The connection between Ekman and Stewartson layers for a rotating disk

When a disk of finite radius and the surrounding medium rotate coaxially with slightly different angular velocities, a so-called Stewartson layer exists at the edge of the disk. The properties of this layer outside the boundary layer of the disk have been given in a previous publication. In the present paper it is shown how the radial flow of the Ekman boundary layer turns into the axial flow of the Stewartson layer. This happens in a region of which both the radial and axial dimensions are O(E^1/2), where E is the Ekman number.     

On the development of characteristics in an oseillating,rotating fluid

Summary A source of fluid with an oscillatory strength, which is situated on the axis of a rotating fluid, commences to act at time t=0. We describe how inviscid, geostrophic forces lead to the development of the characteristic cone when the frequency of oscillation is less than twice the frequency of rotation. Eventually, viscous forces become important when the time is O(E ^-1/3), where E is the small Ekman number, in forming the thin shear layer along the surface of the cone.     

On compressible flow in a rotating cylinder

Summary Axisymmetric steady flow of a perfect gas in a rotating cylinder is studied by applying a linearised analysis to a small perturbation about isothermal rigid body rotation. Motivated by present day gas centrifuges, special attention is focussed on the effect of a length-to-radius ratio which increases from unit magnitude to infinity and on the effect of a strong radial density gradient associated with the isothermal rigid body rotation. The Ekman number E _*based on the small radial density scale and the density at the cylinder wall is taken to be small. It appears that the flow outside Ekman boundary layers at the end caps consists of three types. These correspond to 1 L _* E _* ^{% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS% qaaSqaaiaaigdaaeaacaaIYaaaaaaa!386D!\[ - \tfrac{1}{2}\]} E _* ^{% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS% qaaSqaaiaaigdaaeaacaaIYaaaaaaa!386D!\[ - \tfrac{1}{2}\]} L _*, E _* ^–1 andE _* ^–1 L _* where L _*is the ratio of the cylinder-length to the radial density scale. For 1 L _* E _* ^{% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS% qaaSqaaiaaigdaaeaacaaIYaaaaaaa!386D!\[ - \tfrac{1}{2}\]} an inviscid flow in a region of limited thickness near the cylinder wall is found. Due to the strong decrease of the density, radial diffusion is not confined to Stewartson boundary layers at the wall (typical for incompressible flow) but extends in the core. This finds expression in two layers in the centre of the cylinder, parallel to the rotation axis, having a structure similar to both Stewartson layers and adjusting the inviscid flow near the wall to a flow dominated by radial diffusion near the rotation axis. For L _* E _* ^{% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS% qaaSqaaiaaigdaaeaacaaIYaaaaaaa!386D!\[ - \tfrac{1}{2}\]} and L _* E _* ^–1 both Stewartson layers become successively of the same thickness as the density scale. At the same time the corresponding layers in the core go to the wall and join. As a result, for L _* E _* ^–1 radial diffusive processes are significant in the entire cylinder, a situation also known from studies of flows in semi-infinite gas centrifuges.     

Rotating flow in a cylinder with a circular barrier on the bottom

Summary The relative flow of a homogeneous, slightly viscous fluid in a rotating cylinder is induced by differential rotation of the bottom disk, on which a thin circular strip of small height is fixed. The axis of symmetry of the strip coincides with the rotation axis of the cylinder.At the strip a Stewartson layer exists which is partially free, partially attached to the strip. The structure of the Stewartson % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaCa% aaleqabaWaaSGbaeaacaaIXaaabaGaaGinaaaaaaaaaa!3873!\[E^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} \]-layer (E being the Ekman number) is not affected by the height of the strip, but the % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaCa% aaleqabaWaaSGbaeaacaaIXaaabaGaaG4maaaaaaaaaa!3872!\[E^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} \]-layer problem has to be solved in the two separate intervals. The fact that both solutions do not match at the strip edge necessitates the presence of an intermediate region that exhibits some characteristic features of an Ekman layer.     

