Similar solutions and the asymptotics of filtration equations

In this paper we consider the asymptotic behavior of solutions of the quasilinear equation of filtration as t. We prove that similar solutions of the equation u _t = (u )_xx asymptotically represent solutions of the Cauchy problem for the full equation u _t = [(u)]_xx if (u) is close to u for small u.     

Group Classification of Equations of the Form y″ = f(x,y)

The problem of classification of ordinary differential equations of the form y = f(x,y) by admissible local Lie groups of transformations is solved. Standard equations are listed on the basis of the equivalence concept. The classes of equations admitting a oneparameter group and obtained from the standard equations by invariant extension are described.     

Die Torsionsspannungen in Wellen mit tiefen Längsnuten

Übersicht MitF(x, y) als Spannungsfunktion einer Welle ohne Nut und(, y) als Potentialfunktion des Quelle-Senke-Systems erhält man Spannungsfunktionen(, y) =F(x, y) –(, y) für Wellen mit tiefen Längsnuten. Es wird gezeigt, daß sich damit die Schubspannungen in den Läufern von Schraubenverdichtern ermitteln lassen.

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Shearing stresses in shafts with deep longitudinal grooves

Summary The stress functions(, y) of shafts with deep longitudinal grooves may be represented by(, y) =F(x, y) –(, y) whereF(x, y) is the stress function of a cylindrical shaft without grooves and(, y) denotes the potential function of the source-sink system. It is shown that the shearing stresses in rotors of screw-compressors may be obtained in this way.

     

Heat transfer from laminar flow in ducts with elliptic cross-section

Summary The cooling of a hot fluid in laminar Newtonian flow through cooled elliptic tubes has been calculated theoretically. Numerical data have been computed for the two values 1.25 and 4 of the axial ratio of the elliptic cross-section . For =1.25 the influence of non-zero thermal resistance between outmost fluid layer and isothermal surroundings has also been investigated. Special attention has been given to the distribution of heat flux around the perimeter; when increases the flux varies more with the position at the circumference. This positional dependence becomes less pronounced, however, as the (position-independent) thermal resistance of the wall increases.Flattening of the conduit, while maintaining its cross-sectional area constant, improves the cooling. Comparison with rectangular pipes shows that this improvement is not as marked with elliptic as with rectangular pipes.Nomenclature A _k =A _{m, n} coefficients of expansion (6) - a, b half-axes of ellipse, b<a - a _p =a _{r, s} coefficients of representation (V) - D hydraulic diameter, = 4S/P; S = cross-sectional area, P = perimeter - D _e equivalent diameter, according to (13) - n coordinate (outward) normal to the tube wall - T temperature of fluid - T _i temperature of fluid at the inlet - T _s temperature of surroundings - v ₀ mean velocity of fluid - v _z longitudinal velocity of fluid - x, y carthesian coordinates coinciding with axes of ellipse - z coordinate in flow direction - , dimensionless half-axes of ellipse, =a/D and =b/D - _t heat transfer coefficient from fluid at bulk temperature to surroundings; equation (11) - _w heat transfer coefficient at the wall; equation (3) - axial ratio of ellipse, = a/b = / - , , , dimensionless coordinates; =x/D, =y/D, =z/D, =n/D - dimensionless temperature, = (T–T _s)/(T _i–T _s) - ₀ cup-mixing mean value of ; equation (10) - thermal conductivity of fluid - _m,n = _k eigenvalue - c volumetric heat capacity of fluid - _{m, n} = _k = _k eigenfunction; equations (6) and (I) - Nu total Nusselt number, = _t D/ - Nusselt number at large distance from the inlet - Nu _w wall Nusselt number, = _w D/, based on _w - Pé Péclet number, = ₀ Dc/     

Turbulent heat and mass transfer for Pr ≫ 1 and law of turbulent diffusion damping at a solid wall

The question of determining the law of damping for the turbulent diffusion coefficient at a smooth wall according to data on mass and heat transfer for Pr 1 is discussed. It is proved that the hypothesis that this law is determined by the first member of the Taylor series expansion of , namely, / = yⁿ ₊ is valid in the Pr range from 10³ to 10⁵ only under the assumption that the subsequent terms in the expansion have smaller coefficients. A statistical analysis of electrochemical and other experiments devoted to this problem shows that apparently n = 3, but singularities in the experimental results do not permit making a final conclusion. Requirements on a conclusive experiment are formulated on the basis of the analysis made.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 172–175, March–April, 1977.     

