Asymptotic behavior of one-dimensional discrete-velocity models in a slab

We prove results on the asymptotic behavior of solutions to discrete-velocity models of the Boltzmann equation in the one-dimensional slab 0x<1 with=" general=" stochastic=" boundary=" conditions=" at=" x="0" and=" x="1." assuming=" that=" there=" is=" a=" constant=">wall Maxwellian M=(M _i) compatible with the boundary conditions, and under a technical assumption meaning strong thermalization at the boundaries, we prove three types of results:

Normal Momentum Transfer Study by a Dynamic Technique

A novel dynamic technique for normal momentum accommodation study in a gas–solid body system is developed. The method can also be used for pressure measurements and adsorption studies. The experimental setup is described. The data obtained are interpreted with allowance for geometric parameters. The normal momentum accommodation coefficients are estimated on the basis of measurement results in systems hydrogen (nitrogen) gas mixture–singlecrystal silicon.     

A Turbulence Model Sensitivity Study for CH4/H2 Bluff-Body Stabilized Flames

The objective of this work is to assess the performances of different turbulence models in predicting turbulent diffusion flames in conjunction with the flamelet model.The k– model, the Explicit Algebraic Stress Model (EASM) and the k– model withvaried anisotropy parameter C (LEA k– model)are first applied to the inert turbulent flow over a backward-facing step, demonstrating the quality of the turbulence models. Following this, theyare used to simulate the CH₄/H₂ bluff-body flame studied by the University of Sydney/Sandia.The numerical results are compared to experimental values of the mixture fraction, velocity field, temperature and constituent mass fractions.The comparisons show that the overall result depends on the turbulence model used, and indicate that theEASM and the LEA k– models perform better than the k– model and mimic most of the significant flow features.     

Attractors for the Two-Dimensional Navier–Stokes-α Model: An α-Dependence Study

The two-dimensional Navier–Stokes- model is considered on the torus and on the sphere. Upper and lower bounds for the dimension of the global attractors are given. The dependence of the dimension of the global attractors on is studied. Special attention is given for the limiting cases when 0, that is, when the Navier–Stokes- model tends to the Navier–Stokes equations, and to the case when .     

Heteroclinic orbits in singular systems: A unifying approach

We consider singularly perturbed systems , such that=f(, _o, 0). _o ^m , has a heteroclinic orbitu(t). We construct a bifurcation functionG(, ) such that the singular system has a heteroclinic orbit if and only ifG(, )=0 has a solution=(). We also apply this result to recover some theorems that have been proved using different approaches.     

Natural convection in rectangular enclosures with adiabatic fins attached on the heated wall

The heat transfer by natural convection in vertical and inclined rectangular enclosures with fins attached to the heated wall is numerically studied using the energy and Navier-Stokes equations with the Boussinesq approximation. The range of study covers 10⁴Ra2×10⁵,A=H/L=2.5 to ,B=l/L=0 to 1,C=h/L=0.25 to 2 andPr=0.72. The inclination angle from the vertical was from 0 to 60 degree. The variation of the local Nusselt numberNu _loc along the enclosure height and the average Nusselt numberNu as a function ofRa are computed. Streamlines and isotherms in the enclosure are produced. The results show thatB is an important parameter affecting the heat transfer through the cold wall of the enclosure. The heat transfer is reduced for decreasingC and it passes from a maximum for an inclination angle. The results show that the heat transfer can generally be reduced using appropriate geometrical parameters in comparison with a similar enclosure without fins.Die Wärmeübertragung bei freier Konvektion in vertikalen und geneigten rechtwinkligen Behältern mit Rippen an den beheizten Wänden wird unter Verwendung der Energie- und Navier-Stokes-Gleichungen sowie der Boussinesq-Approximation numerisch untersucht. Der Bereich der Studie liegt bei 10⁴Ra2·10⁵,A=H/L=2,5 bis ,B=l/L=0 bis 1,C=h/L=0,25 bis 2 undPr=0.72. Der Neigungswinkel der Wand liegt zwischen 0 und 60 Grad. Die Veränderung der lokalen Nusselt-Zahl entlang der Höhe der Behälterwände und die mittlere Nusselt-Zahl in Abhängigkeit derRa-Zahl werden berechnet. Strömungslinien und Isothermen werden im Behälter erzeugt. Die Ergebnisse zeigen, daßB ein wichtiger Parameter für die Wärmeübertragung an der nicht beheizten Wand des Behälters ist. Die übertragene Wärmemenge verringert sich mit abnehmendemC und durchschreitet ein Maximum für eine bestimmte Wandneigung. Die Ergebnisse zeigen, daß im Vergleich zu einer Anordnung ohne Rippen, die Wärmeübertragung bei geeigneten geometrischen Parametern allgemein reduziert werden kann.     