The Stewartson layer of a rotating disk of finite radius

It is shown that if a disk of finite radius and the surrounding medium rotate coaxially with slightly different angular velocities, an axial layer in the form of a cylindrical shell exists at the edge of the disk. This shell of thickness O(E ^1/3) has length O(E ^–1) in axial direction, where E is the Ekman number. Its most characteristic element is the axial velocity of O(E ^1/6) which is larger than everywhere else in the field. We calculate the velocity components and the pressure in this layer.     

Mass-transport enhancement in regions bounded by rigid walls

The mass transport into a fluid bounded by stationary rigid walls in the limit of large Péclet number, Pe, is examined analytically. Two model systems are considered in detail: a stationary cavity and a model involving two concentric rotating cylinders. A macroscopic gradient is imposed between the top and bottom surfaces. It is demonstrated that mass transport into the fluid is enhanced owing to a recirculation zone which is connected to the solid boundary through a boundary layer of thickness O(Pe ^–1/3) in which cross-stream molecular diffusion is balanced by convection. The associated enhancement is large and scales as Pe ^1/3. Our asymptotic analysis is found to be in good agreement with numerical solutions of the full transport equation.     

The low Rossby number flow of a rotating fluid past a flat plate

Summary In a rotating fluid, the flow between two infinite plates, perpendicular to the rotation axis, is examined when a uniform stream is aligned with a finite flat plate, parallel to the rotation axis. Since the flow in this configuration is depth-independent the motion is analogous to that considered by Blasius in a non-rotating fluid. When the Rossby number Ro is much smaller than E^3/4, where E is the Ekman number, the equations are linear and the flow has been examined by Hocking [5]. However, when RoE^3/4 inertial effects are important in the E^1/4-layer and the boundary-layer equations are non-linear. For Ro of order E^1/2 the boundary-layer flow is calculated numerically and very close to both the leading and trailing edges of the plate the flow is identical to that in the non-rotating case. Goldstein expansions are calculated at both points and the singularity at the trailing edge is examined using triple-deck theory. This demonstrates that for Ro of order E^1/2 the E^1/4-layer exhibits behaviour similar to that of a classical boundary layer.     

The spin-up problem in Helium II

The laminar spin-up of Helium II is studied by solving the linearized equations of motion for the normal and superfluid components and the quantized vortex lines in a simple case. The fluid is taken to be confined between two parallel planes whose angular velocity increases at a small, steady rate. The vortex lines are treated as a continuum. No direct interactions between the vortex lines and the walls are included. Two mechanisms are identified for the transfer of angular momentum from the container to the interior fluid. In the first place, classical Ekman pumping occurs in the normal fluid component. Secondly, mutual friction between the normal Ekman layer and the vortex lines produces an (Ekman-like) secondary flow in the superfluid component. In both mechanisms, mutual friction in the interior couples the normal and superfluid components together, so that both components spin up. Normal-fluid Ekman pumping is found to dominate at temperatures close to the -point (T=2.17 K), while the second mechanism becomes progressively more important at lower temperatures. In the small-Ekman-number limit, when the vertical container dimension 2a is much larger than the Ekman layer thickness, the spin-up time (i.e., the time lag between the container and the interior fluid) for both components ist _spin-upf(T)a ₀ ^–1/2 , where ₀ is the angular velocity andf(T) is a decreasing function of temperature. Although some experimental spin-up times in He II have been reported in the literature, their analysis involves many uncertainties. Thus, new experiments to test this model should be highly desirable.     

