Some measurements in a binary gas jet

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

The râle of microscopic cross flow in idealized porous media

In this paper, results from a combined network/averaging study are presented. The emphasis is placed on understanding the flow phenomena, rather than predicting results for real porous media. Idealized porous media, consisting of networks of tubes, are used to interpret two of the terms in the averaged momentum equation. In particular, it is demonstrated that the pressure term accounts for microscopic cross flow, and that the magnitude of this term is proportional to the variation of the cross-sectional areas of the tubes in the macroscopic flow direction. For one-dimensional macroscopic flow in these idealized porous media, the agreement of network theory and averaging theory permeabilities depends on areosity (a term related to the area open to flow in a direction) remaining constant in the macroscopic flow direction; it may vary in other directions.     

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

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

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

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

A study on the melt viscosity of linear and branched poly(butyleneterephthalate)

Linear and branched PBTP samples were synthesized and characterized in terms of the intrinsic viscosity, the melt-flow-index and, for some, the melt viscosity over a range of shear rates at 250 °C.An exponent of 3.2 in the equation relating to was found for linear samples. Both linear and branched samples exhibited Newtonian behaviour over a wide range of shear rates, but for any given melt-viscosity the branched samples became shear thinning at lower shear rates than the linear ones. Correlation between a branching index,, and melt-visocity ratio (0,b/0,l) was in agreement with a previous theoretical study.     

Relaxation of quasiconvex functional in BV(Ω, ℝp) for integrands f(x,u,∇;u)

In this paper it is shown that if p(x, u,·) is a quasiconvex function with linear growth, then the relaxed functional in BV(, ^p) of

Heat transfer between gas-liquid spray stream flowing perpendicularly to the row of the cylinders

Experimental investigation and analysis of heat transfer process between a gas-liquid spray flow and the row of smooth cylinders placed in the surface perpendicular to the flow has been performed. Among others, there was taken into account in the analysis the phenomenon of droplets bouncing and omitting the cylinder as well as the phenomenon of the evaporation process from the liquid film surface.In the experiments test cylinders were used, which were placed between two other cylinders standing in the row.From the experiments and the analysis the conclusion can be drawn that the heat transfer coefficients values for a row of the cylinders are higher than for a single cylinder placed in the gasliquid spray flow.

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Wärmeübergang an eine senkrecht anf eine Zylinderreihe auftreffende Gas-Flüssigkeits-Sprüh-Strömung

Zusammenfassung Es wurden Messungen und theoretische Analysen des Wärmeübergangs zwischen einer Gas-FlüssigkeitsSprüh-Strömung und den glatten Oberflächen einer Zylinderreihe durchgeführt, die senkrecht zum Sprühstrahl angeordnet waren. Dabei wurde in der Analyse unter anderem das Phänomen betrachtet, daß die Tropfen die Zylinderwand treffen und verfehlen können und daß sich ein Verdampfungsprozeß aus dem flüssigen Film an der Zylinderoberfläche einstellt.Gemessen wurde an einem zwischen zwei Randzylindern befindlichen Zylinder.Die Experimente und die Analyse gestatten die Schlußfolgerung, daß der Wärmeübergangskoeffizient für eine Zylinderreihe höher ist als für einen einzelnen Zylinder in der Sprühströmung.