Oscillatory measurements of silicone oils —Loss and storage modulus master curves

The work describes a way to obtain loss modulus and storage modulus master curves from oscillatory measurements of silicone oils.The loss modulus master curve represents the dependence of the viscous flow behavior on · ₀ ^* and the storage modulus master curve — the dependence of the elastic flow behavior on · ₀ ^* .The relation between the values of the loss modulus and storage modulus master curves (at a certain frequency) is a measurement of the viscoelastic behavior of a system. The G/G-ratio depends on · ₀ ^* which leads to a viscoelastic master curve. The viscoelastic master curve represents the relation between the elastic and viscous oscillatory flow behavior.     

Über die Übertragungseinheiten und ein „Vierfelderdiagramm” zur Berechnung von Wärmeübertragern

Zusammenfassung Es wird dargelegt, wie man nach Einführung des bekannten Begriffes Übertragungseinheit und mit Hilfe eines Vierfelderdiagramms zu einer relativ einfachen und durchsichtigen Berechnung von Gleich- und Gegenstrom-Wärmeübertragern gelangt.

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On transfer units and a four-quadrant-diagram for the calculation of heat exchangers

It is shown that the introduction of the known concept of a transfer unit and the use of a four-quadrant diagram leads to a relatively simple and clear computation method for co-current and counter-current heat exchangers.

     

Nonlinear rheological behavior of a concentrated spherical silica suspension

Linear and nonlinear viscoelastic properties were examined for a 50 wt% suspension of spherical silica particles (with radius of 40 nm) in a viscous medium, 2.27/1 (wt/wt) ethylene glycol/glycerol mixture. The effective volume fraction of the particles evaluated from zero-shear viscosities of the suspension and medium was 0.53. At a quiescent state the particles had a liquid-like, isotropic spatial distribution in the medium. Dynamic moduli G^* obtained for small oscillatory strain (in the linear viscoelastic regime) exhibited a relaxation process that reflected the equilibrium Brownian motion of those particles. In the stress relaxation experiments, the linear relaxation modulus G(t) was obtained for small step strain (0.2) while the nonlinear relaxation modulus G(t, ) characterizing strong stress damping behavior was obtained for large (>0.2). G(t, ) obeyed the time-strain separability at long time scales, and the damping function h() (–G(t, )/G(t)) was determined. Steady flow measurements revealed shear-thinning of the steady state viscosity () for small shear rates (< ^–1; = linear viscoelastic relaxation time) and shear-thickening for larger (>^–1). Corresponding changes were observed also for the viscosity growth and decay functions on start up and cessation of flow, + (t, ) and _– (t, ). In the shear-thinning regime, the and dependence of ₊(t,) and _–(t,) as well as the dependence of () were well described by a BKZ-type constitutive equation using the G(t) and h() data. On the other hand, this equation completely failed in describing the behavior in the shear-thickening regime. These applicabilities of the BKZ equation were utilized to discuss the shearthinning and shear-thickening mechanisms in relation to shear effects on the structure (spatial distribution) and motion of the suspended particles.Dedicated to the memory of Prof. Dale S. Parson     

