Thermal free convection from an ice sphere in water

Summary Thermal free convection in water is studied by melting ice spheres in water, the uniform temperature of which is varied between 0 and 10°C. Flow patterns as well as local heat transfer are examined. In particular, the effect of anomalous thermal expansion is investigated. From our observations a more accurate picture of the flow phenomena can be obtained that agrees not only with our experimental heat transfer data but also with theoretical results from existing literature.Nomenclature a thermal diffusivity, /(c) - c specific heat capacity - d characteristic length, here diameter of sphere - g gravitational acceleration - coefficient of heat transfer - thermal expansion coefficient - as above, at ambient temperature - temperature difference between fluid and melting ice - as above, for fluid at infinity - thermal conductivity - kinematic viscosity - density - angular distance from upper stagnation point (see Figs. 1, 2, and 3) - Nu Nusselt number, d/ - mean value of Nusselt number - Gr Grashof number, - Pr Prandtl number, /a     

Flow of viscoelastic fluid between confocal elliptic cylinders

Summary Using elliptic coordinates, the flow pattern of a fluid of grade four between two elliptic tubes is determined. A comparison between the position of the maximum of the axial velocity in the present case and in the case of two concentric circular tubes shows a basic difference. In the elliptic case the maximum is shifted towards the external wall, while in the case of concentric circular tubes the shift is in the direction of the internal wall. The secondary flow shows dissymmetry with reference to the intermediate line , which itself lies nearer to the external wall.

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Zusammenfassung Unter Benutzung elliptischer Koordinaten wird die Strömung zwischen zwei elliptischen Rohren bestimmt. Ein Vergleich zwischen der Lage des axialen Geschwindigkeitsmaximums im vorliegenden Fall und im Fall zweier konzentrischer Kreisrohre ergibt einen grundsätzlichen Unterschied: Das Maximum ist im elliptischen Fall zur äußeren Wand hin verschoben, während die Verschiebung im Fall der konzentrischen Kreisrohre zur inneren Wand hin erfolgt. Die Sekundärströmung ist unsymmetrisch relativ zur mittleren Stromlinie , die selbst näher zur äußeren Wand liegt.

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A planar domain representing the annular region - vector inx ₁,x ₂-plane - x _i rectangular coordinates - rectangular unit vectors - , elliptic coordinates - ₁, ₂ ellipses representing respectively the internal and external tubes - = ₂ – ₁ annular widthy = ( – ₁)/ - µ 1^st grade material constant - _i 2^nd grade material constants - _i 3^rd grade material constants - _i 4^th grade material constants - I unit tensor - T _E extra stress (T + pI) - V potential of body forces - material density = (p/) + V = –ax ₃ + () - a specific driving force - arbitrary scalar function - A _k Rivlin-Eriksen tensors - S stress scalar defined onA - t stress vector defined onA - P stress tensor defined onA - v axial velocity - v _i i ^th term in the approximation ofv - u velocity vector perpendicular to the axis 4( ₃ + ₄ + ₅ + 1/2₆) –2/µ(2 ₁ + ₂)( ₂ + ₃) - T stress tensor - p arbitrary hydrostatic pressure - u _i i ^th term in the approximation ofu - stream function definingu - _i i ^th term in the approximation of With 8 figures and 1 table     

Optical co-ordinates in general relativity 2 — the problem of distance

Summary Let denote the congruence of null geodesics associated with a given optical observer inV ₄. We prove that determines a unique collection of vector fieldsM₍₎ ( =1, 2, 3) and ₍₀₎ overV ₄, satisfying a weak version of Killing's conditions.This allows a natural interpretation of these fields as the infinitesimal generators of spatial rotations and temporal translation relative to the given observer. We prove also that the definition of the fieldsM₍₎ and ₍₀₎ is mathematically equivalent to the choice of a distinguished affine parameter f along the curves of, playing the role of a retarded distance from the observer.The relation between f and other possible definitions of distance is discussed.

