Rheological properties of skim milk gels at various temperatures; interrelation between the dynamic moduli and the relaxation modulus

The rheological properties of rennet-induced skim milk gels were determined by two methods, i.e., via stress relaxation and dynamic tests. The stress relaxation modulusG _c (t) was calculated from the dynamic moduliG andG by using a simple approximation formula and by means of a more complex procedure, via calculation of the relaxation spectrum. Either calculation method gave the same results forG _c (t). The magnitude of the relaxation modulus obtained from the stress relaxation experiments was 10% to 20% lower than that calculated from the dynamic tests.Rennet-induced skim milk gels did not show an equilibrium modulus. An increase in temperature in the range from 20° to 35 °C resulted in lower moduli at a given time scale and faster relaxation. Dynamic measurements were also performed on acid-induced skim milk gels at various temperatures andG _c (t) was calculated. The moduli of the acid-induced gels were higher than those of the rennet-induced gels and a kind of permanent network seemed to exist, also at higher temperatures. G storage shear modulus,N·m^–2; - G loss shear modulus,N·m^–2; - G _c calculated storage shear modulus,N·m^–2; - G _c calculated loss shear modulus,N·m^–2; - G _e equilibrium shear modulus,N·m^–2; - G _ec calculated equilibrium shear modulus,N·m^–2; - G(t) relaxation shear modulus,N·m^–2; - G _c (t) calculated relaxation shear modulus,N·m^–2; - G ^*(t) pseudo relaxation shear modulus,N·m^–2; - H relaxation spectrum,N·m^–2; - t time,s; - relaxation time,s; - angular frequency, rad·s^–1. Partly presented at the Conference on Rheology of Food, Pharmaceutical and Biological Materials, Warwick, UK, September 13–15, 1989 [33].     

Measurement of the Coefficient of Transverse Dispersion in Flow Through Packed Beds for a Wide Range of Values of the Schmidt Number

Experimental values of the coefficient of transverse dispersion (D _T) were measured with the system 2-naphthol/water, over a range of temperatures between 293K and 373K, which corresponds to a range of values of viscosity () between 2.83×10^–4 Ns/m² and 1.01×10^–3 Ns/m² and of molecular diffusion coefficient (D _m) between 1.03×10^–9 m²/s and 5.49×10^–9 m²/s. Since the density () of water is close to 10³ kg/m³, the corresponding variation of the Schmidt number (Sc=/D _m) was in the range 1000 – 50. More than 200 experimental values of the transverse dispersion coefficient were obtained using beds of silica sand with average particle sizes (d) of 0.297 and 0.496mm, operated over a range of interstitial liquid velocities (u) between 0.1mm/s and 14mm/s and this gave a variation of the Reynolds number (Re=du/) between 0.01 and 3.5.Plots of the dimensionless coefficient of transverse dispersion (D _T/D _m) vs. the Peclet number (Pe_m=ud/D _m) based on molecular diffusion bring into evidence the influence of Sc on transverse dispersion. As the temperature is increased, the value of Sc decreases and the values of D _T/D _m gradually approach the line corresponding to gas behaviour (i.e. Sc 1), which is known to be well approximated by the equation D _T/D _m=1/+ud/12D _m, where is the tortuosity with regard to diffusion.     

Investigation of chemically reacting and radiating supersonic flow in channels

The two-dimensional time dependent Navier-Stokes equations are used to investigate supersonic flows undergoing finite rate chemical reaction and radiation interaction for a hydrogen-air system. The explicit multi-stage finite volume technique of Jameson is used to advance the governing equations in time until convergence is achieved. The chemistry source term in the species equation is treated implicitly to alleviate the stiffness associated with fast reactions. The multidimensional radiative transfer equations for a nongray model are provided for general configuration, and then reduced for a planar geometry. Both pseudo-gray and nongray models are used to represent the absorption-emission characteristics of the participating species.The supersonic inviscid and viscous, nonreacting flows are solved by employing the finite volume technique of Jameson and the unsplit finite difference scheme of MacCormack to determine a convenient numerical procedure for the present study. The specific problem considered is of the flow in a channel with a 10° compression-expansion ramp. The calculated results are compared with the results of an upwind scheme and no significant differences are noted. The problem of chemically reacting and radiating flows are solved for the flow of premixed hydrogen-air through a channel with parallel boundaries, and a channel with a compression corner. Results obtained for specific conditions indicate that the radiative interaction can have a significant influence on the entire flowfield.Nomenclature A band absorptance (m^–1) - A _o band width parameter (m^–1) - C _j concentration of thejth species (kg mol/m³) - C _o correlation parameter ((N/m²)^–1m^–1) - C _p constant pressure specific heat (J/kgK) - e Planck's function (J/m²S) - E total internal energy (J/kg) - f _j mass fraction of thejth species - h static enthalpy of mixture (J/kg) - H total enthalpy (J/kg) - I identity matrix - I _v spectral intensity (J/m s) - I _bv spectral Planck function - k thermal conductivity (J/m sK) - K _b backward rate constant - K _f forward rate constant - I unit vector in the direction of - M _j molecular weight of thejth species (kg/kg mol) - P pressure (N/m²) - P _j partial pressure of thejth species (N/m²) - P _e equivalent broadening pressure ratio - Pr Prandtl number - P _w a point on the wall - q _R total radiative heat flux (J/m² s) - spectral radiative heat flux (J/m³ s) - R gas constant (J/KgK) - r _w distance between the pointsP andP _w(m) - S integrated band intensity ((N/m²)^–1/m^–2) - S integrated band intensity ((N/m²)^–1 m^–2) - T temperature (K) - u, v velocity inx andy direction (m/s) - production rate of thejth species (kg/m³ s) - x, y physical coordinate - z dummy variable in they direction Greek symbols ratio of specific heats - t _ch chemistry time step (s) - t _f fluid-dynamic time step (s) - absorption coefficient (m^–1) - ,_v spectral absorption coefficient (m^–1) - _p Planck mean absorption coefficient - second coefficient of viscosity, wavelength (m) - dynamic viscosity (laminar flow) (kg/m s) - , computational coordinates - density (kg/m³) - Stefan-Boltzmann constant (erg/s cm² K³) - shear stress (W/m²) - equivalence ratio - wave number (m^–1) - _c frequency at the band center     