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Nomenclature a distance between axes of cylinders, m - c _l specific heat capacity of liquid, J/kg K - c _g specific heat capacity of gas, J/kg K - D cylinder diameter, m - g _l mass velocity of liquid, kg/m²s - ¯k average volume ratio of liquid entering film to amount of liquid directed at the cylinder in gas-liquid spray flow, dimensionless - k() local volume ratio of liquid entering film to amount of liquid directed at the cylinder in gas-liquid spray flow, dimensionless - L specific latent heat of vaporisation, J/kg - m mass fraction of water in gas-liquid spray flow, dimensionless - M constant in Eq. (9) - p pressure, Pa - p _g statical pressure of gas, Pa - p _w pressure of gas on the cylinder surface, Pa - p external pressure on the liquid film surface, Pa - r cylindrical coordinate, m - R radius of cylinder, m - T temperature, K, °C - T _l, t_l liquid temperature in the gas-liquid spray, K, °C - T _w,t_w temperature of cylinder surface, K, °C - T temperature of gas-liquid film interface, K - U liquid film velocity, m/s - w gas velocity on cylinder surface, m/s - w _g gas velocity in free stream, m/s - W _l liquid vapour mass ratio in free stream, dimensionless - W liquid vapour mass ratio at the edge of a liquid film, dimensionless - x coordinate, m - y coordinate, m - z complex variable, dimensionless - average heat transfer coefficient, W/m²K - local heat transfer coefficient, W/m² K - average heat transfer coefficient between cylinder surface and gas, W/m² K - _g, local heat transfer coefficient between cylinder surface and gas, W/m² K - mass transfer coefficient, kg/m²s - liquid film thickness, m - _lg dynamic diffusion coefficient of liquid vapour in gas, kg/m s - pressure distribution function on a cylinder surface - function defined by Eq. (3) - _l liquid dynamic viscosity, kg/m s - _g gas dynamic viscosity, kg/m s - cylindrical coordinate, rad, deg - _l thermal conductivity of liquid, W/m K - _g thermal conductivity of gas, W/m K - mass transfer driving force, dimensionless - _l density of liquid, kg/m³ - _g density of gas, kg/m³ - _w shear stress on the cylinder surface, N/m² - _w shear stress exerted by gas at the liquid film surface, N/m² - air relative humidity, dimensionless - T -T _w - _w =T _wT_l Dimensionless parameters I= enhancement factor of heat transfer - m ^*=M _l/M_g molar mass of liquid to the molar mass of gas ratio - Nu _g= D/ _g gas Nusselt number - Pr _g=c _{g g}/_g gas Prandtl number - Pr _l=c_lll liquid Prandtl number - ¯r=(r–R)/ dimensionless coordinate - Re _g=w_gD _g/_g gas Reynolds number - Re _g,max=w_g,max D _g/_g gas Reynolds number calculated for the maximal gas velocity between the cylinders - Sc=m ^* _g/_l–g Schmidt number =/R dimensionless film thickness     

Dehnungsverhalten von Polyäthylen-Schmelzen

Zusammenfassung In einem Dehnungsrheometer werden Spannungs-Dehnungs-Diagramme von Polyäthylen-Schmelzen bei 150 °C und bei konstanter Dehnungsgeschwindigkeit gemessen ( zwischen 0,001 und 1 sec^–1). Weiterhin wird der reversible (elastische) Dehnungsanteil bestimmt. Messungen mit einem Dehnungstester für Kunststoff-Schmelzen ergänzen die Ausführungen.Die Ergebnisse zeigen deutlich, daß bei Dehnung mit zunehmender Verformungsgeschwindigkeit die Dehnungsviskosität nicht abnimmt, im Gegensatz zu dem bekannten strukturviskosen Verhalten bei Scherung.Bei Dehnungen bis zu=1 kann das Verhalten unabhängig von beschrieben werden, wenn als viskoelastische Materialfunktion die Dehnungs-Spannviskosität betrachtet wird. In diesem Bereich von gilt dabei die BeziehungT(t)=3 _s (t) mit _s (t) als zeitabhängige Scherviskosität im linear-viskoelastischen Bereich.Bei größeren Dehnungen und nicht zu kleinen Dehnungsgeschwindigkeiten zeigt verzweigtes Polyäthylen eine zusätzliche starke Spannungszunahme. In dem Bereich dieser zusätzlichen Verfestigung ist das Verhalten im wesentlichen eine Funktion der Dehnung und fast unabhängig von . Die zusätzliche Verfestigung scheint eine Folge der Verzweigungsstruktur des verzweigten Polyäthylens zu sein, da bei Linear-PE ein derartiger Verlauf des Spannungs-Dehnungs-Diagramms nicht beobachtet wird.Die Betrachtung des reversiblen Dehnungsanteils _R zeigt bei der ausführlich untersuchten Schmelze I (verzweigtes PE) drei verschiedene Bereiche: Unterhalb einer Grenzdehnungsgeschwindigkeit ist _R =0, unterhalb einer Versuchszeitt ^** ist _R =. Im dazwischenliegenden Bereich treten elastische und viskose Dehnungsanteile auf,= _R + _V , wobei für niedrige gilt, daß _R lg . Die Grenze wird der Frequenz der thermisch aktivierten Platzwechsel zugeordnet,t ^** erscheint als Zeit, innerhalb der die Verhakungen wie fixierte Vernetzungen wirken.In dem Anhang wird der Einfluß der Grenzflächenspannung zwischen PE-Schmelze und Silikonöl auf die Ergebnisse der Dehnungsversuche diskutiert.

