The investigation on the properties and structures of starting vortex flow past a backward-facing step by WBIV technique

An experimental investigation of a starting vortex flow around a backward-facing step was conducted in a water channel. The properties and structures of the flow were investigated by qualitative flow visualization using the hydrogen bubble method and by quantitative velocity and vorticity measurements using White-light Bubble Image Velocimetry (WBIV) — a newly developed PIV method. Some invariant properties and 4-stage structures of starting vortex flow were observed.List of symbols a flow acceleration during starting stage - h height of backward-facing step - d _v dimensionless vortex size - t time - t dimensionless time - U free uniform velocity - u, v streamwise and spanwise velocity components respectively - Re Reynolds number based on a and h - x, y streamwise and spanwise coordinates respectively in flow field - x _c , y _c dimensionless vortex center position - vorticity - ov dimensionless vorticity - _max maximum vorticity - ov _max dimensionless maximum vorticity - circulation - dimensionless circulation - kinematic viscosity This work was supported by the CNSF Grant 1939 100-1-3     

On the cross-effects of coupled macroscopic transport equations in porous media

In this paper, the derivation of macroscopic transport equations for this cases of simultaneous heat and water, chemical and water or electrical and water fluxes in porous media is presented. Based on themicro-macro passage using the method of homogenization of periodic structures, it is shown that the resulting macroscopic equations reveal zero-valued cross-coupling effects for the case of heat and water transport as well as chemical and water transport. In the case of electrical and water transport, a nonsymmetrical coupling was found.Notations b mobility - c concentration of a chemical - D rate of deformation tensor - D molecular diffusion coefficient - D _ij ^eff macroscopic (or effective) diffusion tensor - electric field - E ₀ initial electric field - k _ij molecular tensor - j, j ^*, current densities - K _ij macroscopic permeability tensor - l characteristic length of the ERV or the periodic cell - L characteristic macroscopic length - L _ijkl coupled flows coefficients - n _i unit outward vector normal to - p pressure - q _t ,q _t ⁺ , heat fluxes - q _c ,q _c ⁺ , chemical fluxes - s specific entropy or the entropy density - S entropy per unit volume - t time variable - t _ij local tensor - T absolute temperature - v _i velocity - V ₀ initial electric potential - V electric potential - x macroscopic (or slow) space variable - y microscopic (or fast) space variable - _i local vectorial field - _i local vectorial field - electric charge density on the solid surface - , bulk and shear viscosities of the fluid - _ij local tensor - _ij local tensor - _i local vector - _ij molecular conductivity tensor - _ij ^eff effective conductivity tensor - homogenization parameter - fluid density - ₀ ion-conductivity of fluid - _ij dielectric tensor - _i ¹ , _i ² , _i ³ local vectors - ⁴ local scalar - _S solid volume in the periodic cell - _L volume of pores in the periodic cell - boundary between _S and _L - ^s rate of entropy production per unit volume - total volume of the periodic cell - _l volume of pores in the cell On leave from the Politechnika Gdanska; ul. Majakowskiego 11/12, 80-952, Gdask, Poland.     

Nonlocal dispersion in media with continuously evolving scales of heterogeneity

General nonlocal diffusive and dispersive transport theories are derived from molecular hydrodynamics and associated theories of statistical mechanical correlation functions, using the memory function formalism and the projection operator method. Expansion approximations of a spatially and temporally nonlocal convective-dispersive equation are introduced to derive linearized inverse solutions for transport coefficients. The development is focused on deriving relations between the frequency-and wave-vector-dependent dispersion tensor and measurable quantities. The resulting theory is applicable to porous media of fractal character.Nomenclature C _v (t) particle velocity correlation function - C _v ,(t) particle fluctuation velocity correlation function - C _j (x,t) current correlation function - D(x,t) dispersion tensor - D(x,t) fluctuation dispersion tensor - f ₀(x,p) equilibrium phase probability distribution function - f(x, p;t) nonequilibrium phase probability distribution function - G(x,t) conditional probability per unit volume of finding a particle at (x,t) given it was located elsewhere initially - (k,t) Fourier transform ofG(x,t) - G(x,t) fluctuation conditional probability per unit volume of finding a particle at (x,t) given it was located elsewhere initially - k wave vector - K(t) memory function - L Liouville operator - m mass - p(t) particle momentum coordinate - P = (0)( , (0)) projection operator - Q =I-P projection operator - s real Laplace space variable - S(k, ) time-Fourier transform of(k,t) - t time - v(t) particle velocity vector - v(t) particle fluctuation velocity vector - V phase space velocity - time-Fourier variable - ^(itn)(k) frequency moment of(k,t) - x(t) particle displacement coordinate - x(t) particle displacement fluctuation coordinate - friction coefficient - (t) normalized correlation function General Functions () Dirac delta function - () Gamma function Averages ₀ Equilibrium phase-space average - Nonequilibrium phase-space average - (,) L ² inner product with respect tof ₀     