Dissolution kinetics and diffusivity of silver in glassy layers for hybrid microelectronics

Silver and palladium/silver compositions are widely used in hybrid microelectronics, as electrodes for dielectric layers and multilayers, terminations of thick film resistors and interconnections. Interactions between Ag and the adjacent films are known to affect the microcircuit performances. The present study is aimed at collecting data on the behavior of Ag-based films in contact with glassy layers. Most experiments were performed with a glass with composition 68.2 PbO : 30.5 SiO₂ : 1.3 Al₂O₃ wt %. Two different systems were analyzed. The first system consists of thick films prepared from a paste containing glass and either 3 or 15 wt % silver particles; both fine (spherical grains, 0.5–1 m diameter) and coarse (flakes, 2–5 m, <1 m thick) Ag powders were used for these pastes. The distribution of Ag in the film was studied with X-ray diffraction, scanning electron microscopy and fluorescence analysis. The results show that Ag floats on the glassy layer. Diffraction of X-rays generated by a synchrotron radiation source allowed us to study the kinetics of silver dissolution in the glass; this phenomenon is consistent with the Avrami theory, with an apparent activation energy E _dis=0.69±0.04 eV. The second system analyzed, Ag-based terminations of glass layers fired at various peak temperatures, enabled us to obtain quantitative values for both Ag solid solubility (about 2.5 wt %) and Ag diffusion coefficients D _Ag(T ). Typical values of D _Ag(850 °C) are 30.3±11.9 10^–8 cm²/s; an apparent activation energy of the diffusion process is E _a=0.6±0.1 eV.     

A time—memory trade-off frontwidth reduction algorithm for finite element analysis

A frontwidth reduction algorithm is presented with an execution time which may be traded against its primary memory requirement, making it possible to optimize the performance of the algorithm on a particular computer. With an amount of primary memory O(E^1/2), where E is the number of elements, the execution time of the algorithm is O(E^3/2), in two or three dimensions. The algorithm has two parts: first, new node-based data structures are derived from the conventional element list, then these structures are used to reorder the elements for reduced frontwidth.     

Synthesis and ferroelectric properties of SrBi2Ta2O9/Bi4Ti3O12/p-Si multilayer thin films by Sol-Gel

SrBi₂Ta₂O₉(SBT)/Bi₄Ti₃O₁₂(BIT) multilayer thin films were prepared on p-Si substrates by Sol-Gel method, the effect of thickness of SBT and annealing temperature on structure, morphology, ferroelectric and fatigue properties of SBT/BIT ferroelectric films were investigated. The SBT/BIT multilayer films annealed at above 600 ^∘C were uniform and crack free as well as exhibited no pyrochlore phase. The remanent polarization and the coercive field of SBT/BIT multilayer films both increases with the increase of annealing temperature due to better crystallization and larger grain size. The SBT/BIT multilayer thin films consisting of 1 layer of SBT and 3 layers of BIT annealed above 650 ^∘C obtained its best ferroelectric properties with a P_r of 8.1 μC/cm² and a E_c of 130 kV/cm which is comparable to that of pure BIT films and had a fatigue-free property up to 10¹¹ switching cycles, but P_nv appeared under the cycle field of 175 kV/cm and increased with the decrease of cycle field.     

Derivation and analysis of a McPhee-like damping term for inertially oscillating ice drift

Summary It is shown that the empirical McPhee [8] damping term for inertial oscillations in time-dependent ice-ocean motion can be derived as a first-order correction when the air stress is quadratic in air relative to ice velocity. Analytical expressions are derived for the leading term transient ice and surface oceanic boundary-layer velocities and the mass transport function of a perturbation expansion in a small parameter [O(10^-1)-O(10^-2)], the ratio of scale ice speed to air speed.     

Infrared and Raman spectra of binary tellurite glasses containing boron and indium oxides

The constitution of glasses in the systems M₂O₃-TeO₂ (M = B and In) was investigated by Raman and infrared spectroscopy. From the relation between the M₂O₃ content and the intensity ratios of the deconvoluted Raman peaks I(720)/I(665) and I(780)/I(665), it was concluded that In₂O₃ behaves as a network modifier to yield TeO ₃ ^– units and that discrete BO₃ and BO₄ units construct a network of glasses containing boron oxide. A structural model for those glasses was derived which involves three-coordinated oxygen atoms and TeO₄ units of an intermediate configuration, O₃Te⁺ ...O^-.     