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Sommario Sia la congruenza di geodetiche nulle associata ad un osservatore ottico assegnato nello spazio-tempoV ₄. Dimostriamo che determina un'unica collezione di campi vettorialiM₍₎ ( =1, 2, 3) e ₍₀₎ inV ₄ che soddisfano una versione in forma debole delle equazioni di Killing. Ciò suggerisce una naturale interpretazione di questi campi come generatori infinitesimi di rotazioni spaziali e traslazioni temporali relative all'osservatore assegnato. Dimostriamo anche che la definizione dei campiM₍₎, ₍₀₎ è matematicamente equivalente alla scelta di un parametro affine privilegiato f lungo le curve di, che gioca il ruolo di distanza ritardata dall'osservatore. Successivamente si esaminano i legami tra f ed altre possibili definizioni di distanza in grande.

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Work performed in the sphere of activity of: Gruppo Nazionale per la Fisica Matematica del CNR.     

The dynamic performance of the Weissenberg rheogoniometer III. Large amplitude oscillatory shearing — harmonic analysis

The harmonic content of the nonlinear dynamic behaviour of 1% polyacrylamide in 50% glycerol/water was studied using a standard Model R 18 Weissenberg Rheogoniometer. The Fourier analysis of the Oscillation Input and Torsion Head motions was performed using a Digital Transfer Function Analyser.In the absence of fluid inertia effects and when the amplitude of the (fundamental) Oscillation Input motion _I is much greater than the amplitudes of the Fourier components of the Torsion Head motion _Tn empirical nonlinear dynamic rheological propertiesG _n (, ⁰),G _n (, ⁰) and/or _n (, ⁰), _n (, ⁰) may be evaluated without a-priori-knowledge of a rheological constitutive equation. A detailed derivation of the basic equations involved is presented.Cone and plate data for the third harmonic storage modulus (dynamic rigidity)G ₃ (, ⁰), loss modulusG ₃ (, ⁰) and loss angle ₃ (, ⁰) are presented for the frequency range 3.14 × 10^–2 1.25 × 10² rad/s at two strain amplitudes, _CP ⁰ = 2.27 and 4.03. Composite cone and plate and parallel plates data for both the third and fifth harmonic dynamic viscosities ₃ (, ⁰), _S (, ⁰) and dynamic rigiditiesG ₃ (, ⁰),G ₅ (, ⁰) are presented for strain amplitudes in the ranges 1.10 _CP ⁰ 4.03 and 1.80 _PP ⁰ 36 for a single frequency, = 3.14 × 10^–1 rad/s. Good agreement was obtained between the results from both geometries and the absence of significant fluid inertia effects was confirmed by the superposition of the data for different gap widths.     

Filament stretching equation in Lagrangian coordinates

The general integral of the very simple equation ^21/n/() was found to describe the cross sectional area of filaments of isothermal power law fluids while in transient stretching where is time and is the initial location of fluid molecules at time = 0 given as the distance from a reference point fixed in space. Any such stretching transient given as a solution of the above equation is physically realizable subject to the restrictions > 0 and/ < 0.     

The effect of the sheet rigidity on strength of the spot weld tension shear joint

This paper presents some test and analysis results for a spot welded joint subjected to tensile and alternate load. The effect of sheet rigidity on the tensile strength and fatigue life of the spot welded joint is studied by using the stress intensity factorsK _I,K _II,K _III and an effective stress intensity factor K_max calculated by the finite element method for crack around the nugget. The results show that the effective stress intensity factor K_max is an essential parameter for estimating the fatigue life of the spot welded joint.     