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Summary Stress-strain relations for different PE melts are measured at 150 °C in an extensional rheometer under the condition of a constant extensional strain rate ( between 0,001 and 1 sec^–1). Further, the recoverable (elastic) portion _R of the total strain ( in Hencky's measure) is determined and additional measurements with a tensile tester for polymer melts are described.The results show clearly that in extension there is no decrease of the tensile viscosity with increasing deformation rate, in contrast to the well-known pseudoplastic behaviour in shear. Within total strains<1 the tensile behaviour can be described independently from by means of a viscoelastic material function called stressing viscosity . In this range of the relation _T (t)=3 _s (t) holds, where _s (t) is the stressing viscosity in shear in the linear viscoelastic range. For larger tensile strains and not too small branched PB shows a remarkable increase in stress. This work-hardening behaviour is mainly a function of and almost independent from . This additional hardening seems to be due to the branches in branched PE, because linear PE does not show this effect.The discussion of the recoverable tensile strain _R gives three regions of tensile rate: Below a critical there is _R =0. At times shorter thant ^** the equation _R = is valid. Within these limits both elastic and viscous portions of the total strain= _R + _V exist. may correlate with the frequency of the thermally activated position changes of the molecular segments.t ^** is assumed to be the time for the entanglements to act as fixed cross-links.In the appendix the influence of the interface tension between PE melt and silicone oil on the results of the tensile experiments is discussed.

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Vorgetragen auf der Deutschen Rheologen-Tagung, Berlin, 11.-13. Mai 1970.

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An der Weiterentwicklung des Dehnungsrheometers, an der Durchführung und Auswertung der Messungen waren die HerrenB. Kienle, F. Landmesser, M, Reuther undF. Scherr beteiligt. Herr Dr.F.Ramsteiner und HerrH. Schroeck haben sich um die Herstellung der Stränge aus Linear-PE bemüht. Herr Dr.W. Ball besorgte die GPC-Messungen und Herr Dr.P. Simak die Ultrarot-Untersuchung. Den vorgenannten Herren sei für ihre Hilfe beim Zustandekommen dieser Arbeit gedankt. Herrn Dr.H. Baur danke ich für wertvolle Diskussionen.     

Production of temperature fluctuations in grid turbulence: Wiskind's experiment revisited

Measurements have been made in nearly-isotropic grid turbulence on which is superimposed a linearly-varying transverse temperature distribution. The mean-square temperature fluctuations, , increase indefinitely with streamwise distance, in accordance with theoretical predictions, and consistent with an excess of production over dissipation some 50% greater than values recorded in previous experiments. This high level of production has the effect of reducing the ratio,r, of the time scales of the fluctuating velocity and temperature fields. The results have been used to estimate the coefficient,C, in Monin's return-to-isotropy model for the slow part of the pressure terms in the temperature-flux equations. An empirical expression by Shih and Lumley is consistent with the results of earlier experiments in whichr 1.5, C 3.0, but not with the present data where r 0.5, C 1.6. Monin's model is improved when it incorporates both time scales.List of symbols C coefficient in Monin model, Eq. (5) - M grid mesh length - m exponent in power law for temperature variance, x ^m - n turbulence-energy decay exponent,q ² x ^-n - p production rate of - p pressure - q ² - R microscale Reynolds number - r time-scale ratiot/t - T mean temperature - U mean velocity - mean-square velocity fluctuations (turbulent energy components) - turbulent temperature flux - x, y, z spatial coordinates - temperature gradient dT/dy - thermal diffusivity - dissipation rate ofq ²/2 - dissipation rate of - Taylor microscale (₂=5q₂/) - temperature microscale - _v temperature-flux correlation coefficient, /v - dimensionless distance from the grid,x/M     

