On the boundary conditions at the macroscopic level

We study the problem of the boundary conditions specified at the boundary of a porous domain in order to solve the macroscopic transfer equations obtained by means of the volume-averaging method. The analysis is limited to the case of conductive transport but the method can be extended to other cases. A numerical study enables us to illustrate the theoretical results in the case of a model porous medium. Roman Letters _sf interfacial area of the s-f interface contained within the macroscopic system m² - A _sf interfacial area of the s-f interface contained within the averaging volume m² - C _p mass fraction weighted heat capacity, kcal/kg/K - d _s , d _f microscopic characteristic length m - g vector that maps to _s, m - h vector that maps to _f , m - K _eff effective thermal conductivity tensor, kcal/m s K - l REV characteristic length, m - L macroscopic characteristic length, m - n _fs outwardly directed unit normal vector for the f-phase at the f-s interface - n _e outwardly directed unit normal vector at the dividing surface - T ^* macroscopic temperature field obtained by solving the macroscopic equation (3), K - V averaging volume, m³ - V _s , V _f volume of the considered phase within the averaging volume, m³ - volume of the macroscopic system, m³ - _s , _f volume of the considered phase within the volume of the macroscopic system, m³ - dividing surface, m² Greek Letters _s , _f volume fraction - ratio of thermal conductivities - _s , _f thermal conductivities, kcal/m s K - spatial average density, kg/m³ - microscopic temperature, K - ^* microscopic temperature corresponding to T ^* , K - spatial deviation temperature K - error on the temperature due to the macroscopic boundary conditions, K - spatial average - ^s , ^f intrinsic phase average     

关于瑞利-李兹方法边界条件的讨论

 在应用瑞利-李兹方法时, 一般教材仅提及假设的挠曲线应满足位移边界条件(挠度$y$与转角$\d y/\d x$), 而没有强调另外两个边界条件$\d^{2}y/\d x^{2}$及$\d^{3}y/\d x^{3}$的重要性. 这两个边界条件经胡克定律可与弯矩及剪力关联起来, 称为力边界条件. 通过例子指出当力边界条件不满足时, 可能造成误差很大. 亦对两个力边界条件的相对重要性作了扼要的讨论.     

Compatibility conditions in the linear micropolar theory

Various representations of compatibility (continuity) conditions are obtained in terms of strain and bending-torsion tensors as well as in terms of stress and couple-stress tensors.     

Numerical outflow boundary conditions for the shallow-water equations

A simple explanation is given of the occurrence of wiggles in the flow field near outflow boundaries. If the shallow-water equations are solved numerically spurious solutions with an oscillatory character turn out to exist, which can be generated by certain additional numerical boundary conditions on the downstream side. The wiggles usually damp quickly with the distance from the boundary. Some ways of handling the downstream boundary are given which largely avoid the occurrence of wiggles.     

Multi-periodic boundary conditions and the Contact Dynamics method

For investigating the mechanical behavior of granular materials by means of the discrete element approach, it is desirable to be able to simulate representative volume elements with macroscopically homogeneous deformations. This can be achieved by means of fully periodic boundary conditions such that stresses or displacements can be applied in all space directions. We present a general framework for periodic boundary conditions in granular materials and its implementation more specifically in the Contact Dynamics method.     

On the compatibility conditions in a curved membrane

Summary Compatibility conditions for a membrane taking on the shape of a surface with given non-constant Gaussian curvature K are deduced. The system for the unknown displacement in general coordinates is integrated, by making use of the non-holonomic basis formed by the unit vectors tangent to the lines K=const. and to the associated orthogonal trajectories. This method unifies the discussion for rotation surfaces, geodesically parallel surfaces and general surfaces. In particular it is proved that, in the case of geodesically parallel surfaces, the integrability conditions reduce to two and involve explicitly the strain and its derivatives up to the third and fifth order respectively.

---

Sommario Vengono dedotte le condizioni di congruenza per una membrana tesa su una superficie a curvatura gaussiana assegnata non costante. Il procedimento utilizzato consente di integrare il sistema nello spostamento in un generico sistema di coordinate, usando la base anolonoma costituita dai versori delle linee di ugual curvatura e delle traiettorie ortogonali. Tale metodo unifica la discussione per le superficie di rotazione, le superficie geodeticamente parallele e le superficie generali. In particolare nel caso di superficie geodeticamente parallel vengono stabilite due sole condizioni di integrabilità nel tensore di deformazione, una alle derivate terze ed una alle derivate quinte.

---

---

This work has been realized within the activities of the Italian Council for the Research (C.N.R.), Gruppo Nazionale per la Fisica Matematica (GNFM).     

On the boundary conditions for differential turbulence models

The distinctive features of the formulation of boundary conditions at the outer edge of a boundary layer and near a wall are considered. Approximate wall functions of turbulence parameters are proposed for three turbulence models.     

Determining the rubber-cord moduli under plane stress conditions

A numerical-experimental approach to determine the elastic rubber-cord properties under plane stress conditions is described. The experimental part of this approach consists of uniaxial tension tests performed with two-ply rubber-cord samples; these experiments allow one to obtain the effective longitudinal Young’s moduli for various cord angles. Such samples correspond to a breaker that forms the basis of the running part of a radial tire. The numerical part of this approach consists of the following two stages: (i) determining the averaged in-plane elastic moduli and (ii) solving the corresponding inverse problem to determine the rubber and cord elastic moduli. The approach is illustrated by several examples. All computations were performed in the framework of the linear elasticity theory; the effects of nonlinearity and viscosity are not taken into account, which is reasonable for the operating range of tire deformations.     