The dynamics of solid-solid phase transitions 2. Incoherent interfaces

Incoherent phase transitions are more difficult to treat than their coherent counterparts. The interface, which appears as a single surface in the deformed configuration, is represented in its undeformed state by a separate surface in each phase. This leads to a rich but detailed kinematics, one in which defects such as vacancies and dislocations are generated by the moving interface. In this paper we develop a complete theory of incoherent phase transitions in the presence of deformation and mass transport, with phase interface structured by energy and stress. The final results are a complete set of interface conditions for an evolving incoherent interface.Frequently used symbols A_i,C_i generic subsurface of S_t - B_i undeformed phase-i region - C configurational bulk stress, Eshelby tensor - F deformation gradient - G inverse deformation gradient - H relative deformation gradient - J bulk Jacobian of the deformation - ¯K, K_i total (twice the mean) curvature of and S_i - Lin (U, V) linear transformations from U into V - Lin⁺ linear transformations of ³ with positive determinant - Orth⁺ rotations of ³ - Q^a external bulk mass supply of species a - ¯S bulk Cauchy stress tensor - S bulk Piola-Kirchhoff stress tensor - S_i undeformed phase i interface - U_i relative velocity of S_i - Unim⁺ linear transformations of ³ with unit determinant - ¯V, V_i normal velocity of and S_i - intrinsic edge velocity of S and A _i S - W_i volume flow across the phase-i interface - X material point - b external body force - e internal bulk configurational force - f_i external interfacial force (configurational) - ¯g external interfacial force (deformational) - grad, div spatial gradient and divergence - gradient and divergence on - h relative deformation - h^a, diffusive mass flux of species a and list of mass fluxes - ¯m outward unit normal to a spatial control volume - ¯n, n_i unit normal to and S_i - n subspace of ³ orthogonal to n - ¯q^a external interfacial mass supply of species a - s ......... - ¯v, v_i compatible velocity fields of and S_i - ¯w, w_i compatible edge velocity fields for and A_i - x spatial point - y_i deformation or motion of phase i - y^. material velocity - generic subsurfaces of - , _i deformed body and deformed phase-i region - () energy supplied to by mass transport - symmetry group of the lattice - _i, surface jacobians - lattice - () power expended on - spatial control volume - S deformed phase interface - lattice point density - interfacial power density - , A total surface stress - C configurational surface stress for phase 1 (material) - ¯C_i configurational surface stress (spatial) - F_i tangential deformation gradient - G_i inverse tangential deformation gradient - H incoherency tensor - ¯1(x), 1_i(X) inclusions of ¯n(x) and n _i (X) into ³ - K configurational surface stress for phase 2 (material) - ¯L, l_i curvature tensor of and S_i - ¯P(x), P_i(X) projections of ³ onto ¯n(x) and n_i (X) - ¯S, S deformational surface stress (spatial and material) - ¯a, a normal part of total surface stress - c normal part of configurational surface stress for phase 1 (material) - e_i internal interfacial configurational force - ¯v, v_i unit normal to and A _i - (x),_i(X) projections of ³ onto ¯n(x) and n _i (X) - _i normal internal force (material) - bulk free energy - slip velocity - _i=(–1)ⁱ _i ......... - ^a, chemical potential of species a and list of potentials - ^a, bulk molar density of species a and list of molar densities - _i normal internal force (spatial) - surface tension - , _i effective shear - referential-to-spatial transform of field - interfacial energy - grand canonical potential - l unit tensor in ³ - x, vector and tensor product in ³ - (...)^., _t(...) material and spatial time derivative - , Div material gradient and divergence - gradient and divergence on S_i - (...), (...) normal time derivative following and S_i - (...) limit of a bulk field asx ,x_i - [...],...> jump and average of a bulk field across the interface - (...)_ext extension of a surface tensor to ³ - tangential part of a vector (tensor) on and S_i     