Helical flow of molten polymers in a cylindrical annulus

Summary The paper is concerned with an analytical investigation of helical flow of a non-Newtonian fluid through an annulus with a rotating inner cylinder. The shear dependence of viscosity is described by a power law and the temperature dependence by an exponential function.Velocity and temperature profiles, energy input and shear along the stream lines, pressure drop, and torque are presented for the range of input parameters encountered in polymer extrusion.The results of the study can be applied to a mixing element in a screw extruder and for a device to control extrudate temperature and output.Nomenclature a thermal diffusivity [m²/s] - b temperature coefficient [K^–1], see eq. [4] - c heat capacity [J/kg K] - h slot width [m] - I ₁,I ₂,I ₃ invariants of the rate of deformation tensor, see eq. [5] - k thermal conductivity [J/m s K] - l, L = 1/h length of the slot - l _T ,l _K thermal and kinematic entrance length - m power law exponent, see eq. [3] - M torque [m N] - p pressure [N/m²] - P dimensionless pressure gradient, see eq. [24] - P _R,P _RZ dimensionless components of the shear stress tensor, see eq. [25] and eq. [26] - r, R = r/r _wa radial coordinate - r _wa, r_wi outer and inner radius of annulus [m] - t time [s]; dwell time in the annulus - T temperature [K] - v , v_r, V_z velocity components [m/s] - v ₀ angular velocity at inner wall [m/s] - average velocity inz-direction [m/s] - V , V_R, V_Z dimensionless velocity components,v /v₀, v_r/v₀, v_z/v₀ - V _z velocity ratio, helical parameter - Y coordinate inr-direction, see eq. [20] - z, Z = z/h Pe axial coordinate - deformation - rate of deformation tensor [s^–1] - apparent viscosity [N s/m²], see eq. [3] - dimensionless temperature,b (T – T ₀) - azimuth coordinate - ratio of radii,r _wi/r_wa - density [kg/m³] - , kl shear stress tensor [N/m²] - fluidity [m²w/Nw s], see eq. [4] - Gf Griffith number, see eq. [12] - Pe Péclet number, see eq. [13] - Re Reynolds number, - 0 initial state, reference state - equilibrium state - e entrance - wi, wa at surface of inner or outer wall - r, R, z, Z, coordinates - i, j radial and axial position of nodal point in the grid - k, l tensor components Presented at Euromech 37, Napoli 6. 20–23. 1972.With 15 figuresDedicated to Prof. Dr.-Ing. G. Schenkel on his 60th birthday     

Laminar source flow between parallel porous disks with equal suction and injection