Concentration-dependent behaviour of the shear viscosity of coal-fuel oil suspensions

A function correlating the relative viscosity of a suspension of solid particles in liquids to their concentration is derived here theoretically using only general thermodynamic ideas, with out any consideration of microscopic hydrodynamic models. This function ( _r = exp (1/2B ^* C ²)) has a great advantage over the many different functions proposed in literature, for it depends on a single parameter,B ^*, and is therefore concise. To test the validity of this function, a least-squares regression analysis was undertaken of available data on the viscosity and concentration of suspensions of coal particles in fuel oil, which promise to be a useful alternative to fuel oil in the near future. The proposed function was found to accurately describe the concentration-dependent behaviour of the relative viscosity of these suspensions. Furthermore, an attempt was made to obtain information about the factors affecting the value ofB ^*, however the results were only qualitative because of, among other things, the inaccuracy of the viscosity measurements in such highly viscous fluids. shear viscosity of the suspension - ₀ shear viscosity of the Newtonian suspending medium - _r = /₀ relative viscosity - solid volume concentration - c solid weight concentration - _m maximum attainable volume concentration of solids - solid volume concentration at which the relative viscosity of the suspension becomes infinite - c _m maximum attainable solid weight concentration - _s density of the solid phase - _l density of the liquid phase - _m density of the suspension - k _n coefficients of theø-power series expansion of _r - { _j } sets of parameters specifying the thermodynamic state of the solid phase of a suspension - T absolute temperature (K) - f (c, T, _j) formal expression for the relative variation of the viscosity with concentration = [1 / (/c)] _T,j - d median size of the granulometric distribution - _B plastic or Bingham viscosity - K consistency factor - n flow index - g ([c _m –c],T, _j ) function including an asymptotic divergence asc tends toc _m , formally describing the concentration dependent behaviour of the shear viscosity of a suspension - A (T, _j) regression analysis parameters - B (T, _j) regression analysis parameters - B ^* (T, _j ) regression analysis parameters     

Approximate solutions for drag coefficient of bubbles moving in shear-thinning elastic fluids

The drag coefficient for bubbles with mobile or immobile interface rising in shear-thinning elastic fluids described by an Ellis or a Carreau model is discussed. Approximate solutions based on linearization of the equations of motion are presented for the highly elastic region of flow. These solutions are in reasonably good agreement with the theoretical predictions based on variational principles and with published experimental data. C _D Drag coefficient - E ^* Differential operator [E ^{* 2} = ²/² + (sin/ ²)/(1/sin /)] - El Ellis number - F _D Drag force - K Consistency index in the power-law model for non-Newtonian fluid - n Flow behaviour index in the Carreau and power-law models - P Dimensionless pressure [=(p – p ₀)/₀ (U /R)] - p Pressure - R Bubble radius - Re ₀ Reynolds number [= 2R U /₀] - Re Reynolds number defined for the power-law fluid [= (2R) ⁿ U ^2–n /K] - r Spherical coordinate - t Time - U Terminal velocity of a bubble - u Velocity - Wi Weissenberg number - Ellis model parameter - Rate of deformation - Apparent viscosity - ₀ Zero shear rate viscosity - Infinite shear rate viscosity - Spherical coordinate - Parameter in the Carreau model - ^* Dimensionless time [=/(U /R)] - Dimensionless length [=r/R] - Second invariant of rate of deformation tensors - ^* Dimensionless second invariant of rate of deformation tensors [=/(U /R)²] - Second invariant of stress tensors - ^* Dimensionless second invariant of second invariant of stress tensor [= / ₀ ² (U /R)²] - Fluid density - Shear stress - ^* Dimensionless shear stress [=/ ₀ (U /R)] - _1/2 Ellis model parameter - ₁ ^2/* Dimensionless Ellis model parameter [= _1/2/ ₀(U /R)] - Stream function - ^* Dimensionless stream function [=/U R ²]     

On the effects of a periodic, spanwise variation of suction upon asymptotic suction flow