Some measurements in a binary gas jet

Simultaneous measurements of species volume concentration and velocities in a helium/air binary gas jet with a jet Reynolds number of 4,300 and a jet-to-ambient fluid density ratio of 0.64 were carried out using a laser/hot-wire technique. From the measurements, the turbulent axial and radial mass fluxes were evaluated together with the means, variances and spatial gradients of the mixture density and velocity. In the jet near field (up to ten diameters downstream of the jet exit), detailed measurements of u/ ₀ U ₀, v/ ₀ U₀, u v/ ₀ U ₀ ² , u ² / ₀ U ₀ ² and v ² / ₀ U ₀ ² reveal that the first three terms are of the same order of magnitude, while the last two are at least one order of magnitude smaller than the first three. Therefore, the binary gas jet in the near field cannot be approximated by a set of Reynolds-averaged boundary-layer equations. Both the mean and turbulent velocity and density fields achieve self-preservation around 24 diameters. Jet growth and centerline decay measurements are consistent with existing data on binary gas jets and the growth rate of the velocity field is slightly slower than that of the scalar field. Finally, the turbulent axial mass flux is found to follow gradient diffusion relation near the center of the jet, but the relation is not valid in other regions where the flow is intermittent.     

Pure bending problem for a vibrating elastic plate with non-uniform temperature

This work was performed in the University of Delaware Center for Composite Materials under the auspices of the University/Industry Research Program Application of Composite Materials to Industrial Products. The author(s) acknowledge the support of the University of Delaware/Rutgers University National Engineering Research Center for Composites Manufacturing Science and Engineering. The Center is a National Science Foundation cross-disciplinary research program.     

Free vibrations of an unevenly tensioned tape

Vocational College at the Leningradskii Metallicheskii Zavod Industrial Association. Translated from Prikladnaya Mekhanika, Vol. 26, No. 4, pp. 72–79, April, 1990.     

Two-phase flow in heterogeneous porous media III: Laboratory experiments for flow parallel to a stratified system