Some remarks on differential equations of quadratic type

In this paper we study differential equations of the formx(t) + x(t)=f(x(t)), x(0)=x ₀ C HereC is a closed, bounded convex subset of a Banach spaceX,f(C) C, and it is often assumed thatf(x) is a quadratic map. We study the differential equation by using the general theory of nonexpansive maps and nonexpansive, non-linear semigroups, and we obtain sharp results in a number of cases of interest. We give a formula for the Lipschitz constant off: C C, and we derive a precise explicit formula for the Lipschitz constant whenf is quadratic,C is the unit simplex inR ⁿ, and thel ₁ norm is used. We give a new proof of a theorem about nonexpansive semigroups; and we show that if the Lipschitz constant off: CC is less than or equal to one, then lim_tf(x(t))–x(t)=0 and, if {x(t):t 0} is precompact, then lim_tx(t) exists. Iff¦C=L¦C, whereL is a bounded linear operator, we apply the nonlinear theory to prove that (under mild further conditions on C) lim_t f(x(t))–x(t)=0 and that lim_t x(t) exists if {x(t):t 0} is precompact. However, forn 3 we give examples of quadratic mapsf of the unit simplex ofR ⁿ into itself such that lim_t x(t) fails to exist for mostx ₀ C andx(t) may be periodic. Our theorems answer several questions recently raised by J. Herod in connection with so-called model Boltzmann equations.     

An application of nonsymmetric Lax-Milgram lemma to nonassociated plasticity

Based on a general assumption for plastic potential and yield surface, some properties of the nonassociated plasticity are studied, and the existence and uniqueness of the distribution of incremental stress and displacement for work-hardening materials are proved by using nonsymmetric Lax-Milgram lemma, when the work-hardening parameter A>F/Q/–F/, Q/.     

Calculation of heat transfer accompanying hypersonic three-dimensional flow of air over slender blunt-nosed cones

The effective length method [1, 2] has been used to make systematic calculations of the heat transfer for laminar and turbulent boundary layers on slender blunt-nosed cones at small angles of attack ( + 5° in a separationless hypersonic air stream dissociating in equilibrium (half-angles of the cones 0 20°, angles of attack 0 15°, Mach numbers 5 M 25). The parameters of the gas at the outer edge of the boundary layer were taken equal to the inviscid parameters on the surface of the cones. Analysis of the results leads to simple approximate dependences for the heat transfer coefficients.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 173–177, September–October, 1981.     

A Strongly Degenerate Quasilinear Equation: the Parabolic Case

We prove the existence and uniqueness of entropy solutions of the Neumann problem for the quasilinear parabolic equation u_t=÷ a(u, Du), where a(z,)=f(z,), and f is a convex function of with linear growth as ||||, satisfying other additional assumptions. In particular, this class includes the case where f(z,)=(z)(), >0, and is a convex function with linear growth as ||||.     

Reservoir transport equations by compositional approach

The objective of this paper is to present an overview of the fundamental equations governing transport phenomena in compressible reservoirs. A general mathematical model is presented for important thermo-mechanical processes operative in a reservoir. Such a formulation includes equations governing multiphase fluid (gas-water-hydrocarbon) flow, energy transport, and reservoir skeleton deformation. The model allows phase changes due to gas solubility. Furthermore, Terzaghi's concept of effective stress and stress-strain relations are incorporated into the general model. The functional relations among various model parameters which cause the nonlinearity of the system of equations are explained within the context of reservoir engineering principles. Simplified equations and appropriate boundary conditions have also been presented for various cases. It has been demonstrated that various well-known equations such as Jacob, Terzaghi, Buckley-Leverett, Richards, solute transport, black-oil, and Biot equations are simplifications of the compositional model.Notation List B reservoir thickness - B formation volume factor of phase - Cⁱ mass of component i dissolved per total volume of solution - C ⁱ mass fraction of component i in phase - C heat capacity of phase at constant volume - C_p heat capacity of phase at constant pressure - D ⁱ hydrodynamic dispersion coefficient of component i in phase - D_MTf thermal liquid diffusivity for fluid f - F = F(x, y, z, t) defines the boundary surface - f_p fractional flow of phase - g gravitational acceleration - H_p enthalpy per unit mass of phase - J_p volumetric flux of phase - k_rf relative permeability to fluid f - k₀ absolute permeability of the medium - M_p ⁱ mass of component i in phase - n porosity - N rate of accretion - P_f pressure in fluid f - p_ca capillary pressure between phases and =p-p - Rⁱ rate of mass transfer of component i from phase to phase - Rⁱ _source source rate of component i within phase - S saturation of phase - s gas solubility - T temperature - t time - U displacement vector - u velocity in the x-direction - v velocity in the y-direction - V volume of phase - V_s velocity of soil solids - W_i body force in coordinate direction i - x horizontal coordinate - z vertical coordinate Greek Letters _p volumetric coefficient of compressibility - _T volumetric coefficient of thermal expansion - _ij Kronecker delta - volumetric strain - _m thermal conductivity of the whole matrix - internal energy per unit mass of phase - gf suction head - density of phase - _ij tensor of total stresses - _ij tensor of effective stresses - volumetric content of phase - f viscosity of fluid f     