Summary The effects of superposing streamwise vorticity, periodic in the lateral direction, upon two-dimensional asymptotic suction flow are analyzed. Such vorticity, generated by prescribing a spanwise variation in the suction velocity, is known to play an important role in unstable and turbulent boundary layers. The flow induced by the variation has been obtained for a freestream velocity which (i) is steady, (ii) oscillates periodically in time, (iii) changes impulsively from rest. For the oscillatory case it is shown that a frequency can exist which maximizes the induced, unsteady wall shear stress for a given spanwise period. For steady flow the heat transfer to, or from a wall at constant temperature has also been computed.Nomenclature (x, y, z) spatial coordinates - (u, v, w) corresponding components of velocity - (, , ) corresponding components of vorticity - t time - stream function for v and w - v _w mean wall suction velocity - nondimensional amplitude of variation in wall suction velocity - characteristic wavenumber for variation in direction of z - T temperature - P pressure - density - coefficient of kinematic viscosity - coefficient of thermal diffusivity - (/v _w)² - frequency of oscillation of freestream velocity - nondimensional amplitude of freestream oscillation - /v _w ² - z z - y –v _w y/ - v _w ² t/4 - /v _w - U ₀ characteristic freestream velocity - u/U ₀ - coefficient of viscosity - _w wall shear stress - Prandtl number (/) - q heat transfer to wall - T _w wall temperature - T (T _w–T)/(T _w–)     

On surface waves in generalized thermoelasticity

Knowles' representation theorem for harmonically time-dependent free surface waves on a homogeneous, isotropic elastic half-space is extended to include harmonically time-dependent free processes for thermoelastic surface waves in generalized thermoelasticity of Lord and Shulman and of Green and Lindsay.r , , r , , .This work was done when author was unemployed.     

Lp-estimates for the nonlinear spatially homogeneous Boltzmann equation

This paper studies L^p-estimates for solutions of the nonlinear, spatially homogeneous Boltzmann equation. The molecular forces considered include inverse k^th-power forces with k > 5 and angular cut-off.The main conclusions are the following. Let f be the unique solution of the Boltzmann equation with f(v,t)(1 + ¦v²¦)^(s ₁ ^{+ /p)/2} L¹, when the initial value f ₀ satisfies f ₀(v) 0, f ₀(v) (1 + ¦v¦²)^(s ₁ ^{+ /p)/2} L¹, for some s₁ 2 + /p, and f ₀(v) (1 + ¦v¦²)^s/2 L^p. If s 2/p and 1 < p < , then f(v, t)(1 + ¦v¦²)^{(s s} ₁)^/2 L^p, t > 0. If s >2 and 3/(1+ ) < p < , thenf(v,t) (1 + ¦v¦²)^(s(s ₁ ^{+ 3/p))/2} L^p, t > 0. If s >2 + 2C₀/C₁ and 3/(l + ) < p < , then f(v,t)(1 + ¦v¦²)^s/2 L^p, t > 0. Here 1/p + 1/p = 1, x y = min (x, y), and C₀, C₁, 0 < 1, are positive constants related to the molecular forces under consideration; = (k – 5)/ (k – 1) for k^th-power forces.Some weaker conclusions follow when 1 < p 3/ (1 + ).In the proofs some previously known L-estimates are extended. The results for L^p, 1 < p < , are based on these L-estimates coupled with nonlinear interpolation.     

Correlations between porous medium flow data and turbulent tube flow results for aqueous hydroxypropylguar solutions (Part 2)

Turbulent tube flow and the flow through a porous medium of aqueous hydroxypropylguar (HPG) solutions in concentrations from 100 wppm to 5000 wppm is investigated. Taking the rheological flow curves into account reveals that the effectiveness in turbulent tube flow and the efficiency for the flow through a porous medium both start at the same onset wall shear stress of 1.3 Pa. The similarity of the curves = ( _w ) and = ( _w ), respectively, leads to a simple linear relation / =k, where the constantk or proportionality depends uponc. This offers the possibility to deduce (for turbulent tube flow) from (for flow through a porous medium). In conjunction with rheological data, will reveal whether, and if yes to what extent, drag reduction will take place (even at high concentrations).The relation of our treatment to the model-based Deborah number concept is shown and a scale-up formula for the onset in turbulent tube flow is deduced as well.     

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     

Effects of MHD free convection and mass transfer on the flow past an oscillating infinite coaxial vertical circular cylinder

The effects of MHD free convection and mass transfer are taken into account on the flow past oscillating infinite coaxial vertical circular cylinder. The analytical expressions for velocity, temperature and concentration of the fluid are obtained by using perturbation technique.