An analysis is presented for laminar source flow between parallel stationary porous disks with suction at one of the disks and equal injection at the other. The solution is in the form of an infinite series expansion about the solution at infinite radius, and is valid for all suction and injection rates. Expressions for the velocity, pressure, and shear stress are presented and the effect of the cross flow is discussed.Nomenclature a distance between disks - A, B, ..., J functions of R _w only - F static pressure - p dimensionless static pressure, p(a ²/ ²) - Q volumetric flow rate of the source - r radial coordinate - r dimensionless radial coordinate, r/a - R radial coordinate of a point in the flow region - R dimensionless radial coordinate of a point in the flow region, R - Re source Reynolds number, Q/2a - R _w wall Reynolds number, Va/ - reduced Reynolds number, Re/r ² - critical Reynolds number - velocity component in radial direction - u dimensionless velocity component in radial direction, a/ - average radial velocity, Q/2a - u dimensionless average radial velocity, Re/r - ratio of radial velocity to average radial velocity, u/u - velocity component in axial direction - v dimensionless velocity component in axial direction, v - V magnitude of suction or injection velocity - z axial coordinate - z dimensionless axial coordinate, z a - viscosity - density - kinematic viscosity, / - shear stress at lower disk - shear stress at upper disk - ₀ dimensionless shear stress at lower disk, - ₁ dimensionless shear stress at upper disk, - dimensionless stream function     

Material functions generated by the complex viscosity

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

The contribution of internal degrees of freedom to the non-Newtonian rheology of model polymer fluids

We report non-equilibrium molecular dynamics simulations of rigid and non-rigid dumbbell fluids to determine the contribution of internal degrees of freedom to strain-rate-dependent shear viscosity. The model adopted for non-rigid molecules is a modification of the finitely extensible nonlinear elastic (FENE) dumbbell commonly used in kinetic theories of polymer solutions. We consider model polymer melts — that is, fluids composed of rigid dumbbells and of FENE dumbbells. We report the steady-state stress tensor and the transient stress response to an applied Couerte strain field for several strain rates. We find that the rheological properties of the rigid and FENE dumbbells are qualitatively and quantitatively similar. (The only exception to this is the zero strain rate shear viscosity.) Except at high strain rates, the average conformation of the FENE dumbbells in a Couette strain field is found to be very similar to that of FENE dumbbells in the absence of strain. The theological properties of the two dumbbell fluids are compared to those of a corresponding fluid of spheres which is shown to be the most non-Newtonian of the three fluids considered.Symbol Definition b dimensionless time constant relating vibration to other forms of motion - F force on center of mass of dumbbell - F _i force on bead i of dumbbell - F force between center of masses of dumbbells and - F _ij force between beads i and j - h vector connecting bead to center of mass of dumbbell - H dimensionless spring constant for dumbbells, in units of / ² - I moment of inertia of dumbbell - J general current induced by applied field - k _B Boltzmann's constant - L angular momentum - m mass of bead, (= m/2) - M mass of dumbbell, g - N number of dumbbells in simulation cell - P translational momentum of center of mass of dumbbell - P pressure tensor - P _xy xy component of pressure tensor - Q separation of beads in dumbbell - Q _eq equilibrium extension of FENE dumbbell and fixed extension of rigid dumbbell - Q ₀ maximum extension of dumbbell - r _ij vector connecting beads i and j - r position vector of center of mass dumbbell - R vector connecting centers of mass of two dumbbells - t time - t ^* dimensionless time, in units of m/ - T ^* dimensionless temperature, in units of /k - u potential energy - u velocity vector of flow field - u _x x component of velocity vector - V volume of simulation cell - X general applied field - strain rate, s^–1 - ^* dimensionless shear rate, in units of /m ² - general transport property - Lennard-Jones potential well depth - friction factor for Gaussian thermostat - shear viscosity, g/cms - ^* dimensionless shear viscosity, in units of m/ ² - ^* dimensionless number density, in units of ^–3 - Lennard-Jones separation of minimum energy - relaxation time of a fluid - angular velocity of dumbbell - orientation angle of dumbbell      

Transient flow towards a well in an aquifer including the effect of fluid inertia