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.     

Limiting analytical solutions for complex saturated and unsaturated transport problems

Non-linear diffusion and velocity-dependent dispersion problems are under consideration. The necessary and sufficient conditions allowing the comparison of solutions to the two dimensional convection-dispersion equations with different coefficients are obtained. These conditions provide a framework within which solutions to the complex non-linear problems mentioned above can be estimated by solutions to the problems possessing analytical solvability.Nomenclature c(x, y, t) concentration of solute in solution,ML ^–3 - C(h)=d/dh moisture capacity function - D,D _ij hydrodynamic dispersion coefficient, a second order tensor,L ² T ^–1 - D _L longitudinal hydrodynamic dispersion coefficient,L ² T ^–1 - D _m molecular diffusion coefficient,L ² T ^–1 - D _T transverse hydrodynamic coefficient,L ² T ^–1 - G flow domain for the unsaturated flow problem - G _z , G _w flow domain and complex potential domain, respectively, for the hydrodynamic dispersion problem - h piezometric head,L - I _n given mass flux normal to the boundary,MLT ^–1 - k hydraulic conductivity,LT ^–1 - K(h) unsaturated hydraulic conductivity,LT ^–1 - L continuously differentiable function with respect to all arguments - m porosity - n(x,t) outer normal vector to the boundary - t time,T - V(x, y, t) seepage velocity vector withV=V,LT ^–1 - x Cartesian coordinate system - x horizontal coordinate,L - y vertical coordinate (elevation),L - (x),(x,t) given functions in initial and boundary conditions (3), (4) - ₁(,) angle between vectors ₁c andV - boundary of the flow domain - _L , _T longitudinal and transverse dispersivities, respectively,L - water mass density,ML ^–3 - v _i components of a unit vector in the direction of the outward normal to the boundary - =–kh velocity potential - =/m - stream function defined such thatw=+i is the complex potential - =/m     

Two-phase geothermal flows with conduction and the connection with Buckley-Leverett theory