Effects of temperature and pressure on the rheological behavior of montmorillonite sols

Summary The effects of high temperatures up to 180 C and high pressures up to 560 kg/cm² on the rheological properties of pure montmorillonite suspensions with well-defined base-exchange cations have been investigated. The suspensions behave very much asBingham plastics according to the equation=+, in which is the shear stress,D the shear rate, the plastic viscosity, and the yield stress which is largely a measure of residual flocculation.The observed effects depend strongly on the interparticle forces that govern the colloidal stability and the rheological behavior of the suspensions. One can distinguish between two categories of suspensions:P-type sols in which the clay particles are associated through Coulombic attraction between positive edges and negative faces and are located in a primary potential energy minimum, andS-type sols in which the particles are associated edge-to-edge and are located in a weaker secondary potential minimum obtained by the summation ofvan der Waals attraction and double layer repulsion.Both theBingham yield stress and the plastic viscosity of theP-type sols decrease with increasing temperature. The temperature dependence of follows theArrhenius equation. TheP-type suspensions are either weakly or not at all thixotropic at room temperature and are definitely non-thixotropic at higher temperatures. Pressure slightly increases both the plastic viscosity and the yield stress.TheS-type sols, on the other hand, display an increase in yield stress and degree of thixotropy with increasing temperature and generally a decrease in the plastic viscosity. This behavior is modified in the case of Ca-montmorillonite suspensions, in which both and pass through a maximum at 150C, followed by a decline. The maximum can be explained by disaggregation of face-to-face aggregated clay packets.Pressure causes a decrease both in and the degree of thixotropy in theS-type suspensions, while it causes a slight increase in the plastic viscosity. This behavior is a consequence of the destruction of the hydration shell caused by high pressure.

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Zusammenfassung Der Einfluß von hohen Temperaturen (bis 180 C) und hohen Drucken (bis zu 560 kp/cm²) auf die rheologischen Eigenschaften reiner Suspensionen von Montmorillonit mit definierten Austauschionen wurde untersucht. Die Suspensionen folgen dem Binghamschen Flie\gesetz gemÄ\ der Gleichung=+, worin die Schubspannung,D das GeschwindigkeitsgefÄlle, dieBingham- ViskositÄt und die Flie\spannung darstellen. Letztere zeigt in erster Linie das Ausma\ der Koagulation an.Die beobachteten Effekte hÄngen stark von den KrÄften zwischen den Teilchen ab, welche die StabilitÄt der Kolloide und das rheologische Verhalten der Suspensionen bestimmen. Man kann zwei Kategorien von Suspensionen unterscheiden: Sole vom sog.P-Typ, in denen die Tonteilchen durch Coulombsche AnziehungskrÄfte zwischen positiven Kanten und negativen FlÄchen assoziiert sind und sich in einem primÄren Potentialminimum befinden, und Sole vomS-Typ, in denen die Teilchen Kante-zu-Kante assoziiert sind und sich in einem flacheren sekundÄren Potentialminimum befinden, welches durch das Zusammenwirken vonvan der Waalsschen AnziehungskrÄften und von Absto\ungskrÄften infolge der Wechselwirkung der elektrischen Doppelschichten entsteht.Sowohl die Flie\grenze als auch dieBingham-ViskositÄt der Sole vomP-Typ nehmen mit wachsender Temperatur ab. Die TemperaturabhÄngigkeit von folgt derArrhenius-Gleichung. Die Suspensionen vomP-Typ sind bei Zimmertemperatur entweder gar nicht oder nur schwach thixotrop, wÄhrend sie bei hoher Temperatur in keinem Fall thixotrop sind. Mit wachsendem Druck erhöht sich sowohl dieBingham-ViskositÄt als auch die Flie\grenze ein wenig.Bei den Solen vomS-Typ andererseits steigt die Flie\spannung und der Grad der Thixotropie mit steigender Temperatur, wÄhrend dieBingham-ViskositÄt im allgemeinen abnimmt. Bei Ca-Montmorillonit-Suspensionen ist das Verhalten etwas anders: Sowohl als auch erreichen bei 150 C ein Maximum und fallen dann wieder ab. Das Maximum kann durch Desaggregation flÄchenhaft aggregierter Tonpartikel erklÄrt werden. Bei steigendem Druck fÄllt sowohl als auch der Grad der Thixotropie in den Suspensionen vomS-Typ ab, wÄhrend dieBingham-YiskositÄt leicht ansteigt. Dieses Verhalten ist eine Folge der dann eintretenden Zerstörung der Solvathüllen.