An assessment of expressions for the apparent thermal conductivity of cellular materials

Diverse expressions for the thermal conductivity of cellular materials are reviewed. Most expressions address only the conductive contribution to heat transfer; some expressions also consider the radiative contribution. Convection is considered to be negligible for cell diameters less than 4 mm. The predicted results are compared with measured conductivities for materials ranging from fine-pore foams to coarse packaging materials. The dependencies of the predicted conductivities on the material parameters which are most open to intervention are presented graphically for the various models.Nomenclature a Absorption coefficient - C _v (Jmol^–1 K^–1) Specific heat - E Emissivity - E _L Emissivity of hypothetical thin parallel layer - E ₀ Boundary surfaces emissivity - f Fraction of solid normal to heat flow - fics Fraction of total solid in struts of cell - K(m^–1) Mean extinction coefficient - k(W m^–1 K^–1) Effective thermal conductivity of foam - k _cd(W m^–1 K^–1) Conductive contribution - k _cr(W m^–1 K^–1) Convective contribution - k _g(W m^–1 K^–1) Thermal conductivity of cell gas - k _r(W m^–1 K^–1) Radiative contribution - k _s(W m^–1 K^–1) Thermal conductivity of solid - L(m) Thickness of sample - L _g(m) Diameter of cell - L _s(m) Cell-wall thickness - n Number of cell layers - r Reflection coefficient - t Transmission coefficient - T(K) Absolute temperature - T _m(K) Mean temperature - T _N Fraction of energy passing through cell wall - T ₁(K) Temperature of hot plate - T ₂(K) Temperature of cold plate - V _g Volume fraction of gas - V _w Volume fraction of total solid in the windows - w Refractive index - (m) Effective molecular diameter - (Pa s) Gas viscosity - Structural angle with respect to rise direction - (W m^–2 K^–4) Stefan constant     

An assessment of expressions for the apparent thermal conductivity of cellular materials

Diverse expressions for the thermal conductivity of cellular materials are reviewed. Most expressions address only the conductive contribution to heat transfer; some expressions also consider the radiative contribution. Convection is considered to be negligible for cell diameters less than 4 mm. The predicted results are compared with measured conductivities for materials ranging from fine-pore foams to coarse packaging materials. The dependencies of the predicted conductivities on the material parameters which are most open to intervention are presented graphically for the various models.Nomenclature a Absorption coefficient - C _itv(J mol^–1 K^–1) Specinc heat - E Emissivity - E _L Emissivity of hypothetical thin parallel layer - E _o Boundary surfaces emissivity - f Fraction of solid normal to heat flow - f _s Fraction of total solid in struts of cell - K(m^–1) Mean extinction coefficient - k(Wm^–1 K^–1) Effective thermal conductivity of foam - k _cd(Wm^–1 K^–1) Conductive contribution - k _cr(Wm^–1 K^–1) Convertive contribution - k _g(Wm^–1K^–1) Thermal conductivity of cell gas - k _r(Wm^–1 K^–1) Radiative contribution - k _s(Wm^–1 K^–1) Thermal conductivity of solid - L(m) Thickness of sample - L _g(m) Diameter of cell - L _s(m) Cell-wall thickness - n Number of cell layers - r Reflection coefficient - t Transmission coefficient - T(K) Absolute temperature - T _m(K) Mean temperature - T _N Fraction of energy passing through cell wall - T ₁(K) Temperature of hot plate - T ₂(K) Temperature of cold plate - V _g Volume fraction of gas - V _w Volume fraction of total solid in the windows - w Refractive index - (m) Effective molecular diameter - (Pa s) Gas viscosity - Structural angle with respect to rise direction - (Wm^–2 K^–4) Stefan constant     

In situ production and defect characterization of laser PVD layers from YBaCuO HTSC targets inside a scanning electron microscope