Two-phase flows of boiling water and steam in geothermal reservoirs satisfy a pair of conservation equations for mass and energy which can be combined to yield a hyperbolic wave equation for liquid saturation changes. Recent work has established that in the absence of conduction, the geothermal saturation equation is, under certain conditions, asymptotically identical with the Buckley-Leverett equation of oil recovery theory. Here we summarise this work and show that it may be extended to include conduction. In addition we show that the geothermal saturation wave speed is under all conditions formally identical with the Buckley-Leverett wave speed when the latter is written as the saturation derivative of a volumetric flow.Roman Letters C(P, S,q) geothermal saturation wave speed [ms^–1] (14) - c _t (P, S) two-phase compressibility [Pa^–1] (10) - D(P, S) diffusivity [m s^–2] (8) - E(P, S) energy density accumulation [J m^–3] (3) - g gravitational acceleration (positive downwards) [ms^–2] - h _w (P),h _w (P) specific enthalpies [J kg^–1] - J _M (P, S,P) mass flow [kg m^–2 s^–1] (5) - J _E (P, S,P) energy flow [J m^–2s^–1] (5) - k absolute permeability (constant) [m²] - k _w (S),k _s (S) relative permeabilities of liquid and vapour phases - K formation thermal conductivity (constant) [Wm^–1 K^–1] - L lower sheetC<0 in flow plane - m, c gradient and intercept - M(P, S) mass density accumulation [kg m^–3] (3) - O flow plane origin - P(x,t) pressure (primary dependent variable) [Pa] - q volume flow [ms^–1] (6) - S(x, t) liquid saturation (primary dependent variable) - S ^*(x,t) normalised saturation (Appendix) - t time (primary independent variable) [s] - T temperature (degrees Kelvin) [K] - T _sat(P) saturation line temperature [K] - TdT _sat/dP saturation line temperature derivative [K Pa^–1] (4) - T _c ,T _D convective and diffusive time constants [s] - u _w (P),u _s (P),u _r (P) specific internal energies [J kg^–1] - U upper sheetC > 0 in flow plane - U(x,t) shock velocity [m s^–1] - x spatial position (primary independent variable) [m] - X representative length - x, y flow plane coordinates - z depth variable (+z vertically downwards) [m] Greek Letters _P , _S remainder terms [Pa s^–1], [s^–1] - double-valued saturation region in the flow plane - h =h _s –h _w latent heat [J kg^–1] - = _w – _s density difference [kg m^–3] - line envelope - =D _K /D ₀ diffusivity ratio - porosity (constant) - _w (P), _s (P), _t (P, S) dynamic viscosities [Pa s] - v _w (P),v _s (P) kinematic viscosities [m²s^–1] - v ₀ =kh/KT kinematic viscosity constant [m² s^–1] - ₀ =v ₀ dynamic viscosity constant [m² s^–1] - _w (P), _s (P) density [kg m^–3] Suffixes r rock matrix - s steam (vapour) - w water (liquid) - t total - av average - 0 without conduction - K with conduction     

Bubble growth predictions by the hyperbolic and parabolic heat conduction equations

A model for bubble growth in a uniformly superheated liquid is presented which is valid for both inertia and heat diffusion controlled growth. Two different heat transfer equations are considered: The Fourier (parabolic) equation and the hyperbolic heat conduction equation. It is shown that for short times, bubble growth prediction based on the Fourier equation, differs considerably from that based on the hyperbolic heat conduction equation. For long times, both predictions coincide. Using the hyperbolic heat conduction equation is important for bubble growth prediction in fluids like Helium II, in which thermal disturbances have a low speed of propagation. In such liquids the second sound effects must be considered long after the inertia and dynamic effects become unimportant.The validity of using a semi-infinite approximation to the heat conduction problem during the bubble growth period is investigated. An analytical solution in spherical coordinates reveals that the ratio between the spherical and semi-infinite solutions is a function of the Jakob number. Results of the present model, using the Fourier equation, are shown to be in better agreement with data for bubble growth in water, than other published solutions.

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Beschreibung des Blasenwachstums durch Wärmeleitungs-Gleichungen von hyperbolischer und parabolischer Form

Zusammenfassung Es wird ein Modell für Blasenwachstum in überhitzter Flüssigkeit vorgestellt, das sowohl bei durch Trägheit als auch bei durch Wärmediffusion kontrolliertem Blasenwachstum verwendbar ist. Zwei unterschiedliche Wärmeübertragungsbeziehungen werden in Betracht gezogen: Die Fourier-Gleichung (parabolisch) und eine Wärmeleitungs-Gleichung in hyperbolischer Form.Es wird gezeigt, daß die Modellergebnisse basierend auf der Fourier-Gleichung für schnelle Blasenwachstumszeiten signifikant von vergleichbaren Ergebnissen basierend auf der hyperbolischen Gleichung abweichen, während sie für langsames Wachstum mehr oder weniger identisch sind. Die Verwendung der hyperbolischen Wärmeleitungsgleichung in Blasenwachstumsmodellen ist vor allem in Fluiden wie Helium II von Bedeutung, wo thermische Störungen eine geringe Ausbreitungsgeschwindigkeit haben. Hier müssen die second sound-Effekte noch berücksichtigt werden, wenn die dynamischen und die Einflüsse der Trägheit schon vernachlässigbar sind.Es wurde untersucht, ob die Benutzung einer semi-unendlichen Approximation des Wärmeleitungsproblems während des Blasenwachstums zulässig ist. Eine analytische Lösung in Kugelkoordinaten zeigt, daß das Verhältnis zwischen letzteren und semi-unendlichen Ergebnissen eine Funktion der Jakob-Zahl ist.Schließlich wird gezeigt, daß die Resultate des vorgestellten Modells bei Benutzung der Fourier-Gleichung experimentelle Ergebnisse von Blasenwachstum in Wasser besser wiedergeben als andere bekannte Lösungen.