Two-phase flow in stratified porous media is a problem of central importance in the study of oil recovery processes. In general, these flows are parallel to the stratifications, and it is this type of flow that we have investigated experimentally and theoretically in this study. The experiments were performed with a two-layer model of a stratified porous medium. The individual strata were composed of Aerolith-10, an artificial: sintered porous medium, and Berea sandstone, a natural porous medium reputed to be relatively homogeneous. Waterflooding experiments were performed in which the saturation field was measured by gamma-ray absorption. Data were obtained at 150 points distributed evenly over a flow domain of 0.1 × 0.6 m. The slabs of Aerolith-10 and Berea sandstone were of equal thickness, i.e. 5 centimeters thick. An intensive experimental study was carried out in order to accurately characterize the individual strata; however, this effort was hampered by both local heterogeneities and large-scale heterogeneities.The theoretical analysis of the waterflooding experiments was based on the method of large-scale averaging and the large-scale closure problem. The latter provides a precise method of discussing the crossflow phenomena, and it illustrates exactly how the crossflow influences the theoretical prediction of the large-scale permeability tensor. The theoretical analysis was restricted to the quasi-static theory of Quintard and Whitaker (1988), however, the dynamic effects described in Part I (Quintard and Whitaker 1990a) are discussed in terms of their influence on the crossflow.Roman Letters A interfacial area between the -region and the -region contained within V, m² - a vector that maps onto , m - b vector that maps onto , m - b vector that maps onto , m - B second order tensor that maps onto , m² - C second order tensor that maps onto , m² - E energy of the gamma emitter, keV - f fractional flow of the -phase - g gravitational vector, m/s² - h characteristic length of the large-scale averaging volume, m - H height of the stratified porous medium , m - i unit base vector in the x-direction - K local volume-averaged single-phase permeability, m² - K - {K}, large-scale spatial deviation permeability - { K} large-scale volume-averaged single-phase permeability, m² - K ^* large-scale single-phase permeability, m² - K ^** equivalent large-scale single-phase permeability, m² - K local volume-averaged -phase permeability in the -region, m² - K local volume-averaged -phase permeability in the -region, m² - K - {K } , large-scale spatial deviation for the -phase permeability, m² - K ^* large-scale permeability for the -phase, m² - l thickness of the porous medium, m - l characteristic length for the -region, m - l characteristic length for the -region, m - L length of the experimental porous medium, m - characteristic length for large-scale averaged quantities, m - n outward unit normal vector for the -region - n outward unit normal vector for the -region - n unit normal vector pointing from the -region toward the -region (n = - n ) - N number of photons - p pressure in the -phase, N/m² - p ₀ reference pressure in the -phase, N/m² - local volume-averaged intrinsic phase average pressure in the -phase, N/m² - large-scale volume-averaged pressure of the -phase, N/m² - large-scale intrinsic phase average pressure in the capillary region of the -phase, N/m² - - , large-scale spatial deviation for the -phase pressure, N/m² - p_c , capillary pressure, N/m² - p _c capillary pressure in the -region, N/m² - p capillary pressure in the -region, N/m² - {p _c } ^c large-scale capillary pressure, N/m² - q -phase velocity at the entrance of the porous medium, m/s - q -phase velocity at the entrance of the porous medium, m/s - S_wi irreducible water saturation - S /, local volume-averaged saturation for the -phase - S _i initial saturation for the -phase - S _r residual saturation for the -phase - S ^* { }^*/}^*, large-scale average saturation for the -phase - S saturation for the -phase in the -region - S saturation for the -phase in the -region - t time, s - v -phase velocity vector, m/s - v local volume-averaged phase average velocity for the -phase, m/s - {v } large-scale averaged velocity for the -phase, m/s - v local volume-averaged phase average velocity for the -phase in the -region, m/s - v local volume-averaged phase average velocity for the -phase in the -region, m/s - v -{v } , large-scale spatial deviation for the -phase velocity, m/s - v -{v } , large-scale spatial deviation for the -phase velocity in the -region, m/s - v -{v } , large-scale spatial deviation for the -phase velocity in the -region, m/s - V large-scale averaging volume, m³ - y position vector relative to the centroid of the large-scale averaging volume, m - {y}^c large-scale average of y over the capillary region, m Greek Letters local porosity - local porosity in the -region - local porosity in the -region - local volume fraction for the -phase - local volume fraction for the -phase in the -region - local volume fraction for the -phase in the -region - {}^* { }^*+{ }^*, large-scale spatial average volume fraction - { }^* large-scale spatial average volume fraction for the -phase - mass density of the -phase, kg/m³ - mass density of the -phase, kg/m³ - viscosity of the -phase, N s/m² - viscosity of the -phase, Ns/m² - V /V , volume fraction of the -region ( + =1) - V /V , volume fraction of the -region ( + =1) - attenuation coefficient to gamma-rays, m^-1 - -      

Two-phase flow in heterogeneous porous media II: Numerical experiments for flow perpendicular to a stratified system