Transient propagation of weak pressure perturbations in a homogeneous, isotropic, fluid saturated aquifer has been studied. A damped wave equation for the pressure in the aquifer is derived using the macroscopic, volume averaged, mass conservation and momentum equations. The equation is applied to the case of a well in a closed aquifer and analytical solutions are obtained to two different flow cases. It is shown that the radius of influence propagates with a finite velocity. The results show that the effect of fluid inertia could be of importance where transient flow in porous media is studied.List of symbols b Thickness of the aquifer, m - c ₀ Wave velocity, m/s - k Permeability of the porous medium, m² - n Porosity of the porous medium - p( ,t) Pressure, N/m² - Q Volume flux, m³/s - r Radial coordinate, m - r _w Radius of the well, m - s Transform variable - S Storativity of the aquifer - S _d(r, t) Drawdown, m - t Time, s - T Transmissivity of the aquifer, m²/s - ( ,t) Velocity of the fluid, m/s - Coordinate vector, m - z Vertical coordinate, m - Coefficient of compressibility, m²/N - Coefficient of fluid compressibility, m²/N - Relaxation time, s - (r, t) Hydraulic potential, m - Dynamic viscosity of the fluid, Ns/m² - Dimensionless radius - Density of the fluid, Ns²/m⁴ - (, ) Dimensionless drawdown - Dimensionless time - , x Dummy variables - ₀, ₁ Auxilary functions     

Regularity for the stationary Navier-Stokes equations in bounded domain

Let u, p be a weak solution of the stationary Navier-Stokes equations in a bounded domain ^N, 5N . If u, p satisfy the additional conditions

Non-Linear response of a long,discontinuous fiber/melt system in elongational flows

The authors investigated the transient elongational behavior of a highly-aligned 600% volume fraction long, discontinuous fiber filled poly-ether-ketone-ketone melt with a computer-controlled extensional rheometer at 370°C. Prior experiments at controlled strain rate and stress produced _E ⁺ (t, ) and ^– (t, _E) similar to a shear dominated flow of a non-linear viscoelastic fluid. Stress relaxation following steady extension showed nonlinear effects in the change in stress decay rate with increasing strain rate. Continuous relaxation spectra showed a shift in the spectral peak to smaller values of with increasing strain rate. The Giesekus nonlinear constitutive relation modeled the elongation and stress relaxation with shearing rate at the fiber surface set by a strain rate magnification factor. Suitable for elongation, the model produced insufficient shift in the stress relaxation spectrum to account for the large change in stress decay rate exhibited in the experiments.English alphabet a _r aspect ratio of the fibers or l/d - A ₀ initial uniform cross-section area of the specimen - d fiber diameter - f fiber volume fraction - H() relaxation spectrum found by the method of Ferry and William l length of the fiber - L(t) time function specimen length - L ₀ initial specimen length - r radial coordinate across the shear cell - R _i fiber radius and inner cell dimension - R _o outer cell radius - t time in s - t _max duration of the extension - T _g glass transition temperature of the polymer - v velocity of the moving end of the test specimen - x axial position where is calculated Greek alphabet nonlinearity parameter in the Giesekus relation - axial mass distribution along the specimen major axis - shear strain rate - strain tensor - ₍₁₎ first convected derivative of the strain tensor - ₍₂₎ second convected derivative of the strain tensor - average strain at the end of extension as determined from - extension strain rate - average extension strain rate determined from - ^– transient strain rate under controlled stress, creep, test - _E elongational viscosity - _Eapp apparent elongational viscosity determined from - _E ⁺ transient elongational viscosity - ₀ zero shear rate viscosity - relaxation parameter - ₁ relaxation parameter in either Jeffrey's or Giesekus fluid - ₂ retardation parameter in either Jeffrey's or Giesekus fluid - _max relaxation value at which 99.9% of the H spectrum had occurred - _p relaxation value at which H reaches a maximum - volumetric composite density - _E elongational stress - _E ⁺ transient elongational stress - _E ^– controlled elongational stress, creep stress - _E ^y peak elongational stress in controlled experiment - shear stress at surface of the fiber in a shear cell - _yx simple shear component of the strain rate tensor - stress tensor - ₁ first convected derivative of the stress tensor     

A general model for the effective viscosity of pseudoplastic and dilatant fluids

Summary A new and very general expression is proposed for correlation of data for the effective viscosity of pseudoplastic and dilatant fluids as a function of the shear stress. Most of the models which have been proposed previously are shown to be special cases of this expression. A straightforward procedure is outlined for evaluation of the arbitrary constants.