A new transient method for analysis of solidification in the continuous casting of metals

Solidification processes involve complex heat and mass transfer phenomena, the modelling of which requires state-of-the art numerical techniques. An efficient and accurate transient numerical method is proposed for the analysis of phase change problems. This method combines both the enthalpy and the enhanced specific heat approaches in incorporating the effects of latent heat released due to phase change. The sensitivity and accuracy of the proposed method to both temporal and spatial discretization is shown together with closed-form solutions and the results from the enhanced specific heat approach. In order to explore the proposed method fully, a non-linear heat release, as is the case for binary alloys, is also examined. The number of operations required for the new transient approach is less than or equal to the enhanced heat capacity method depending on the averaging method adopted. To demonstrate the potential of this new finite-element technique, measurements obtained on operating machines for the casting of zinc, aluminum and steel are compared with the model predictions. The death/birth technique, together with the proper heat-transfer coefficients, were employed in order to model the casting process with minimal error due to the modelling itself.Nomenclature [A] conductance matrix - [B] matrix containing the derivative of the element shape functions - c, C _p specific heat (J kg^–1°C^–1) - effective specific heat (J kg^–1°C^–1) - f(T) local liquid fraction - f thermal load vector - H enthalpy (J kg^–1) - [H] capacitance matrix - h, h _r,h _c heat transfer coefficient (W m^–2°C^–1) - K thermal conductivity (W m^–1°C^–1) - L latent heat of solidification (J kg^–1) - l overall length (m) - N _i shape functions - Q rate of heat generation per unit volume (J m^–3) - q heat flux (W m^–2) - R residual temperature (°C) - T temperature (°C) - T _s solidus temperature (°C) - T _l liquidus temperature (°C) - T _pouring pouring temperature (°C) - T _top temperature at the top of the mould (°C) - T _w temperature of the water spray (°C) - approximated temperature (°C) - T surrounding temperature (°C) - cooling rate (°C/s) - t time (seconds) - x _i,x, y, z spatial variables (m) - t time step (s) - x element size (m) - diffusivity (m²s^–1) - density (kg m^–3) - time marching parameter - _n direction cosines of the unit outward normal to the boundary     

Optimum rib size to enhance forced convection in a horizontal triangular duct with ribbed internal surfaces

The optimum rib size to enhance heat transfer had been proposed through an experimental investigation on the forced convection of a fully developed turbulent flow in an air-cooled horizontal equilateral triangular duct fabricated on its internal surfaces with uniformly spaced square ribs. Five different rib sizes (B) of 5 mm, 6 mm, 7 mm, 7.9 mm and 9 mm, respectively, were used in the present investigation, while the separation (S) between the center lines of two adjacent ribs was kept at a constant of 57 mm. The experimental triangular ducts were of the same axial length (L) of 1050 mm and the same hydraulic diameter (D) of 44 mm. Both the ducts and the ribs were fabricated with duralumin. For every experimental set-up, the entire inner wall of the duct was heated uniformly while the outer wall was thermally insulated. From the experimental results, a maximum average Nusselt number of the triangular duct was observed at the rib size of 7.9 mm (i.e. relative rib size ). Considering the pressure drop along the triangular duct, it was found to increase almost linearly with the rib size. Non-dimensional expressions had been developed for the determination of the average Nusselt number and the average friction factor of the equilateral triangular ducts with ribbed internal surfaces. The developed equations were valid for a wide range of Reynolds numbers of 4,000 < Re _D < 23,000 and relative rib sizes of under steady-state condition. A Inner surface area of the triangular duct [m²] - A _C Cross-sectional area of the triangular duct [m²] - B Side length of the square rib [mm] - C _P Specific heat at constant pressure [kJ·kg^–1·K^–1] - C ₁, C ₂, C ₃ Constant coefficients in Equations (10), (12) and (13), respectively - D Hydraulic diameter of the triangular duct [mm] - Electric power supplied to heat the triangular duct [W] - f Average friction factor - F View factor for thermal radiation from the duct ends to its surroundings - h Average convection heat transfer coefficient at the air/duct interface [W·m^–2 ·K^–1] - k Thermal conductivity of the air [W·m^–1 ·K^–1] - L Axial length of the triangular duct [mm] - Mass flow rate [kg·s^–1] - n ₁, n ₂, n ₃ Power indices in Equations (10), (12) and (13), respectively - Nu _D Average Nusselt number based on hydraulic diameter - P Fluid pressure [Pa] - Pr Prandtl number of the airflow - _c Steady-state forced convection from the triangular duct to the airflow [W] - _l Heat loss from external surfaces of the triangular duct assembly to the surroundings [W] - _r Radiation heat loss from both ends of the triangular duct to the surroundings [W] - Re _D Reynolds number of the airflow based on hydraulic diameter - S Uniform separation between the centre lines of two consecutive ribs [mm] - T Fluid temperature [K] - T _a Mean temperature of the airflow [K] - T _ai Inlet mean temperature of the airflow [K] - T _ao Outlet mean temperature of the airflow [K] - T _s Mean surface temperature of the triangular duct [K] - T Ambient temperature [K] - U Mean air velocity in the triangular duct [m·s^–1] - _r Mean surface-emissivity with respect to thermal radiation - Dynamic viscosity of the fluid [kg·m^–1·s^–1] - Kinematic viscosity of the airflow [m²·s^–1] - Density of the airflow [kg·m^–3] - Stefan-Boltzmann constant [W·m^–2·K^–4]     