     

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     

Flow in vicinity of blunt body stagnation point in diverging hypersonic stream

In the hypersonic thin shock layer approximation for a small ratio k of the densities before and after the normal shock wave the solution of [1] for the vicinity of the stagnation point of a smooth blunt body is extended to the case of nonuniform outer flow. It is shown that the effect of this nonuniformity can be taken into account with the aid of the effective shock wave radius of curvature R_*, whose introduction makes it possible to reduce to universal relations the data for different nonuniform outer flows with practically the same similarity criterion k. The results of the study are compared with numerical calculations of highly underexpanded jet flow past a sphere.Notations x, y a curvilinear coordinate system with axes directed respectively along and normal to the body surface with origin at the forward stagnation point - R radius of curvature of the meridional plane of the body surface - uV, vV., , p V ² respectively the velocity projections on the x, y axes, density, and pressure - and V freestream density and velocity The indices =0 and=1 apply to plane and axisymmetric flows Izv. AN SSSR, Mekhanika Zhidkosti i Gaza, Vol. 5, No. 3, pp. 102–105, 1970.     

Analytical Modeling of Nonaqueous Phase Liquid Dissolution with Green's Functions

Equilibrium and bicontinuum nonequilibrium formulations of the advection–dispersion equation (ADE) have been widely used to describe subsurface solute transport. The Green's Function Method (GFM) is particularly attractive to solve the ADE because of its flexibility to deal with arbitrary initial and boundary conditions, and its relative simplicity to formulate solutions for multidimensional problems. The Green's functions that are presented can be used for a wide range of problems involving equilibrium and nonequilibrium transport in semiinfinite and infinite media. The GFM is applied to analytically model multidimensional transport from persistent solute sources typical of nonaqueous phase liquids (NAPLs). Specific solutions are derived for transport from a rectangular source (parallel to the flow direction) of persistent contamination using first, second, or thirdtype boundary or source input conditions. Away from the source, the first and thirdtype condition cannot be expected to represent the exact surface condition. The secondtype condition has the disadvantage that the diffusive flux from the source needs to be specified a priori. Near the source, the thirdtype condition appears most suitable to model NAPL dissolution into the medium. The solute flux from the pool, and hence the concentration in the medium, depends strongly on the mass transfer coefficient. For all conditions, the concentration profiles indicate that nonequilibrium conditions tend to reduce the maximum solute concentration and the total amount of solute that enters the porous medium from the source. On the other hand, during nonequilibrium transport the solute may spread over a larger area of the medium compared to equilibrium transport.     