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Nomenclature a thermal diffusivity - B _s sphericity correction factor - b temperature decay coefficient - c propagation speed of thermal disturbances - E parameter, Eq. (37) - f function of the dimensionless time and bubble radius, Eq. (34) - h _v heat of evaporation - Ja Jakob number, Eq. (35) - k thermal conductivity - N /T - P pressure - P _i initial system pressure - P _v vapour pressure - Q* dimensionless heat flux (Stanton number) - q heat flux - transformed heat flux - q _wL heat flux into the liquid at the bubble boundary - R bubble radius - R* dimensionless bubble radius, Eq. (16) - R ₀ initial (critical) bubble radius - r radial coordinate - s the Laplace transform parameter - T temperature - T _i initial liquid temperature - T _s saturation temperature - T _v instantaneous bubble temperature - T ₀ initial saturation temperature,T _s (0) - T temperature difference,T _i–T _s (0) - t time - t* dimensionless time, Eq. (16) - y dimensionless distance from the bubble surface - Z constant of integration, Appendix A - a proportionality constant - temperature function, Eq. (8) - transformed temperature function - _v vapour density - _L liquid density - _vi initial vapour density - relaxation time,a/c ² - normalized temperature distribution, Eq. (15)     

The energy-integral method: application to linear hyperbolic heat-conduction problems

This paper utilizes the energy-integral method to obtain approximate analytic solutions to a linear hyperbolic heat-conduction problem for a semi-infinite one-dimensional medium. As for the mathematical formulation of the problem, a time-dependent relaxation model for the energy flux is assumed, leading to a hyperbolic differential equation which is solved under suitable initial and boundary conditions. In fact, analytical expressions are derived for uniform as well as varying initial conditions along with (a) prescribed surface temperature, or (b) prescribed heat flux at the surface boundary. The case when a heat source (or sink) of certain type takes place has also been discussed. Comparison of the approximate analytic solutions obtained by the energy-integral method with the corresponding available or obtainable exact analytic solutions are made; and the accuracy of the approximate solutions is generally acceptable.Nomenclature A,C constants - a ₀(t),a ₁(t),...,a _n (t) arbitrary time-dependent coefficients, equation (3.2) - b thermal propagation speed - C _p specific heat of solid at constant pressure - g(x) given function, equation (5.1) - h(t) specified function of time - I _n modified Bessel function of the first kind - K thermal conductivity - j,n positive constants - P _n (x,t) polynomial of degreen - q(x,t) heat flux - Q(t),R(t),H(t),E(t) see equations (3.9), (II.d), (4.10), (4.12), respectively - (t) thermal penetration depth - (t,) approximate thermal penetration depth - T(x,t) temperature distribution - t time - y dimensionless time, equation (3.17) - V(y) dimensionless surface heat flux - W(y) dimensionless surface temperature - U-(t) unit-step function - G(x;t,) Green's function - x spatial variable - ()₀ surface value (atx=0) Greek symbols thermal diffusivity - density of solid - parameter, see equations (3.11) and (3.13) - parameter depending onn and - specified parameter, equations (4.5a) and (5.12b) - (t),(t) given functions of time, equations (4.6) and (5.5b) - , dummy variables - relaxation time - energy integral - (y),(y) specified functions ofy; equations (3.22) and (4.19)     

Flow birefringence of polymer melts: Application to the investigation of time dependent rheological properties

Summary Transient stresses including normal stresses, which are developed in a polymer melt by a suddenly imposed constant rate of shear, are investigated by mechanical measurement and, indirectly, with the aid of the flow birefringence technique. For the latter purpose use is made of the so-called stress-optical law, which is carefully checked.It appears that the essentially linear model of the rubberlike liquid, as proposed byLodge, is capable of describing the behaviour of polymer melts rather well, if the applied total shear does not exceed unity. In order to describe also steady state values of the stresses successfully, one should extend measurements to extremely low shear rates.These statements are verified with the aid of a method which was originally designed bySchwarzl andStruik for the practical calculation of interrelations between linear viscoelastic functions. In the present paper dynamic shear moduli are used as reference functions.