The degeneration of nucleate boiling conditions under conditions of free convection

An experimental investigation was made of the effect of the degeneration of nucleate boiling conditions under conditions of natural convection for alkali metals (potassium, cesium) in working sections with diameters of 14 and 11 mm in a pressure range of 0.04–1.7 bar, and for nonmetallic liquids (ethanol, benzene, and carbon tetrachloride) in working sections with diameters from 0.1 to 10 mm in a pressure range of 0.05–2.5 bar. The regions were determined for thermal flows, with which the appearance of a vapor phase after single-phase natural convection leads to the formation of a stable vapor film. The results of high-speed movingpicture photography of this phenomenon are also given.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 69–73, January–February, 1972.     

Some refinements concerning the boundary conditions at the macroscopic level

The study of boundary effects initiated in a previous paper is continued. New assumptions regarding the geometrical structure of the boundary surface are introduced. Under these assumptions, it is shown that macroscopic Neumann conditions do not generally affect the determination of the macroscopic field in the case of the transport process considered — heat conduction. For this type of boundary condition, the boundary effect is generally confined within a thin layer near the boundary. When heat sources are taken into account within the porous domain, the result is different. In this case, making use of a Neumann boundary condition, expressed in terms of macroscopic variables, amounts to introducing an extra flux. Under normal circumstances, however, this additional flux is negligible.Roman Letters A cross-sectional area of a unit cell - A _e cross-sectional area of a unit cell at the boundary surface - A _sf interfacial area of the s-f interface contained within the averaging volume - surface area per unit volume (A _sf/ ) - A _sf interfacial area of the s-f interface contained within the macroscopic system - g closure vector - h closure vector - k heat transfer coefficient at the s-f interface - K_eff effective thermal conductivity tensor - _x unit cell length - n unit vector - n_e outwardly directed unit normal vector at the boundary - n_sf outwardly directed unit normal vector for thes-phase at f-s interface - q heat flux density - T ^* macroscopic temperature defined by the macroscopic problem - s closure variable - V volume of the macroscopic system - V boundary surface of the macroscopic domain - V ₁ macroscopic sub-surface of the boundary surface - x local coordinate Greek Letters _s,_f volume fraction - _s, gl_f microscopic thermal conductivities - true microscopic temperature - ^* microscopic temperature corresponding toT ^* - microscopic error temperature - vector defined by Equation (34) - < > spatial average     

Necessary conditions for the existence of positive solutions of second-order linear difference equations and sufficient conditions for the oscillation of solutions

We consider the difference equation

Dislocation equilibrium conditions revisited

If there is an equilibrium arrangement of a given collection of dislocations, each having a fixed size and shape, in an externally loaded or unloaded elastic body, the corresponding potential energy will be stationary with respect to infinitesimal perturbations of the dislocation positions. This leads to the dislocation equilibrium conditions: the Peach–Koehler forces along the dislocation line of each dislocation due to externally applied stress and the interaction of the dislocation with other dislocations and its own image field is a set of self-equilibrated forces. The earlier proof of this result presented in the literature was based on an incomplete expression for the elastic strain energy. This is modified here by using the elastic strain energy expression that accounts for all dislocation core energy.     

Yield under plane-stress conditions

Characteristic jagged marks on moiré interference photographs using diffraction gratings show clearly the growth of a yielding region in a mild-steel test specimen under plane-stress conditions. With more accurate measurements, analysis of the displacement contours in the elastic region and extrapolation to the elastic-plastic boundary should provide much needed experimental data for the theory of elasto-plasticity of metals.     

Ellipticity conditions of the static equations of nonlinear elasticity

The relations of the nonlinear model of the theory of elasticity are considered. The Cauchy and the strain gradient tensors are taken to be the characteristics of the stress-strain state of a body. Sufficient conditions under which the static equations of elasticity are of elliptic type are established. These conditions are expressed in the form of constraints imposed on the derivatives of the elastic potential with respect to the strain-measure characteristics. The cases of anisotropic and isotropic bodies are treated, including the case where the Almansi tensor is taken to be the strain measure. The plane strain of a body is investigated using actual-state variables. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 196–203, March–April, 1999.     

Projection conditions on the vorticity in viscous incompressible flows

The problem of establishing appropriate conditions for the vorticity transport equation is considered. It is shown that, in viscous incompressible flows, the boundary conditions on the velocity imply conditions of an integral type on the vorticity. These conditions determine a projection of the vorticity field on the linear manifold of the harmonic vector fields. Some computational consequences of the above result in two-dimensional calculations by means of the nonprimitive variables, stream function and vorticity, are examined. As an example of the application of the discrete analogue of the projection conditions, numerical solutions of the driven cavity problem are reported.     

Evaluating the time to ductile fracture under creep conditions

S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 11, pp. 75–84, November, 1994.     

Investigation of slip boundary conditions in the T-shaped microchannel

Flow regimes and mixing performance in a T-type micromixer at high Reynolds numbers were studied by numerical solution of the Navier–Stokes equations. The Reynolds number was varied from 1 to 1000. The cross section of the mixing channel was 100 μm × 200 μm, and its length was 1400 μm. The transverse inlet channels were symmetric about the mixing channel, and their cross-section was 100 μm × 100 μm, and the total length was 800 μm. Five different flow regimes were identified: (i) stationary vortex-free flow (Re < 5); (ii) stationary symmetric vortex flow with two horseshoe vortices (5 < Re < 150); (iii) stationary asymmetric vortex flow (150 < Re < 240); (iv) non-stationary periodic flow (240 < Re < 400); and (v) stochastic flow (Re > 400). Maximum mixing efficiency is obtained for nonstationary asymmetric vortex flow. In this case, an S-shaped vortex structure is formed in the flow field. The effect of the slip conditions on the flow pattern and mixing efficiency is studied. The slip length varied from 1 to 70 μm in the calculations. It was shown that the mixing can be controlled by hydrophobic coating.