In this work, we make use of numerical experiments to explore our original theoretical analysis of two-phase flow in heterogeneous porous media (Quintard and Whitaker, 1988). The calculations were carried out with a two-region model of a stratified system, and the parameters were chosen be consistent with practical problems associated with groundwater flows and petroleum reservoir recovery processes. The comparison between theory (the large-scaled averaged equations) and experiment (numerical solution of the local volume averaged equations) has allowed us to identify conditions for which the quasi-static theory is acceptable and conditions for which a dynamic theory must be used. Byquasi-static we mean the following: (1) The local capillary pressure,everywhere in the averaging volume, can be set equal to the large-scale capillary pressure evaluated at the centroid of the averaging volume and (2) the large-scale capillary pressure is given by the difference between the large-scale pressures in the two immiscible phases, and is therefore independent of gravitational effects, flow effects and transient effects. Bydynamic, we simply mean a significant departure from the quasi-static condition, thus dynamic effects can be associated with gravitational effects, flow effects and transient effects. To be more precise about the quasi-static condition we need to refer to the relation between the local capillary pressure and the large-scale capillary pressure derived in Part I (Quintard and Whitaker, 1990). Herep _c ¦_y represents the local capillary pressure evaluated at a positiony relative to the centroid of the large-scale averaging volume, and {p _c }¦_x represents the large-scale capillary pressure evaluated at the centroid.In addition to{p _c } ^c being evaluated at the centroid, all averaged terms on the right-hand side of Equation (1) are evaluated at the centroid. We can now write the equations describing the quasi-static condition as , , This means that the fluids within an averaging volume are distributed according to the capillary pressure-saturation relationwith the capillary pressure held constant. It also means that the large-scale capillary pressure is devoid of any dynamic effects. Both of these conditions represent approximations (see Section 6 in Part I) and one of our main objectives in this paper is to learn something about the efficacy of these approximations. As a secondary objective we want to explore the influence of dynamic effects in terms of our original theory. In that development only the first four terms on the right hand side of Equation (1) appeared in the representation for the local capillary pressure. However, those terms will provide an indication of the influence of dynamic effects on the large-scale capillary pressure and the large-scale permeability tensor, and that information provides valuable guidance for future studies based on the theory presented in Part I.Roman Letters A scalar that maps {}*/t onto - A scalar that maps {}*/t onto - A interfacial area between the -region and the -region contained within, m² - A interfacial area between the -region and the -region contained within, m² - A interfacial area between the -region and the -region contained within, m² - a vector that maps ({}*/t) onto , m - a vector that maps ({}*/t) onto , m - b vector that maps ({p}– g) onto , m - b vector that maps ({p}– g) onto , m - B second order tensor that maps ({p}– g) onto , m² - B second order tensor that maps ({p}– g) onto , m² - c vector that maps ({}*/t) onto , m - c vector that maps ({}*/t) onto , m - C second order tensor that maps ({}*/t) onto , m² - C second order tensor that maps ({}*/t) onto . m² - D third order tensor that maps ( ) onto , m - D third order tensor that maps ( ) onto , m - D second order tensor that maps ( ) onto , m² - D second order tensor that maps ( ) onto , m² - E third order tensor that maps () onto , m - E third order tensor that maps () onto , m - E second order tensor that maps () onto - E second order tensor that maps () onto - p _c =(), capillary pressure relationship in the-region - p _c =(), capillary pressure relationship in the-region - g gravitational vector, m/s² - largest of either or - - - i unit base vector in thex-direction - I unit tensor - K local volume-averaged-phase permeability, m² - K local volume-averaged-phase permeability in the-region, m² - K local volume-averaged-phase permeability in the-region, m² - {K } large-scale intrinsic phase average permeability for the-phase, m² - K –{K }, large-scale spatial deviation for the-phase permeability, m² - K –{K }, large-scale spatial deviation for the-phase permeability in the-region, m² - K –{K }, large-scale spatial deviation for the-phase permeability in the-region, m² - K ^* large-scale permeability for the-phase, m² - L characteristic length associated with local volume-averaged quantities, m - characteristic length associated with large-scale averaged quantities, m - I _i i = 1, 2, 3, lattice vectors for a unit cell, m - l characteristic length associated with the-region, m - ; characteristic length associated with the-region, m - l _H characteristic length associated with a local heterogeneity, m - - n unit normal vector pointing from the-region toward the-region (n =–n ) - n unit normal vector pointing from the-region toward the-region (n =–n ) - p pressure in the-phase, N/m² - p local volume-averaged intrinsic phase average pressure in the-phase, N/m² - {p } large-scale intrinsic phase average pressure in the capillary region of the-phase, N/m² - p local volume-averaged intrinsic phase average pressure for the-phase in the-region, N/m² - p local volume-averaged intrinsic phase average pressure for the-phase in the-region, N/m² - p –{p }, large scale spatial deviation for the-phase pressure, N/m² - p –{p }, large scale spatial deviation for the-phase pressure in the-region, N/m² - p –{p }, large scale spatial deviation for the-phase pressure in the-region, N/m² - P _c p –{p }, capillary pressure, N/m² - {p_c}^c large-scale capillary pressure, N/m² - r ₀ radius of the local averaging volume, m - R ₀ radius of the large-scale averaging volume, m - r position vector, m - , m - S /, local volume-averaged saturation for the-phase - S ^* {}*{}*, large-scale average saturation for the-phaset time, s - t time, s - u , m - U , m² - v -phase velocity vector, m/s - v local volume-averaged phase average velocity for the-phase in the-region, m/s - v local volume-averaged phase average velocity for the-phase in the-region, m/s - {v } large-scale intrinsic phase average velocity for the-phase in the capillary region of the-phase, m/s - {v } large-scale phase average velocity for the-phase in the capillary region of the-phase, m/s - v –{v }, large-scale spatial deviation for the-phase velocity, m/s - v –{v }, large-scale spatial deviation for the-phase velocity in the-region, m/s - v –{v }, large-scale spatial deviation for the-phase velocity in the-region, m/s - V local averaging volume, m³ - V volume of the-phase in, m³ - V large-scale averaging volume, m³ - V capillary region for the-phase within, m³ - V capillary region for the-phase within, m³ - V _c intersection of m³ - V volume of the-region within, m³ - V volume of the-region within, m³ - V () capillary region for the-phase within the-region, m³ - V () capillary region for the-phase within the-region, m³ - V () , region in which the-phase is trapped at the irreducible saturation, m³ - y position vector relative to the centroid of the large-scale averaging volume, m Greek Letters local volume-averaged porosity - local volume-averaged volume fraction for the-phase - local volume-averaged volume fraction for the-phase in the-region - local volume-averaged volume fraction for the-phase in the-region - local volume-averaged volume fraction for the-phase in the-region (This is directly related to the irreducible saturation.) - {} large-scale intrinsic phase average volume fraction for the-phase - {} large-scale phase average volume fraction for the-phase - {}* large-scale spatial average volume fraction for the-phase - –{}, large-scale spatial deviation for the-phase volume fraction - –{}, large-scale spatial deviation for the-phase volume fraction in the-region - –{}, large-scale spatial deviation for the-phase volume fraction in the-region - a generic local volume-averaged quantity associated with the-phase - mass density of the-phase, kg/m³ - mass density of the-phase, kg/m³ - viscosity of the-phase, N s/m² - viscosity of the-phase, N s/m² - interfacial tension of the - phase system, N/m - , N/m - , volume fraction of the-phase capillary (active) region - , volume fraction of the-phase capillary (active) region - , volume fraction of the-region ( + =1) - , volume fraction of the-region ( + =1) - {p }– g, N/m³ - {p }– g, N/m³     