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Zusammenfassung Mit der Zeit anwachsende Spannungen, darunter auch Normalspannungen, wie sie sich nach dem plötzlichen Anlegen einer konstanten Schergeschwindigkeit in einer Polymerschmelze entwickeln, werden mit Hilfe mechanischer Messungen und indirekt mit Hilfe der Strömungsdoppelbrechung untersucht. Für den letzteren Zweck wird das sogenannte spannungsoptische Gesetz herangezogen, dessen Gültigkeit sorgfältig überprüft wird.Es ergibt sich, daß das im Wesen lineare Modell der gummiartigen Flüssigkeit, wie es vonLodge vorgeschlagen wurde, sich recht gut zur Beschreibung des Verhaltens von Polymerschmelzen eignet, solange der im ganzen angelegte Schub den Wert Eins nicht überschreitet. Um auch stationäre Werte der Spannungen in die Beschreibung erfolgreich einzubeziehen, sollte man die Messungen bis zu extrem niedrigen Schergeschwindigkeiten ausdehnen.Die gemachten Feststellungen werden mit Hilfe einer Methode verifiziert, die vonSchwarzl undStruik ursprünglich für die praktische Berechnung von Beziehungen zwischen Zustandsfunktionen entwickelt wurde, die dem linear viskoelastischen Verhalten entsprechen. In der vorliegenden Veröffentlichung dienen die dynamischen Schubmoduln als Bezugsfunktionen.

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a _T shift factor - B _ij Finger deformation tensor - C stress-optical coefficient, (m²/N) - f (p _jl ) undetermined scalar function - G shear modulus, (N/m²) - G(t) time dependent shear modulus, (N/m²) - G() shear storage modulus, (N/m²) - G() shear loss modulus, (N/m²) - G _r reduced shear storage modulus, (N/m²) - G _r reduced shear loss modulus, (N/m²) - H() shear relaxation time spectrum, (N/m²) - k Boltzmann constant, (Nm/°K) - n _ik refractive index tensor - p undetermined hydrostatic pressure, (N/m²) - p _ij ,p _ik stress tensor, (N/m²) - p ₂₁ shear stress, (N/m²) - p ₁₁ –p ₂₂ first normal stress difference, (N/m²) - p ₂₂ –p ₃₃ second normal stress difference, (N/m²) - q shear rate, (s^–1) - t, t time, (s) - T absolute temperature, (°K) - T ₀ reference temperature, (°K) - x the ratiot/ - x position vector of a material point after deformation, (m) - x position vector of a material point before deformation, (m) - ₀, ₁ constants in eq. [37] - ₀, ₁ constants in eq. [37] - shear deformation - (t, t) time dependent shear deformation - _ij unity tensor - n flow birefringence in the 1–2 plane - (q) non-Newtonian shear viscosity, (N s/m²) - ^* () complex dynamic viscosity, (N s/m²) - | ^* ()| absolute value of complex dynamic viscosity, (N s/m²) - () real part of complex dynamic viscosity, (N s/m²) - () imaginary part of complex dynamic viscosity, (N s/m²) - (t — t) memory function, (N/m² · s) - v number of effective chains per unit of volume, (m^–3) - temperature dependent density, (kg/m³) - ₀ density at reference temperatureT ₀, (kg/m³) - relaxation time, (s) - integration variable, (s) - (x) approximate intensity function - ₁ (x) error function - extinction angle - _m orientation angle of the stress ellipsoid - circular frequency, (s^–1) - 1 direction of flow - 2 direction of the velocity gradient - 3 indifferent direction - t time dependence The present investigation has been carried out under the auspices of the Netherlands Organization for the Advancement of Pure Research (Z. W. O.).North Atlantic Treaty Organization Science Post Doctoral Fellow.Research Fellow, Delft University of Technology.With 11 figures and 2 tables     