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Einwirkungen von freier MHD-Konvektion und Stoffübertragung auf eine Strömung nach einem schwingenden unendlichen koaxialen vertikalen Zylinder

Zusammenfassung Die Einwirkungen der freien MHD-Konvektion und Stoffübertragung auf eine Strömung nach einem schwingenden, unendlichen, koaxialen, vertikalen Zylinder wurden untersucht. Die analytischen Ausdrücke der Geschwindigkeit, Temperatur und Fluidkonzentration sind durch die Perturbationstechnik erhalten worden.

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Nomenclature C _p Specific heat at constant temperature - C the species concentration near the circular cylinder - C _w the species concentration of the circular cylinder - C the species concentration of the fluid at infinite - * dimensionless species concentration - D chemical molecular diffusivity - g acceleration due to gravity - Gr Grashof number - Gm modified Grashof number - K thermal conductivity - Pr Prandtl number - r _a ,r _b radius of inner and outer cylinder - a, b dimensionless inner and outer radius - r,r coordinate and dimensionless coordinate normal to the circular cylinder - Sc Schmidt number - t time - t dimensionless time - T temperature of the fluid near the circular cylinder - T _w temperature of the circular cylinder - T temperature of the fluid at infinite - u velocity of the fluid - u dimensionless velocity of the fluid - U ₀ reference velocity - z,z coordinate and dimensionless coordinate along the circular cylinder - coefficient of volume expansion - * coefficient of thermal expansion with concentration - dimensionless temperature - H ₀ magnetic field intensity - coefficient of viscosity - _e permeability (magnetic) - kinematic viscosity - electric conductivity - density - M Hartmann number - dimensionless skin-friction - frequency - dimensionless frequency     

Some aspects of dilute suspension theory

A theory proposed by the author as representative of the flow of a general suspension contains three interaction forces, f, S and N. For a quasi-concentrated suspension and for a dilute suspension, N and S, N are omitted, respectively. For the latter special case, we treat diffusion of a fluid through an elastic solid. For a quasi-concentrated suspension, we show that F and S depend on the gradient of the motion gradient. We demonstrate the existence of interesting phenomena: non-simple behavior, dissipative effects, generalized lift and drag forces.Presented at the second conference Recent Developments in Structured Continua, May 23 – 25, 1990, in Sherbrooke, Québec, Canada.     

Methods of Small and Large δ in the Nonlinear Dynamics – A Comparative Analysis

New asymptotic approaches for dynamical systems containing a power nonlinear term x ⁿ are proposed and analyzed. Two natural limiting cases are studied: n 1 + , 1 and n . In the firstcase, the 'small method' (SM)is used and its applicability for dynamical problems with the nonlinearterm sin as well as the usefulness of the SMfor the problem with small denominators are outlined. For n , a new asymptotic approach is proposed(conditionally we call it the 'large method' –LM). Error estimations lead to the followingconclusion: the LM may be used, even for smalln, whereas the SM has a narrow application area. Both of the discussed approaches overlap all values ofthe parameter n.     

Asymptotic analysis of equilibrium states for rotating turbulent flows

The equilibrium states of homogeneous turbulence simultaneously subjected to a mean velocity gradient and a rotation are examined by using asymptotic analysis. The present work is concerned with the asymptotic behavior of quantities such as the turbulent kinetic energy and its dissipation rate associated with the fixed point (/kS)=0, whereS is the shear rate. The classical form of the model transport equation for (Hanjalic and Launder, 1972) is used. The present analysis shows that, asymptotically, the turbulent kinetic energy (a) undergoes a power-law decay with time for (P/)<1, (b) is independent of time for (P/)=1, (c) undergoes a power-law growth with time for 1<(P/)<(C ₂–1), and (d) is represented by an exponential law versus time for (P/)=(C ₂–1)/(C ₁–1) and (/kS)>0 whereP is the production rate. For the commonly used second-order models the equilibrium solutions forP/,II, andIII (whereII andIII are respectively the second and third invariants of the anisotropy tensor) depend on the rotation number when (P/kS)=(/kS)=0. The variation of (P/kS) andII versusR given by the second-order model of Yakhot and Orzag are compared with results of Rapid Distortion Theory corrected for decay (Townsend, 1970).     