Thermal stability of composite superconducting tape under the effect of a two-dimensional dual-phase-lag heat conduction model

Thermal stability of composite superconducting tape subjected to a thermal disturbance is numerically investigated under the effect of a two-dimensional dual-phase-lag heat conduction model. It is found that the dual-phase-lag model predicts a wider stable region as compared to the predictions of the parabolic and the hyperbolic heat conduction models. The effects of different design, geometrical and operating conditions on superconducting tape thermal stability were also studied.a conductor width, (m) - A conductor cross sectional area of, (m²) - As conductor aspect ratio, (a/b) - b conductor thickness, (m) - Bi Biot number - B dimensionless disturbance Intensity - C heat capacity, (J m^–3 K^–1) - D disturbance energy density, (W m^–3) - f volume fraction of the stabilizer in the conductor - g(T) steady capacity of the Ohmic heat source, (W m^–3) - g_max Ohmic heat generation with the whole current in the stabilizer, (W m^–3) - G_max dimensionless maximum Joule heating - h convective heat transfer coefficient, (W m^–2 K^–1) - J current density, (A m^–2) - k thermal conductivity of conductor, (W m^–1 K^–1) - q conduction heat flux vector, (W m^–2) - Q dimensionless Joule heating - R relaxation times ratio (_T/2_q) - t rime, (s) - T temperature, (K) - T_c critical temperature, (K) - T_c1 current sharing temperature, (K) - T_i initial temperature, (K) - T_o ambient temperature, (K) - x, y co-ordinate defined in Fig. 1, (m) - thermal diffusivity (m² s^–1) - dimensionless time - _i dimensionless duration time - dimensionless y-variable - _o superconductor dimensionless thickness - dimensionless temperature - _c1 dimensionless current sharing temperature - 1 dimensionless maximum temperature - dimensionless disturbance energy - numerical tolerance - x width of conductor subjected to heat disturbances, (m) - y thickness of conductor subjected to heat disturbances, (m) - dimensionless x-variable - _o superconductor dimensionless width - stabilizer electrical resistivity, () - _q relaxation time of heat flux, (s) - _T relaxation time of temperature gradient, (s) - i initial - sc current sharing - max maximum - o ambient     

Numerical analysis of simulation accuracy for hypersonic heat transfer in subsonic jets of dissociated nitrogen

One-dimensional problems of the flow in a boundary layer of finite thickness on the end face of a model and in a thin viscous shock layer on a sphere are solved numerically for three regimes of subsonic flow past a model with a flat blunt face exposed to subsonic jets of pure dissociated nitrogen in an induction plasmatron [1] (for stagnation pressures of (10⁴–3·10⁴) N/m² and an enthalpy of 2.1·10⁷ m²/sec²) and three regimes of hypersonic flow past spheres with parameters related by the local heat transfer simulation conditions [2, 3]. It is established that given equality of the stagnation pressures, enthalpies and velocity gradients on the outer edges of the boundary layers at the stagnation points on the sphere and the model, for a model of radius R_m=1.5·10^–2 m in a subsonic jet the accuracy of reproduction of the heat transfer to the highly catalytic surface of a sphere in a uniform hypersonic flow is about 3%. For surfaces with a low level of catalytic activity the accuracy of simulation of the nonequilibrium heat transfer is determined by the deviations of the temperatures at the outer edges of the boundary layers on the body and the model. For this case the simulation conditions have the form: dU_e/dx=idem, p₀=idem, T_e=idem. At stagnation pressuresP ₀2·10⁴ N/m² irrespective of the catalycity of the surface the heat flux at the stagnation point and the structure of the boundary layer near the axis of symmetry of models with a flat blunt face of radius R_m1.5·10^–2 m exposed to subsonic nitrogen jets in a plasmatron with a discharge channel radius R_c=3·10^–2 m correspond closely to the case of spheres in hypersonic flows with parameters determined by the simulation conditions [2, 3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 135–143, March–April, 1990.     