Calculation of intense blowing on a blunt body and an airfoil

Various aspects of the problem of intense blowing through the surface of bodies have, been theoretically studied by a number of authors, within the framework of inviscid flow theory. A detailed bibliography on this topic is given, e.g., in [1, 2]. The well-known approaches to solution of this problem have a limited area of application. For example, asymptotic methods can be used for hypersonic flow regimes only at relatively low levels of the blown gas momentum ( = ² = _ov_o ²/ V ² 1). The same limitation applies to the numerical method of straight lines [2]. The forward Eulerian calculation schemes [3, 4] smear the contact discontinuity severely, and cannot handle the case where the blown gas and the gas in the incident flow have different thermodynamic properties (_o ). This paper presents results of a numerical investigation of supersonic flow over two-dimensional and axisymmetric bodies with intense blowing on the forward surface, performed using a time-dependent finite-difference method [5] with an explicit definition of the contact interface between the two cases. The calculations encompass a family of elliptic cylinders with semiaxis ratio 0.5 4, a flat-face cylinder, and a flat plate with rounding near the midsection, with variations in the blowing law, the incident flow Mach number M (3 M 10), the adiabatic indices, and the blowing parameter 0 0.5.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 117–124, January–February, 1977.In conclusion, the authors thank T. S. Novikov and I. D. Sandomirskii, who took part In the present calculations.     

On the closure problem for Darcy's law

In a previous derivation of Darcy's law, the closure problem was presented in terms of an integro-differential equation for a second-order tensor. In this paper, we show that the closure problem can be transformed to a set of Stokes-like equations and we compare solutions of these equations with experimental data. The computational advantages of the transformed closure problem are considerable.Roman Letters A interfacial area of the- interface contained within the macroscopic system, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A _e area of entrances and exits for the-phase contained within the averaging volume, m² - B second-order tensor used to respresent the velocity deviation - b vector used to represent the pressure deviation, m^–1 - C second-order tensor related to the permeability tensor, m^–2 - D second-order tensor used to represent the velocity deviation, m² - d vector used to represent the pressure deviation, m - g gravity vector, m/s² - I unit tensor - K C ^–1,–D, Darcy's law permeability tensor, m² - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - l _i i=1, 2, 3, lattice vectors, m - n unit normal vector pointing from the-phase toward the-phase - n _e outwardly directed unit normal vector at the entrances and exits of the-phase - p pressure in the-phase, N/m ² - p intrinsic phase average pressure, N/m² - p –p , spatial deviation of the pressure in the-phase, N/m² - r position vector locating points in the-phase, m - r ₀ radius of the averaging volume, m - t time, s - v velocity vector in the-phase, m/s - v intrinsic phase average velocity in the-phase, m/s - v phase average or Darcy velocity in the \-phase, m/s - v –v , spatial deviation of the velocity in the-phase m/s - V averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ Greek Letters V /V volume fraction of the-phase - mass density of the-phase, kg/m³ - viscosity of the-phase, Nt/m²     

Analysis of slip and temperature jump coefficients in a binary gas mixture

The possibility of simplifying the formulas obtained by the Maxwell-Loyalka method for the velocity _u, temperature _T and diffusion _d slip coefficients and the temperature jump coefficient in a binary gas mixture with frozen internal degrees of freedom of the molecules is considered. Special attention is paid to gases not having sharply different physicochemical properties. The formulas are written in a form convenient for use without linearization in the thermal diffusion coefficient. They are systematically analyzed for mixtures of inert gases, N₂, O₂, CO₂, and H₂ at temperatures extending from room temperature to 2500°K. It is shown that for the molecular weight ratios m^* = m₂/m₁ considered the expressions for _u and can be radically simplified. With an error acceptable for practical purposes (up to 10%) it is possible to employ expressions of the same structural form as for a single-component gas: for _u if 1 m^* 6, and for if 1 m^* 3. When 1 m^* 2 the expression for _T can be simplified with a maximum error of 5%. Within the limits of accuracy of the method the expression for _t can be linearized in the thermal diffusion coefficient. An approximate expression convenient for practical calculations is proposed for _d Finally, the , _u, and _T for a single-component polyatomic gas with easy excitation of the internal degrees of freedom of the molecules are similarly analyzed; it is shown that these expressions can be considerably simplified.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 152–159, November–December, 1990.