Effect of slip on porous-walled squeeze films in the presence of a transverse magnetic field

The present investigation is devoted to study the effect of viscous resistance, arising due to sparse distribution of particles in porous media, on the load capacity and thickness time response of porous-walled squeeze films in the presence of a uniform magnetic field. The results of the analysis obtained by using Beavers and Joseph [1] slip-boundary condition show that the viscous resistance increases the load capacity and thickness time response of squeeze films when compared with the results of Chandrasekhara [2] obtained in the absence of viscous resistance. Hence, for efficient performance of a porous walled squeeze film a suitable porous media in which the material is loosely packed may be used.Nomenclature p pressure in the squeeze film - h thickness of the squeeze film at time t - h ₀ thickness of the film at t=0 - u streamwise velocity component in the squeeze film - v transverse velocity component in the squeeze film - P pressure in the porous material - H thickness of the porous material - U streamwise velocity component in the porous material - V transverse velocity component in the porous material - B B ₀+b - B ₀ impressed uniform magnetic field - b induced magnetic field - E electric field vector (E _x , E _y , E _z ) - m ₀ constant defined in (6), (B ₀ ² / _{m f m})^1/2 - v _h value of v at y=h - h/h ₀, the non-dimensional variable - _n eigen values - _f viscosity - _m magnetic permeability - density - _m magnetic diffusivity, 1/ _{m e} - dimensionless parameter, - _e electrical conductivity - q velocity vector (u, v) - L load capacity - I _n integral defined in (37) - M Hartmann number defined in (7), (m ₀ ² h ²)^1/2 - l length of the strips in x-direction - K permeability of the porous material - J current density vector (J _x , J _y , J _z ) - t time - G _n series coefficient appearing in equation (27)     

Finite elastic straining

Summary Finite elastic straining is analysed with all quantities referred consistently to the deformed body taken as the function domain. The straining-displacement of a typical point is relative to a set of axes imbedded in the body at one arbitrary point and rotating in fixed space with that neighborhood if necessary in a particular problem. The resulting |plane stress' equations have precisely the same form as in the classical theory but relate to |true' quantities in the deformed body.The solution of a circular hole in a deformed sheet under simple tension is given and checks closely with experiment on rubber. Cauchy strains of order 65% and local rotation of order 30° are found to occur at the hole boundary.The solution of a deformed quadrantal cantilever is given. Cauchy strains of several hundred percent and local rotation of order 90° occur.Any boundary value problem already solved for the classical infinitesimal strains theory can be applied directly as a finite strains solution for the deformed body.Notation x, y, z, r, , z Cartesian and polar co-ordinates respectively - , Normal and shear true stresses respectively - , Normal and shear true strains respectively - r Position vector - Airy stress function - S Simple tensile stress applied to sheet - a Radius of circular hole in deformed sheet - a, b Inner and outer radii of quadrantal cantilever - u Straining-displacement vector - u, v Straining-displacement scalar components - E, True Young's modulus and Poisson's ratio respectively - c ₁, c ₂ Local unit vectors in principal normal strains directions - i, j Cartesian axes constant unit vectors - Stress dyadic or tensor - First stress invariant - I Idemfactor or spherical tensor - P Shear load per unit thickness applied to quadrantal cantilever - A, B, D, N, H, K, L Arbitrary constants of integration     

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.     