Heat transfer to superheated steam flowing in tubes and annular channels

Heat transfer measurements with superheated steam have been performed in the pressure range of 10.5–33.3 bar, steam temperatures: 229–470 °C and heat fluxes: 2.6·10⁴–5.7·10⁵ W/m². The measured data with steam flowing in an electrically heated tube of 10 mm inner diameter and 1 m length could be correlated with two different expressions, the mean steam temperature resp. the film temperature being used as reference for physical properties. The results of measurements in annular channels, with a heated rod of 14 mm outer diameter and channel diameters of 21 and 28 mm, were 10–15% lower than the proposed tube correlations. This could be explained by the differences in velocity and temperature distributions between tubes and annular channels. A satisfactory agreement was achieved between the tube correlations and transformed annular channel results.

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Zusammenfassung Wärmeübergangsmessungen an überhitztem Wasserdampf wurden im folgenden Bereich durchgeführt: Druck: 10,5–33,3 bar, Dampftemperatur: 229–470 °C, Wärmestromdichte: 2,6 · 10⁴-5,7 · 10⁵ W/m². Die Eesultate der in einem elektrisch beheizten Rohr von 10 mm Innendurchdurchmesser und 1 m Länge durchgeführten Meß-Serie konnten durch zwei verschiedene Beziehungen je nach Bezugstemperatur der thermodynamischen Stoffwerte des Kühlmittels beschtieben werden. Als Bezugsgröße wurde die mittlere Dampf — temperatur bzw. die Filmtemperatur verwendet. Messungen, die in einem Ringspalt mit einem beheizten Stab von 14 mm Außendurchmesser und mit einem Kanaldurchmesser von 21 bzw. 28 mm durchgeführt wurden, ergaben 10–15% niedrigere Wärmeübergangszahlen, als die vorgeschlagenen Gleichungen für Rohrgeometrie. Dies kann durch den Unterschied der Geschwindigkeitsverteilungen in Rohren und Ringspalten erklärt werden. Die auf Rohrgeometrie umgerechneten Resultate der Ringspaltmessungen zeigten gute Übereinstimmung mit den vorgeschlagenen Gleichungen.

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Nomenclature

Symbol Dimension A m² surface area - C constant - c _p J/kg °C heat capacity - D m channel outer wall diameter - d m channel inner wall diameter - F m² flow area - f friction coefficient - h W/°C m² heat transfer coefficient - k W/°C m thermal conductivity - L, l m lenght - m kg/s mass flow - m exponent ofRe-number - n exponent of Pr-number - p N/m², bar pressure - Q W power - q W/m² heat flux - r m radius - t °C temperature - w m/s velocity - dimensionless factor for reference temperature - y exponent of the temperature ratio - z exponent of the temperature ratiot _w/t _s - Nusselt-Number - Prandtl-Number - Reynolds-Number - Stanton-Number - _B m boundary layer thickness - kg/ms dynamic viscosity - W/°C m heat conductivity including the influence of turbulence - =t _w–t _s°C temperature difference - m²/s kinematic viscosity - kg/m³ density - N/m² shear stress Indices B boundary layer - F film - S steam - SS stainless steel - T turbulent core - W wall - a outside - h hydraulic - i inside - x based on reference temperaturet _x - 0 based on zero shear surface - 1 based on inner subchannel of an annulus - 2 based on outer subchannel of an annulus - - (over a symbol): average - * (over a symbol): transformed     

Diffusion of14C in dense saturated bentonite under steady-state conditions

Diffusion coefficients are critical parameters for predicting migration rates and fluxes of contaminants through clay-based barrier materials used in many waste containment strategies. Cabon-14 is present in high-level nuclear fuel waste and also in many low-level wastes such as those generated from some medical research activities. Diffusion coefficients were measured for¹⁴C (in the form of carbonate) in bentonite compacted to a series of dry bulk densities, _b, ranging from about 0.9 to 1.6 Mg/m³. The clay was saturated with a Na-Ca-Cl-dominated groundwater solution typical of those found deep in plutonic rock on the Canadian Shield. Both effective,D _e, and apparent,D _a, diffusion coefficients were determined.D _e is defined asD _{0 a} n _e, where D₀ is the diffusion coefficient in pure bulk water, _a the apparent tortuosity factor, andn _e the effective porosity available for diffusion; andD _a is defined asD _{0 a} n _e/(n _e + _b K _d ), where K_d is the solid/liquid distribution coefficient. BothD _e andD _a decrease with increasing _b:D _e values range from about 10×10^–12 m²/s at _b0.9 Mg/m³ to 0.6×10^–12 m²/s at 1.6 Mg/m³, andD _a values vary from approximately 40×10^–12 to 4×10^–12 m²/s over the same density range. The decrease inD _e andD _a is attributed to a decrease in both _a andn _e as _b increases. The data indicate thatn _e is <10% of the total solution-filled porosity of the clay at all densities.K _d values for¹⁴C with the clay range from about 0.3 to <0.1 m³/Mg; this indicates there is a small amount of¹⁴C sorbed on the clay and/or some¹⁴C is isotopically exchanged with¹²C in carbonate phases present in the clay. Finally, theD _e values for¹⁴C are lower than those of other diffusants — I^–, Cl^–, TcO₄ ^–, and Cs⁺ — that have been measured in this clay and pore-water solution. This is attributed to lower values for bothn _e andD ₀ for¹⁴C species relative to those of the other diffusants.     