In situ generation and defect characterization of YBa₂Cu₃O_7–x layers are carried out inside a scanning electron microscope (SEM) to investigate the process of pulsed laser PVD. A Nd-glass laser with a wavelength =1.06 µm and an energy densityE<200 MJ m^–2 was combined with a SEM and used for experiments. By the irradiation of a YBa₂Cu₃O_7–x target the laser PVD process and the deposition on Si substrates were originated. At high pulse energies and/or long laser pulses heating and evaporation processes of target material dominated. For laser irradiation without Q-switch (=10 ms) the defect particles incorporated into laser PVD layers were mainly big droplets originated from the liquid phase. Otherwise, for very short pulses (Q-switch mode) the layer defects were also particles of irregular size generated by material ablation on the target surface. In contrast to the droplets these irregular particles and also the layer itself showed the same stoichiometry as the target. Both the particle density and the layer thickness showed the same plane distribution function. Therefore the particle density in the plasma seemed to possess the same solid angle distribution as the plasma itself.     

NiaSi2O5（OH）4 Multi-Walled Nanotubes with Tunable Magnetic Properties and Their Application as Anode Materials for Lithium Batteries

Highly crystalline and thermally stable pure multi-walled Ni₃Si₂O₅(OH)₄ nanotubes with a layered structure have been synthesized in water at a relatively low temperature of 200–210 °C using a facile and simple method. The nickel ions between the layers could be reduced in situ to form size-tunable Ni nanocrystals, which endowed these nanotubes with tunable magnetic properties. Additionally, when used as the anode material in a lithium ion battery, the layered structure of the Ni₃Si₂O₅(OH)₄ nanotubes provided favorable transport kinetics for lithium ions and the discharge capacity reached 226.7 mA·h·g⁻¹ after 21 cycles at a rate of 20 mA·g⁻¹. Furthermore, after the nanotubes were calcined (600 °C, 4 h) or reduced (180 °C, 10 h), the corresponding discharge capacities increased to 277.2 mA·h·g⁻¹ and 308.5 mA·h·g⁻¹, respectively.      

Thermochemical studies on SrFe12O19(s)

The citrate-nitrate gel combustion route was used to prepare SrFe₁₂O₁₉(s) powder sample and the compound was characterized by X-ray diffraction analysis. A solid-state electrochemical cell of the type: (−)Pt, O₂(g)/{CaO(s) + CaF₂(s)}//CaF₂(s)//{SrFe₁₂O₁₉(s) + SrF₂(s) + Fe₂O₃(s)}/O₂(g), Pt(+) was used for the measurement of emf as a function of temperature from 984 to 1151 K. The standard molar Gibbs energy of formation of SrFe₁₂O₁₉(s) was calculated as a function of temperature from the emf data and is given by: (SrFe₁₂O₁₉, s, T)/kJ mol⁻¹ (±1.3) = −5453.5 + 1.5267 × (T/K). Standard molar heat capacity of SrFe₁₂O₁₉(s) was determined in two different temperature ranges 130-325 K and 310-820 K using a heat flux type differential scanning calorimeter (DSC). A heat capacity anomaly was observed at 732 K, which has been attributed to the magnetic order-disorder transition from ferrimagnetic state to paramagnetic state. The standard molar enthalpy of formation, (298.15 K) and the standard molar entropy, (298.15 K) of SrFe₁₂O₁₉(s) were calculated by second law method and the values are −5545.2 kJ mol⁻¹ and 633.1 J K⁻¹ mol⁻¹, respectively.     

Unsteady boundary-layer induced pressures at high Mach numbers

Summary A flat plate moving at a high Mach number and zero angle of attack is stepwise accelerated in the direction of its initial motion by a small amount. The time history of the associated flow, till the final steady state is reached, is analysed and the unsteady flow solutions determining the temporal variations in the boundary layer induced weakinteraction pressure and shear stress are numerically obtained for an insulated surface. In the analysis, the fluid is assumed to be a perfect gas. The obtained results are for a free stream Mach numberM of 10 and a given plate lengthL ^* with various values of impulsive velocity increments. It is also found that by suitable formulation of the variables and in particular by avoiding the variable of the typet ^* u ^*/x ^*, the discontinuity quoted by Stewartson and others is avoided.With 8 Figures