special types of aggregates on the empirical Richardson-Zaki （RZ） equation, originally proposed to describe the expansion of fluidized beds of non-aggregated particles. We have addressed two important issues, namely the flow regime, which is a function of the size of the aggregates, and the effect of shape and orientation of the chain-like aggregates with respect to gas flow on fluid drag. We propose a modified RZ equation （MRZE） in which the velocity scale, given by the terminal settling velocity of the individual aggregates, and the RZ exponent are predetermined as a function of the chain length. The chain length depends on the ratio of the magnetic energy to gravitational energy, and is estimated from the magnetic field intensity, and particle magnetization, size and density. Predictions of the MRZE are successfully compared with published results in the literature on the expansion of magnetic particles in the presence of externally applied magnetic fields.     

理论力学习题课的一种尝试

理论力学习题课“一题多解”训练是一种行之有效的方法。     

The thermoelastic material-momentum equation

The equation of material momentum, or pseudomomentum, is obtained for thermoelastic materials. This is done in the classic theory, based on the heat conduction hypothesis, and also in the framework of a thermoelasticity approach involving no dissipation of energy, as recently proposed by Green and Naghdi. The results are applied to the thermoelastic fracture problem. When the pseudomomentum equation is written in global form, for a fractured body, it provides path-domain invariant expressions for the thermoelastic energy-release rate.Dedicated to the memory of Paul M. Naghdi.     

Poroelastic toughening in polymer gels: A theoretical and numerical study

We explore the Mode I fracture toughness of a polymer gel containing a semi-infinite, growing crack. First, an expression is derived for the energy release rate within the linearized, small-strain setting. This expression reveals a crack tip velocity-independent toughening that stems from the poroelastic nature of polymer gels. Then, we establish a poroelastic cohesive zone model that allows us to describe the micromechanics of fracture in gels by identifying the role of solvent pressure in promoting poroelastic toughening. We evaluate the enhancement in the effective fracture toughness through asymptotic analysis. We confirm our theoretical findings by means of numerical simulations concerning the case of a steadily propagating crack. In broad terms, our results explain the role of poroelasticity and of the processes occurring in the fracturing region in promoting toughening of polymer gels.     

The Benjamin-Ono-Burgers equation: An application in solar physics

Equations describing nonlinear, long wavelength motions in a ducted weakly dissipative mhd system are derived and found to belong to a class of which the Benjamin-Ono-Burgers equation is a particular example. The slow decay of the solitary wave solutions is investigated. An illustration of the theory is provided by photospheric flux tubes on the Sun.     

A theoretical study on fiber spinnability

Summary A theoretical investigation on fiber spinnability is undertaken, by formulating the force balance equation for a steady liquid thread under axial tension. In order to investigate the effect of the elastic properties of a spin dope, a simple model of a three-constantOldroyd fluid is chosen to solve the force balance equation. In the present study the effect of a hardening process (coagulation in wet spinning and cooling in melt spinning) is not considered, which confines our investigation to a very short distance from the face of the spinnerette. However, the approach taken in the present investigation is justifiable because in practice the breakdown of a thread in fiber spinning is believed to occur at a very short distance from the spinnerette. Prom the solution of the force balance equation a relation between the pullaway tension and jet stretch is obtained, with the elasticity of the spin dope as parameter. A criterion for spinnability is introduced in terms of the critical tensile stress which is the characteristic of a given spin dope. Some of the results are presented and discussed.

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Zusammenfassung Durch Formulierung einer Gleichung für das Kräftegleichgewicht einer stationären Flüssigkeit unter axialer Spannung wird die Spinnfähigkeit von Fasern theoretisch untersucht. Um den Einfluß der elastischen Eigenschaften auf eine Spinnflüssigkeit zu bestimmen, wird ein einfaches Modell einer dreikonstantigen Oldroyd-Flüssigkeit zur Lösung der Gleichung benutzt. In der vorliegenden Arbeit wird der Einfluß des Erstarrungsprozesses (Koagulation beim Naßspinnen und Abkühlung beim Schmelzspinnen) nicht betrachtet, so daß sich die Untersuchungen auf einen kurzen Bereich nach Austritt aus der Spinndüse beschränken. Dennoch erscheint die bei diesen Untersuchungen vorgenommene Näherung gerechtfertigt, weil man in der Praxis annimmt, daß ein Fadenbruch beim Spinnen in einem kurzen Abstand von der Spinndüse auftritt. Aus der Lösung dieser Gleichung wird eine Beziehung zwischen der Abzugsspannung und Strahldehnung abgeleitet mit der Elastizität der Spinnflüssigkeit als Parameter. Ein Kriterium für die Spinnbarkeit wird in Abhängigkeit von der kritischen Dehnspannung, die charakteristisch ist für eine gegebene Spinnflüssigkeit, eingeführt. Einige Ergebnisse werden mitgeteilt und diskutiert.

     

A theoretical investigation of the development of stationary crossflow vortices in the boundary layer on a swept wing

Crossflow instability plays very important role in the transition of the boundary layer on a swept wing, typical in the engineering applications. Experiments revealed that the linear stability theory well predicted the form of the crossflow vortices, but usually much overpredicted their growth rate. Using nonlinear theory of hydrodynamic stability, combined with some other considerations, we were able to obtain the growth rate in good agreement with experimental observations. The project supported by the National Natural Science Foundation of China, Grant No. 19572048     

理论力学教材的发展趋势

从历史发展的角度讨论理论力学教材的发展趋势．列举一些近50年来的优秀教材，并总结出理论力学教材内容改进的4种趋势：即突出分析力学、理论体系的几何化、加强动力系统与动力系统定性理论和突出变换群与不变量理论．并且提出了一些作为基础教材改革的一些设想，还附带了一个改革内容的简要提纲．