Injection moulding of plastics

Summary In continuation of a previous investigation a simple analytical expression is derived in closed form for the thickness distribution of the freeze-off layer which is vitrified at the (flat) wall of an oblong rectangular cavity. As has been pointed out previously, this layer is marked for amorphous polymers by the molecular orientation (birefringence pattern) in the moulded sample. One can show that a more detailed study with the aid of the coupled equations of energy and of motion will not furnish essential improvements. Problems of polymer physics like glass transition or crystallization kinetics at extreme rates of cooling and shearing must be solved first.

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Zusammenfassung In Fortsetzung einer früheren Untersuchung wurde ein einfacher analytischer Ausdruck in geschlossener Form für die Dickenverteilung der eingefrorenen Schicht abgeleitet, die an der (flachen) Wand eines langgestreckten rechteckigen Formnestes während des Einspritzvorgangs glasig erstarrt. Wie früher auseinandergesetzt wurde, wird diese Schicht bei amorphen Polymeren durch die Molekülorientierung (Doppelbrechungsmuster) im gespritzten Formteil markiert. Man kann zeigen, daß eine eingehendere Studie mit Hilfe der gekoppelten Energie- und Impulsgleichungen keine essentiellen Verbesserungen bringt. Probleme der Polymerphysik, wie Glasübergang oder Kristallisationskinetik bei extremen Abkühlungs- und Schergeschwindigkeiten, müssen erst gelöst werden.

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List of Symbols a heat diffusivity of polymer melt (averaged overT) [m²s^–1] - B breadth of mould cavity [m] - Br Brinkman number ( ) - c heat capacity of polymer melt (averaged overT) [J kg^–1 K^–1] - F ₀ Fourier number (at _i/4H ²) - h heat transfer coefficient by melt flow [J K^–1 s^–1 m^–2] - h heat transfer coefficient by layer growth [J K^–1 s^–1 m^–2] - H half height of mould cavity [m] - L length of mould cavity [m] - n exponent in eq. [18] (= 0.417) - Nu Nußelt number (2Hh/) - P pressure gradientdP/dz in mould [N m^–3] - t time [s] - t _i injection time [s] - T _g glass transition temperature of polymer [K] - T _i injection temperature of polymer melt [K] - T _l stagnation temperature [K] - T _m mould wall temperature [K] - speed of flow front during mould filling [m s^–1] - x coordinate perpendicular to mould wall [m] - z coordinate in the injection direction [m] - thickness of stagnant layer (atT _l) [m] - ₀ optically detectable freeze-off thickness [m] - ⁺ apparent layer thickness (atT _i) [m] - dimensionless freeze-off thickness (= ₀/2H) - dimensionless distance from entrance (=z/L) - _m dimensionless coordinate of layer maximum - _g dimensionless temperature (= (T _i –T _l)/(T _g –T _m)) - _i dimensionless temperature (= (T _i –T _l)/(T _i –T _m)) - _l dimensionless temperature (= (T _i –T _l)/(T _l –T _m)) - _i viscosity of polymer atT _i [N s m^–3] - _l viscosity of polymer atT _l [N s m^–3] - heat conductivity of polymer melt (averaged) [J K^–1 s^–1 m^–1] - density of polymer melt (averaged) [kg m^–3] - dimensionless time (eq. [11]) - ⁺ dimensionless parameter (eqs. [19a] and [19b]) - dimensionless layer thickness (eq. [12]) - ⁺ dimensionless parameter (eq. [20a]) - dimensionless parameter (eqs. [11a] and [11b]) Formerly at Delft University of Technology, Delft (The Netherlands).Paper presented at the Conference on Chemical Engineering Rheology, Annual Meeting of the Deutsche Rheologische Gesellschaft in Aachen, March 5–7, 1979.With 3 figures and 1 table     

MOTIONS, FORCES AND MODE TRANSITIONS IN VORTEX-INDUCED VIBRATIONS AT LOW MASS-DAMPING

These experiments, involving the transverse oscillations of an elastically mounted rigid cylinder at very low mass and damping, have shown that there exist two distinct types of response in such systems, depending on whether one has a low combined mass-damping parameter (low m^*ζ), or a high mass-damping (highm^*ζ ). For our low m^*ζ, we find three modes of response, which are denoted as an initial amplitude branch, an upper branch and a lower branch. For the classical Feng-type response, at highm^*ζ , there exist only two response branches, namely the initial and lower branches. The peak amplitude of these vibrating systems is principally dependent on the mass-damping (m^*ζ), whereas the regime of synchronization (measured by the range of velocity U^*) is dependent primarily on the mass ratio, m^*ζ. At low (m^*ζ), the transition between initial and upper response branches involves a hysteresis, which contrasts with the intermittent switching of modes found, using the Hilbert transform, for the transition between upper–lower branches. A 180° jump in phase angle φ is found only when the flow jumps between the upper–lower branches of response. The good collapse of peak-amplitude data, over a wide range of mass ratios (m^*=1–20), when plotted against (m^*+C_A) ζ in the “Griffin” plot, demonstrates that the use of a combined parameter is valid down to at least (m^*+C_A)ζ 0·006. This is two orders of magnitude below the “limit” that had previously been stipulated in the literature, (m^*+C_A) ζ>0·4. Using the actual oscillating frequency (f) rather than the still-water natural frequency (f_N), to form a normalized velocity (U^*/f^*), also called “true” reduced velocity in recent studies, we find an excellent collapse of data for a set of response amplitude plots, over a wide range of mass ratiosm^* . Such a collapse of response plots cannot be predicted a priori, and appears to be the first time such a collapse of data sets has been made in free vibration. The response branches match very well the Williamson–Roshko (Williamson & Roshko 1988) map of vortex wake patterns from forced vibration studies. Visualization of the modes indicates that the initial branch is associated with the 2S mode of vortex formation, while the Lower branch corresponds with the 2P mode. Simultaneous measurements of lift and drag have been made with the displacement, and show a large amplification of maximum, mean and fluctuating forces on the body, which is not unexpected. It is possible to simply estimate the lift force and phase using the displacement amplitude and frequency. This approach is reasonable only for very low m^*.     