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/     

Temperature distribution in a thin rotating disk

The problem of heat conduction in a thin rotating disk with heat input at a fixed point is considered. The disk is cooled by forced convection from its lateral surfaces. By defining a complex temperature, the temperature throughout the disk is presented as a series of Bessel functions of complex argument. Results are given for a range of rotational speeds.Nomenclature R radial coordinate - angular coordinate - a radius of disk - b thickness of disk - T temperature - T ambient temperature - rotational speed of disk - q heat flux into disk - k thermal conductivity of disk - density of disk - c specific heat of disk - h coefficient of convective heat transfer - r dimensionless radial coordinate, R/a - T* characteristic temperature, q ₀ a/ k - t dimensionless temperature, (T–T )/T* - C ₁, C ₂ dimensionless parameters defined in (3)     

An optical method for the measurement of unsteady film thickness

This paper describes a measurement technique to quantify temporal variations in the thickness of an unsteady liquid film with a resolution which is independent of the thickness. The optical transformation function has been derived for fringes of equal inclination and, for a temporally varying film, allows the unsteady component of film thickness to be measured in terms of frequency modulated signal analysis of light intensity variations. As it does not require calibration, the method is suited to in-situ measurements of complex and rapidly varying films as encountered in engineering two-phase flows. It requires the inversion of the frequency time-series of the light intensity observed by a photodetector which represents the absolute values of the time-derivative of the thickness variation.The technique has been used to measure the thickness of the film formed as a result of impingement of a pulsating two-phase jet onto a heated flat plate with surface temperatures of 150 °C and 240 °C and located 143 nozzle exit diameters downstream of the nozzle. The angle between the jet axis and the surface normal was 20 degrees and the injection frequency was 16.7 Hz corresponding to a flow rate of 7.2 mm³ per injection. The results along the line of incidence showed that the ensemble-averaged space-time structure of the film was qualitatively independent of the plate temperature with three peaks, two of which occurred at large radial distances and disappeared in less than 10 ms. The third peak was close to the impingement region and persisted for more than 50 ms due to the small velocities of the incoming two-phase jet as the nozzle needle closed and the low momentum wall jet which was unable to transport the droplets radially outwards. At the higher surface temperature, the rate of evaporation and the amplitude variation of the unsteady component of the overall film thickness increased, and the film covered a smaller area.List of symbols A amplitude of electric field - E electric field produced at a point - f _i (t) instantaneous frequency - f _PMT (t) frequency observed by photomultiplier tube - h thickness resolution - h _m minimum thickness that can be measured - h (t, x) unsteady film thickness - h _s (x) steady film thickness - h _t (t,x) overall film thickness - H nozzle-to-plate distance - i - I light intensity at a point - interference term - k wavenumber of the monochromatic point source - n refractive index of the air - refractive index of thin film material - r radial distance from the geometrical impingement point - rms root-mean-square - t time - t ₀ zero-crossing time defined in Fig. 2 - t _r rise-time defined in Fig. 2 - T _w wall temperature - x position vector in 3-D space Greek symbols angle of impingement - angle of reflection - integration constant defined in Eq. (15) - _c integration constant at r = 0 - phase angle - P optical path difference - P minimum optical path difference - azimuthal coordinate defined in Fig. 3 a - initial phase angle defined as = h _s (x) - wavelength of the illuminating source - initial phase of an electric field produced by a source - _f (t) ensemble-averaged rms series for f _PMT (t) - angle of incidence - angle of refraction - angular frequency of the electric field produced by a source - optical transformation function     

The wedge subjected to tractions: a paradox resolved

The classical two-dimensional solution provided by Lévy for the stress distribution in an elastic wedge, loaded by a uniform pressure on one face, becomes infinite when the opening angle 2 of the wedge satisfies the equation tan 2 = 2. Such pathological behavior prompted the investigation in this paper of the stresses and displacements that are induced by tractions of O(r ^–) as r0. The key point is to choose an Airy stress function which generates stresses capable of accommodating unrestricted loading. Fortunately conditions can be derived which pre-determine the form of the necessary Airy stress function. The results show that inhomogeneous boundary conditions can induce stresses of O(r ^–), O(r ^– ln r), or O(r ^– ln² r) as r0, depending on which conditions are satisfied. The stress function used by Williams is sufficient only if the induced stress and displacement behavior is of the power type. The wedge loaded by uniform antisymmetric shear tractions is shown in this paper to exhibit stresses of O(ln r) as r0 for the half-plane or crack geometry. At the critical opening angle 2, uniform antisymmetric normal and symmetric shear tractions also induce the above type of stress singularity. No anticipating such stresses, Lévy used an insufficiently general Airy stress function that led to the observed pathological behavior at 2.     