On the Ellipticity of the Middle Surface of a Shell and its Application to the Asymptotic Analysis of ‘Membrane’ Shells

We consider a surface S = (), where ² is a bounded, connected, open set with a smooth boundary and : ³ is a smooth map; let () denote the components of the two-dimensional linearized strain tensor of S and let 0 with length 0 > 0. We assume the the norm ,|| ()||0, in the space V0() = { H¹() × H¹() × L²(); = 0 on 0 } is equivalent to the usual product norm on this space. We then establish that this assumption implies that the surface S is uniformly elliptic and that we necessarily have 0 = .     

Determination of permeability tensors for two-phase flow in homogeneous porous media: Theory

In this paper we continue previous studies of the closure problem for two-phase flow in homogeneous porous media, and we show how the closure problem can be transformed to a pair of Stokes-like boundary-value problems in terms of pressures that have units of length and velocities that have units of length squared. These are essentially geometrical boundary value problems that are used to calculate the four permeability tensors that appear in the volume averaged Stokes' equations. To determine the geometry associated with the closure problem, one needs to solve the physical problem; however, the closure problem can be solved using the same algorithm used to solve the physical problem, thus the entire procedure can be accomplished with a single numerical code.Nomenclature a a vector that maps V onto , m^-1. - A a tensor that maps V onto . - A area of the - interface contained within the macroscopic region, m². - A area of the -phase entrances and exits contained within the macroscopic region, m². - A area of the - interface contained within the averaging volume, m². - A area of the -phase entrances and exits contained within the averaging volume, m². - Bo Bond number (= (=(–)g²/). - Ca capillary number (= v/). - g gravitational acceleration, m/s². - H mean curvature, m^-1. - I unit tensor. - permeability tensor for the -phase, m². - viscous drag tensor that maps V onto V. - ^* dominant permeability tensor that maps onto v , m². - ^* coupling permeability tensor that maps onto v , m². - characteristic length scale for the -phase, m. - l characteristic length scale representing both and , m. - L characteristic length scale for volume averaged quantities, m. - n unit normal vector directed from the -phase toward the -phase. - n unit normal vector representing both n and n . - n unit normal vector representing both n and n . - P pressure in the -phase, N/m². - p superficial average pressure in the -phase, N/m². - p intrinsic average pressure in the -phase, N/m². - p –p , spatial deviation pressure for the -phase, N/m². - r ₀ radius of the averaging volume, m. - r position vector, m. - t time, s. - v fluid velocity in the -phase, m/s. - v superficial average velocity in the -phase, m/s. - v intrinsic average velocity in the -phase, m/s. - v –v , spatial deviation velocity in the -phase, m/s. - V volume of the -phase contained within the averaging volmue, m³. - averaging volume, m³. Greek Symbols V /, volume fraction of the -phase. - viscosity of the -phase, Ns/m². - density of the -phase, kg/m³. - surface tension, N/m. - (v +v ^T ), viscous stress tensor for the -phase, N/m².     