On the laminar forced convection with axial conduction in a circular tube with exponential wall heat flux

The values of the fully developed Nusselt number for laminar forced convection in a circular tube with axial conduction in the fluid and exponential wall heat flux are determined analytically. Moreover, the distinction between the concepts of bulk temperature and mixing-cup temperature, at low values of the Peclet number, is pointed out. Finally it is shown that, if the Nusselt number is defined with respect to the mixing-cup temperature, then the boundary condition of exponentially varying wall heat flux includes as particular cases the boundary conditions of uniform wall temperature and of convection with an external fluid.

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Über laminare Zwangskonvektion mit Längswärmeleitung in einem Kreisrohr mit exponentiell veränderlichem Wandwärmefluß

Zusammenfassung Es werden die Endwerte der Nusselt-Zahlen für vollausgebildete laminare Zwangskonvektion in einem Kreisrohr mit Längswärmeleitung und exponentiell veränderlichem Wandwärmefluß analytisch ermittelt. Besondere Betonung liegt auf dem Unterschied zwischen den Konzepten für die Mittel- und die Mischtemperatur bei niedrigen Peclet-Zahlen. Schließlich wird gezeigt, daß bei Definition der Nusselt-Zahl bezüglich der Mischtemperatur die Randbedingung exponentiell veränderlichen Randwärmeflusses die Spezialfälle konstanter Wandtemperatur und konvektiven Wärmeaustausches mit einem umgebenden Fluid einschließt.

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Nomenclature A _n dimensionless coefficients employed in the Appendix - Bi Biot numberBi=h _e r ₀/ - c _n dimensionless coefficients defined in Eq. (17) - c _p specific heat at constant pressure of the fluid within the tube, [J kg^–1 K^–1] - f solution of Eq. (15) - h ₁,h ₂ specific enthalpies employed in Eqs. (2) and (4), [J kg^–1] - h _e convection coefficient with a fluid outside the tube, [W m^–2 K^–1] - rate of mass flow, [kg s^–1] - Nu bulk Nusselt number,2r ₀ q _w /[(T _w –T _b )] - Nu _H fully developed value of the bulk Nusselt number for the boundary condition of uniform wall heat flux - Nu _T fully developed value of the bulk Nusselt number for the boundary condition of uniform wall temperature - Nu ^* mixing Nusselt number,2r ₀ q _w /[(T _w –T _m )] - Nu _C ^* fully developed value of the mixing Nusselt number for the boundary condition of convection with an external fluid - Nu _H ^* fully developed value of the mixing Nusselt number for the boundary condition of uniform wall heat flux - Nu _T ^* fully developed value of the mixing Nusselt number for the boundary condition of uniform wall temperature - Pe Peclet number, 2r ₀/ - q ₀ wall heat flux atx=0, [W m^–2] - q _w wall heat flux, [W m^–2] - r radial coordinate, [m] - r ₀ radius of the tube, [m] - s dimensionless radius,s=r/r ₀ - T temperature, [K] - T ₀ temperature constant employed in Eq. (14), [K] - T reference temperature of the fluid external to the tube, [K] - T _b bulk temperature, [K] - T _m mixing or mixing-cup temperature, [K] - T _w wall temperature, [K] - u velocity component in the axial direction, [m s^–1] - mean value ofu, [m s^–1] - x axial coordinate, [m] Greek symbols thermal diffusivity of the fluid within the tube, [m² s^–1] - exponent in wall heat flux variation, [m^–1] - dimensionless parameter - dimensionless temperature =(T _w –T)/(T _w –T _b ) - ^* dimensionless temperature ^*=(T _w –T)/(T _w –T _m ) - thermal conductivity of the fluid within the tube, [W m^–1 K^–1] - density of the fluid within the tube, [kg m^–3]     

Application of Hildebrand's fluidity model to non-Newtonian solutions

Summary The fluidity model ofHildebrand has been applied to dilute aqueous polymer solutions exhibiting non-Newtonian behaviour. As for Newtonian fluids, it is found that the temperature dependence of apparent viscosity of non-Newtonian fluids is determined entirely by the temperature dependence of fluid density. However, whereas the parameters ofHildebrand's equation are constants, independent of the conditions of shear for Newtonian fluids, these parameters become shear-rate dependent for non-Newtonian fluids. The magnitude of the parameters and their variation with shear rate can be explained qualitatively in terms of interactions of polymer molecules and their behaviour in a shear field.