Conversion of longitudinal waves into electromagnetic radiation in a slowly varying plasma under W.K.B. approximation

This paper investigates the conversion of a dispersive longitudinal oscillation into reflected and transmitted electromagnetic radiation fields in slowly varying unmagnetized warm fluid plasmas, using W.K.B. approximations. The expressions for the power of the transmitted and reflected electromagnetic radiations, generated by electron acoustic waves, have also been obtained. It is shown that this conversion process becomes most efficient under certain conditions.

Nomenclature

In § 2 H magnetic field - H ₁ - u electron fluid velocity - k _t wave number of the transverse wave - k ₁ wave number of the longitudinal wave in electron fluid - m electronic mass - N ₀ number density of electrons in the unperturbed state - N perturbation in the electron number density - p perturbation in the electron fluid pressure - v _e adiabatic sound velocity of the electron fluid - K _t ² c ^{2 2}–e² - K ₁ ² v _e ^{2 2}–e² - wave frequency - _e electron plasma frequency - 1– _e ² / ² - c velocity of light in vacuum In § 3 K ₀ wave number in the 0X direction - K ₁ ² K ₁ ² –K ₀ ² - K ₂ ² K _t ² –K ₀ ² - K ₃ K ₁–K ₂ - K ₄ K ₁+K ₂ - K ₅ (K ₁ K ₂)^1/2 See Appendix A - A ₁ pressure amplitude of the reflected part of the incident wave - B ₁ pressure amplitude of the transmitted part of the incident wave - L characteristic length of variation ofN ₀ - e _x unit vector along 0X - e _z unit vector along 0Z In § 4 S _t Poynting flux of the transverse electromagnetic radiation - S _tZ ^/t Average of the transmitted part of the poynting flux along 0Z over the time period 2/ - S _tZ ^/r Average of the reflected part of the poynting flux along 0Z over the time period 2/ In § 5 S ₁ Energy flux carried by the longitudinal pressure wave - S _1Z ^/t Average of the transmitted part ofS ₁ along 0Z over the time period 2/     

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     

Positively invariant regions for a problem in phase transitions

Positively invariant regions for the system v _t + p(W) _x = V _xx , W _t –V _x = W _xx are constructed where p < 0, w < , w > , p(w) = 0, w , > 0. Such a choice of p is motivated by the Maxwell construction for a van der Waals fluid. The method of an analysis is a modification of earlier ideas of Chueh, Conley, & Smoller [1]. The results given here provide independent L bounds on the solution (w, v).Dedicated to Professor James Serrin on the occasion of his sixtieth birthday     

Translatory accelerating motion of a circular disk in a viscous fluid

The exact solution of the equation of motion of a circular disk accelerated along its axis of symmetry due to an arbitrarily applied force in an otherwise still, incompressible, viscous fluid of infinite extent is obtained. The fluid resistance considered in this paper is the Stokes-flow drag which consists of the added mass effect, steady state drag, and the effect of the history of the motion. The solutions for the velocity and displacement of the circular disk are presented in explicit forms for the cases of constant and impulsive forcing functions. The importance of the effect of the history of the motion is discussed.Nomenclature a radius of the circular disk - b one half of the thickness of the circular disk - C dimensionless form of C ₁ - C ₁ magnitude of the constant force - D fluid drag force - f(t) externally applied force - F() dimensionaless form of applied force - F ₀ initial value of F - g gravitational acceleration - H() Heaviside step function - k magnitude of impulsive force - K dimensionless form of k - M a dimensionless parameter equals to (1+37#x03C0;_s/4_f) - S displacement of disk - t time - t ₁ time of application of impulsive force - u velocity of the disk - V dimensionless velocity - V ₀ initial velocity of V - V _t terminal velocity - parameter in (13) - parameter in (13) - (t) Dirac delta function - ratio of b/a - () function given in (5) - dynamical viscosity of the fluid - kinematic viscosity of the fluid - _f fluid density - _s mass density of the circular disk - dimensionless time - _i dimensionless form of t _i - dummy variable - dummy variable