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Zusammenfassung Eine neue und sehr allgemeine Formel wird für die Korrelation der Werte der effektiven Viskosität von strukturviskosen und dilatanten Flüssigkeiten in Abhängigkeit von der Schubspannung vorgeschlagen. Die meisten schon früher vorgeschlagenen Methoden werden hier als Spezialfälle dieser Gleichung gezeigt. Ein einfaches Verfahren für die Auswertung der willkürlichen Konstanten wird beschrieben.

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Nomenclature b arbitrary constant inSisko model (eq. [5]) - n arbitrary exponent in eq. [1] - x independent variable - y(x) dependent variable - y ₀(x) limiting behavior of dependent variable asx 0 - y(x) limiting behavior of dependent variable asx - z original dependent variable - arbitrary constant inSisko model (eq. [5]) andBird-Sisko model (eq. [6]) - arbitrary exponent in eqs. [2] and [8] - effective viscosity = shear stress/rate of shear - _A effective viscosity at = _A - _B empirical constant in eqs. [2] and [8] - ₀ limiting value of effective viscosity as 0 - ₀() limiting behavior of effective viscosity as 0 - limiting value of effective viscosity as - () limiting behavior of effective viscosity as - rate of shear - arbitrary constant inBird-Sisko model (eq.[6]) - shear stress - _A arbitrary constant in eqs. [2] and [8] - ₀ shear stress at inBingham model - _1/2 shear stress at = ( ₀ + )/2 With 8 figures     

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

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

Simultaneous measurements of velocity and temperature fluctuations in thermal boundary layer in a drag-reducing surfactant solution flow

The mechanism of turbulent heat transfer in the thermal boundary layer developing in the channel flow of a drag-reducing surfactant solution was studied experimentally. A two-component laser Doppler velocimetry and a fine-wire thermocouple probe were used to measure the velocity and temperature fluctuations simultaneously. Two layers of thermal field were found: a high heat resistance layer with a high temperature gradient, and a layer with a small or even zero temperature gradient. The peak value of was larger for the flow with the drag-reducing additives than for the Newtonian flow, and the peak location was away from the wall. The profile of was depressed in a similar manner to the depression of the profile of in the flow of the surfactant solution, i.e., decorrelation between v and compared with decorrelation between u and v. The depression of the Reynolds shear stress resulted in drag reduction; similarly, it was conjectured that the heat transfer reduction is due to the decrease in the turbulent heat flux in the wall-normal direction for a flow with drag-reducing surfactant additives.List of symbols ensemble averaged value - (·)⁺ normalized by the inner wall variables - (·) root-mean-square value - C concentration of cetyltrimethyl ammonium chloride (CTAC) solution - c _p heat capacity - D hydraulic diameter - f friction factor - H channel height - h heat transfer coefficient - j _H Colburn factor - l length - Nu Nusselt number, h - Pr Prandtl number, c _p/ - q _w wall heated flux - Re Reynolds number, U _b/ - T temperature - T _b bulk temperature - T _i inlet temperature - T _w wall temperature - T friction temperature, q _w /c _p u - U local time-mean streamwise velocity - U ₁ velocity signals from BSA1 - U ₂ velocity signals from BSA2 - U _b bulk velocity - u streamwise velocity fluctuation - u1 velocity in abscissa direction in transformed coordinates - u friction velocity, - v wall-normal velocity fluctuation - v1 velocity in ordinate direction in transformed coordinates - var(·) variance - x streamwise direction - y wall-normal direction - z spanwise direction - _j junction diameter of fine-wire TC - _w wire diameter of fine-wire TC - angle of principal axis of joint probability function p(u,v) - _f heat conduction of fluid - _w heat conduction of wire of fine-wire TC - kinematic viscosity - local time-mean temperature difference, T _w –T - temperature fluctuation - standard deviation - density - _w wall shear stress