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Zusammenfassung Das Fließmodell vonHildebrand wird auf verdünnte wäßrige Polymerlösungen mit nicht-newtonschem Verhalten angewandt. Wie bei newtonschen Flüssigkeiten findet man hierfür, daß die Temperaturabhängigkeit der schergeschwindigkeitsabhängigen Viskosität ausschließlich durch die Temperaturabhängigkeit der Flüssigkeitsdichte bestimmt ist. Während jedoch die Parameter derHildebrandschen Gleichung für newtonsche Flüssigkeiten von den Scherbedingungen unabhängige Konstanten darstellen, werden diese bei nichtnewtonschen Flüssigkeiten schergeschwindigkeitsabhängig. Die Größe dieser Parameter und ihrer Variation mit der Schergeschwindigkeit kann qualitativ als eine Folge der Wechselwirkung der Polymermoleküle und ihres Verhaltens im Scherfeld gedeutet werden.

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Notation B Parameter inHildebrand's fluidity equation, eq. [1] (N^–1 m² s^–1) - K Consistency parameter in the power-law (Nm^–2 s ⁿ ) - n Flow behaviour index in the power-law (–) - T Temperature (K) - V Molar volume (m³ mol^–1) - V ₀ Intrinsic molar volume; parameter inHildebrand's fluidity equation, eq. [1] (m³ mol^–1) - Shear rate (s^–1) - Viscosity or apparent viscosity (Nm^–2 s) - Density (kg m^–3) - ₀ Intrinsic density; modified parameter inHildebrand's fluidity equation; eq. [5] (kg m^–3) - Shear stress (Nm^–2) With 4 figures and 1 table     

Transients in flow and local heat transfer due to a pressure wave in pipe flow

The effect of a pressure wave on the turbulent flow and heat transfer in a rectangular air flow channel has been experimentally studied for fast transients, occurring due to a sudden increase of the main flow by an injection of air through the wall. A fast response measuring technique using a hot film sensor for the heat flux, a hot wire for the velocities and a pressure transducer have been developed. It was found that in the initial part of the transient the heat transfer change is independent of the Reynolds number. For the second part the change in heat transfer depends on thermal boundary layer thickness and thus on the Reynolds number. Results have been compared with a simple numerical turbulent flow and heat transfer model. The main effect on the flow could be well predicted. For the heat transfer a deviation in the initial part of the transient heat transfer has been found. From the turbulence measurements it has been found that a pressure wave does not influence the absolute value of the local turbulent velocity fluctuations. They could be considered to be frozen.Nomenclature A surface area (m²) - D diameter (m) - h heat transfer coefficient (Wm^–2 K^–1) - p pressure drop (Pa) - P pressure (Pa) - Q heat flow (W) - R tube radius (m) - T bulk temperature (K) - T _s surface temperature (K) - t time (s) - u velocity (m/s) - V voltage (V) - y distance from wall (m) - viscosity (N s m^–2) - kinematic viscosity (m^–2 s^–1) - density (kg m^–3) - _w wall shear stress (N m^–2) - Nu Nusselt number - Re Reynolds number     

Atmospheric boundary layer simulation in a wind tunnel, using air injection

The inner part of a neutral atmospheric boundary layer has been simulated in a wind tunnel, using air injection through the wind tunnel floor to thicken the boundary layer. The flow over both a rural area and an urban area has been simulated by adapting the roughness of the wind tunnel floor. Due to the thickening of the boundary layer the scaling factor of atmospheric boundary layer simulation with air injection is considerably smaller than that without air injection. This reduction of the scaling factor is very important for the simulation of atmospheric dispersion problems in a wind tunnel.The time-mean velocity distribution, turbulence intensity, Reynolds stress and turbulence spectra have been measured in the inner part of the wind tunnel boundary layer. The results are in rather good agreement with atmospheric measurements.Nomenclature d Zero plane displacement, m - h Height of roughness elements, m - k Von Kármán's constant - n Frequency of turbulence velocity component, s^–1 - S _u(n) Energy spectrum for longitudinal turbulence velocity component, m² s^–1 - S _v(n) Energy spectrum for lateral turbulence velocity component, m² s^–1 - S _w(n) Energy spectrum for vertical turbulence velocity component, m² s^–1 - U _o Free stream velocity outside the boundary layer, m s^–1 - Time-mean velocity inside the boundary layer, m s^–1 - u* Wall-friction velocity, m s^–1 - u Longitudinal turbulence intensity, m s^–1 - v Lateral turbulence intensity, m s^–1 - w Vertical turbulence intensity, m s^–1 - Reynolds stress, m² s^–2 - z Height above earth's surface or wind tunnel floor, m - z _o Roughness length, m - Thickness of inner part of boundary layer, m - Thickness of boundary layer, m - Kinematic viscosity, m² s^–1     

Mixed convection flow over a vertical plate with localized heating (cooling), magnetic field and suction (injection)