Hypersonic flow past blunt-edged delta wings

A method is proposed for calculating hypersonic ideal-gas flow past blunt-edged delta wings with aspect ratios = 100–200. Systematic wing flow calculations are carried out on the intervals 6 M 20, 0 20, 60 80; the results are analyzed in terms of hypersonic similarity parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 175–179, September–October, 1990.     

Indentation of the semi-infinite elastic solid by a concave rigid punch

A solution is obtained for the relationship between load, displacement and inner contact radius for an axisymmetric, spherically concave, rigid punch, indenting an elastic half-space. Analytic approximations are developed for the limiting cases in which the ratio of the inner and outer radii of the annular contact region is respectively small and close to unity. These approximations overlap well at intermediate values. The same method is applied to the conically concave punch and to a punch with a central hole. , , . , . . .     

Transport in ordered and disordered porous media II: Generalized volume averaging

In this paper we develop the averaged form of the Stokes equations in terms of weighting functions. The analysis clearly indicates at what point one must choose a media-specific weighting function in order to achieve spatially smoothed transport equations. The form of the weighting function that produces the cellular average is derived, and some important geometrical theorems are presented.Roman Letters A interfacial area of the- interface associated with the local closure problem, m² - A _e area of entrances and exits for the-phase contained within the averaging system, m² - A _p surface area of a particle, m² - d _p 6V _p/A_p, effective particle diameter, m - g gravity vector, m/s² - I unit tensor - K _m permeability tensor for the weighted average form of Darcy's law, m² - L general characteristic length for volume averaged quantities, m - L _p general characteristic length for volume averaged pressure, m - L characteristic length for the porosity, m - L _v characteristic length for the volume averaged velocity, m - l characteristic length (pore scale) for the-phase - l _i i=1, 2, 3 lattice vectors, m - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - m _v special convolution product weighting function associated with the traditional averaging volume - m _g general convolution product weighting function - m _V unit cell convolution product weighting function - m _C special convolution product weighting function for ordered media which produces the cellular average - m _D special convolution product weighting function for disordered media - m _M master convolution product weighting function for ordered and disordered media - n unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p_m superficial weighted average pressure, N/m² - p _m intrinsic weighted average pressure, N/m² - p traditional intrinsic volume averaged pressure, N/m² - p p p _m , spatial deviation pressure, N/m² - r ₀ radius of a spherical averaging volume, m - r _m support of the convolution product weighting function, m - r position vector, m - r position vector locating points in the-phase, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - V velocity vector in the-phase, m/s - vm superficial weighted average velocity, m/s - v _m intrinsic weighted average velocity, m/s - V volume of the-phase contained in the averaging volume, m³ - V _p volume of a particle, m³ - v traditional superficial volume averaged velocity, m/s - v v p _m spatial deviation velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - _m m * . weighted average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m² - V /V, volume fraction of the-phase     

Material functions generated by the complex viscosity

Based on the complex viscosity model various steady-state and transient material functions have been completed. The model is investigated in terms of a corotational frame reference. Also, BKZ-type integral constitutive equations have been studied. Some relations between material functions have been derived. C ^–1 Finger tensor - F[], (F ^–1[]) Fourier (inverse) transform - rate of deformation tensor in corotating frame - h(I, II) Wagner's damping function - J (x) Bessel function - m parameter inh (I, II) - m(s) memory function - m _k, n_k integers (powers in complex viscosity model) - P principal value of the integral - parameter in the complex viscosity model - rate of deformation tensor - shear rates - [], [] incomplete gamma function - (a) gamma function - steady-shear viscosity - ^* complex viscosity - , real and imaginary parts of ^* - ₀ zero shear viscosity - ⁺, ₁ ⁺ stress growth functions - ^–, ₁ ^- stress relaxation functions - (s) relaxation modulus - ₁(s) primary normal-stress coefficient - ø(a, b; z) degenerate hypergeometric function - ₁, ₂ time constants (parameters of ^*) - frequency - extra stress tensor