Comparison between uniaxial and biaxial elongational flow behavior of viscoelastic fluids as predicted by differential constitutive equations

Behavior of polymer melts in biaxial as well as uniaxial elongational flow is studied based on the predictions of three constitutive models (Leonov, Giesekus, and Larson) with single relaxation mode. Transient elongational viscosities in both flows are calculated for three constitutive models, and steady-state elongational viscosities are obtained as functions of strain rates for the Giesekus and the Larson models.Change of elongational flow behavior with adjustable parameter is investigated in each model. Steady-state viscosities _E and _B are obtained for the Leonov model only when the strain-hardening parameter is smaller than the critical value _cr determined in each flow. In this model, uniaxial elongational viscosity _E increases with increasing strain rate , while biaxial elongational viscosity _B decreases with increasing biaxial strain rate _B . The Giesekus model predictions depend on the anisotropy parameter . _E and _B increase with strain rates for small _B while they decrease for large . When is 0.5, _E in increasing, but _B is decreasing. The Larson model predicts strain-softening behavior for both flows when the chain-contraction parameter > 0.5. On the other hand, when is small, the steady-state viscosities of this model show distinct maximum around = _B = 1.0 with relaxation time . The maximum is more prominent in _E than in _B .     

Transport in ordered and disordered porous media IV: Computer generated porous media for three-dimensional systems

In the method of volume averaging, the difference between ordered and disordered porous media appears at two distinct points in the analysis, i.e. in the process of spatial smoothing and in the closure problem. In theclosure problem, the use of spatially periodic boundary conditions isconsistent with ordered porous media and the fields under consideration when the length-scale constraint,r ₀L is satisfied. For disordered porous media, spatially periodic boundary conditions are an approximation in need of further study.In theprocess of spatial smoothing, average quantities must be removed from area and volume integrals in order to extractlocal transport equations fromnonlocal equations. This leads to a series of geometrical integrals that need to be evaluated. In Part II we indicated that these integrals were constants for ordered porous media provided that the weighting function used in the averaging process contained thecellular average. We also indicated that these integrals were constrained by certain order of magnitude estimates for disordered porous media. In this paper we verify these characteristics of the geometrical integrals, and we examine their values for pseudo-periodic and uniformly random systems through the use of computer generated porous media.

Nomenclature

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 _i i=1, 2, 3 gaussian probability distribution used to locate the position of particles - I unit tensor - L general characteristic length for volume averaged quantities, m - L characteristic length for , m - L characteristic length for , m - characteristic length for the -phase particles, m - ⁰ reference characteristic length for the-phase particles, m - characteristic length for the-phase, m - _i i=1, 2, 3 lattice vectors, m - m convolution product weighting function - m _v special convolution product weighting function associated with the traditional volume average - n _i i=1, 2, 3 integers used to locate the position of particles - 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 - r _p position vector locating the centroid of a particle, m - r gaussian probability distribution used to determine the size of a particle, m - r ₀ characteristic length of an averaging region, m - r position vector, m - r _m support of the weighting functionm, m - averaging volume, m³ - V volume of the-phase contained in the averaging volume,, m³ - x positional vector locating the centroid of an averaging volume, m - x ₀ reference position vector associated with the centroid of an averaging volume, m - y position vector locating points 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 - /L, small parameter in the method of spatial homogenization - standard deviation ofa _i - _r standard deviation ofr - _r intrinsic phase average of      

On shock layer method for the hypersonic flow problems

Chernyi’s series method[1] is not proper for the case that(γ-l)/(γ+l)<<2/(γ+1)×M²sin²β (γ=c_p/c_v-adiabatic index number, M-Much number, β-shock incidence). In this paper, we only suppose that in the neighbour of the shock, there exists a shock layer in which the density of the gas is very big, but we do not remove the case that (γ-1)/(γ+1)<<2/(γ+1)M²sin²β.