An analysis is carried out to study the effects of localized heating (cooling), suction (injection), buoyancy forces and magnetic field for the mixed convection flow on a heated vertical plate. The localized heating or cooling introduces a finite discontinuity in the mathematical formulation of the problem and increases its complexity. In order to overcome this difficulty, a non-uniform distribution of wall temperature is taken at finite sections of the plate. The nonlinear coupled parabolic partial differential equations governing the flow have been solved by using an implicit finite-difference scheme. The effect of the localized heating or cooling is found to be very significant on the heat transfer, but its effect on the skin friction is comparatively small. The buoyancy, magnetic and suction parameters increase the skin friction and heat transfer. The positive buoyancy force (beyond a certain value) causes an overshoot in the velocity profiles.A mass transfer constant - B magnetic field - C_fx skin friction coefficient in the x-direction - C_p specific heat at constant pressure, kJ.kg^–1.K - C_v specific heat at constant volume, kJ.kg^–1.K^–1 - E electric field - g acceleration due to gravity, 9.81 m.s^–2 - Gr Grashof number - h heat transfer coefficient, W.m².K^–1 - Ha Hartmann number - k thermal conductivity, W.m^–1.K - L characteristic length, m - M magnetic parameter - Nu_x local Nusselt number - p pressure, Pa, N.m^–2 - Pr Prandtl number - q heat flux, W.m^–2 - Re Reynolds number - Re_m magnetic Reynolds number - T temperature, K - T_o constant plate temperature, K - u,v velocity components, m.s^–1 - V characteristic velocity, m.s^–1 - x,y Cartesian coordinates - thermal diffusivity, m².s^–1 - coefficient of thermal expansion, K^–1 - , transformed similarity variables - dynamic viscosity, kg.m^–1.s^–1 - ₀ magnetic permeability - kinematic viscosity, m².s^–1 - density, kg.m^–3 - buoyancy parameter - electrical conductivity - stream function, m².s^–1 - dimensionless constant - dimensionless temperature, K - w, conditions at the wall and at infinity     

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     

Exact solution of eigenvalue problems arising in connection with the study of certain hydrodynamical or quantummechanical wave phenomena

Summary The differential equationd ²/dx ²+{a[cosh (x/c)]^–2+b[sinh (x/c)]^–2–m ²}=0, which is a special case of an equation that is encountered in various physical problems, is solved analytically. The eigenvalue problem which arises when this equation is combined with certain boundary conditions is discussed and solved for various cases.     

Dynamic interaction of various beams with the underlying soil – finite and infinite, half-space and Winkler models

Various beams lying on the elastic half-space and subjected to a harmonic load are analyzed by a double numerical integration in wavenumber domain. The compliances of the beam–soil systems are presented for a wide frequency range and for a number of realistic parameter sets. Generally, the soil stiffness G has a strong influence on the low-frequency beam compliance whereas the beam parameters EI and m^′ are more important for the high-frequency compliance. An important parameter is the elastic length l=(EI/G)^1/4 of the beam–soil system. Around the corresponding frequency ω_l=v_S/l, the wave velocity of the combined beam–soil system changes from the Rayleigh wave v_R≈v_S to the bending wave velocity v_B and the combined beam–soil wave has typically a strong damping. The interaction frequency ω_l is found not far from the characteristic frequency ω₀=(G/m^′)^1/2 where an amplification compared to the static compliance is observed for special parameter constellations. In contrast, real foundation beams show no resonance effects as they are highly damped by the radiation into the soil. At medium and high frequencies, asymptotes for the compliance of the beam–soil system are found, u/P(ρv_Paiω)^−3/4 in case of the dominating damping and u/P(−m^′ω²)^−3/4 for high frequencies. The low-frequency compliance of the coupled beam–soil system can be approximated by u/P1/Gl, but it also depends weakly on the width a of the foundation. All numerical results of different beam–soil systems are evaluated to yield a unique relation u/P₀=f(a/l). The integral transform method is also applied to ballasted and slab tracks of railway lines, showing the influence of train speed on the deformation of the track beam. The presented results of infinite beams on half-space are compared with results of finite beams and with infinite beams on a Winkler support. Approximating Winkler parameters are given for realistic foundation-soil systems which are useful when vehicle-track interaction is analyzed for the prediction of railway induced vibration.     

Wall visualization of swirling decaying flow using a dot-paint method

This paper deals with the visualization of swirling decaying flow in an annular cell fitted with a tangential inlet. A wall visualization method, the so-called dot-paint method, allows the determination of the flow direction on both cylinders of the cell. This study showed the complex structure of the flow field just downstream of the inlet, where a recirculation zone exists, the effects of which are more sensitive on the inner cylinder. The flow structure can be considered as three-dimensional in the whole entrance section. The swirl number and the entrance length were estimated using the measured angle of the streamlines. Experimental correlations of these two parameters, taking into account the Reynolds number and the axial distance from the tangential inlet, are given.List of symbols e = R ₂ – R ₁ thickness of the annular gap (m) - L _ax entrance length of axial flow on the outer cylinder (m) - L _ti length of the three-dimensional flow region on the inner cylinder (m) - L _to length of the three-dimensional flow region on the outer cylinder (m) - Q _v volumetric flowrate in the annular cell (m³s^–) - r radial position (m) - R ₁ external radius of the inner cylinder (m) - R ₂ internal radius of the outer cylinder (m) - Re=2eU _m /v Reynolds number - Sn swirl number - T time average resulting velocity (m s^–) - u time average axial velocity component (ms ^–) - average velocity in the annulus (m s^–) - w time average tangential velocity component (m s^–) - x axial location from the tangential inlet (m) - _e diameter of the tangential inlet (m) - streak angle with respect to the horizontal (degree) - angle with respect to the tangential inlet axis (degree) - gn kinematic viscosity of the working liquid (m²s^–)