Hydrodynamic Region Research Articles

Transient convective diffusion for the hydrodynamic rear stagnant region, with an application in electrochemistry

Time dependence of mean current densities after the step change in concentration of depolarizer on the surface of convective electrode under conditions of limiting diffusion currents for the axially symmetrical configuration of the rear critical region is approximately determined by a new type of the similarity integral method.

Numerical Analysis and Flow Visualization on the Hydrodynamic Entrance Region of Laminar Flow in Curved Square Channels

Numerical Analysis and Flow Visualization on the Hydrodynamic Entrance Region of Laminar Flow in Curved Square Channels

Critical Behavior of a Micellar Solution

The static and dynamic critical properties of a nonionic amphiphile aqueous solution showing a lower consolution point are investigated by laser-light scattering. The osmotic isothermal compressibility and the correlation length of the concentration fluctuations diverge with mean-field-theory critical exponents. The behavior of the mass diffusion constant in the hydrodynamic region is in good agreement with the predictions of the mode-mode-coupling theory.

Rayleigh-Brillouin spectrum of compressed He, Ne, and Ar. II. The hydrodynamic region

The Rayleigh-Brillouin spectra of He, Ne, and Ar are reported for the hydrodynamic region. The experiments show that light scattering may be used to determine thermodynamic and transport properties of gases as a function of density and thus provide another method of probing intermolecular potentials. In addition, the transition from the hydrodynamic to the kinetic regime is examined in terms of generalized hydrodynamics.

Critical dynamics of elastic phase transitions

The critical dynamics at elastic phase transitions of second order is studied by the renormalization-group theory. The dynamical theory is based on a stochastic equation of motion for damped phonons. For systems with one-dimensional soft sectors usual dynamical scaling holds with the critical dynamical exponent $z=2$. For two-dimensional sectors logarithmic corrections appear. Novel features are found for an isotropic-phonon model. The dynamical susceptibility and the characteristic frequency depend singularly on an irrelevant parameter, which, however, diverges at the fixed point. Consequently dynamical scaling breaks down; e.g., the characteristic frequency is no longer a homogeneous function of the wave number $k$ and the inverse correlation length ${\ensuremath{\xi}}^{\ensuremath{-}1}$. Instead it is a homogeneous function of these variables and an irrelevant parameter. Consequently, the critical dynamics can be characterized by the relaxational exponent $z=2+c\ensuremath{\eta}$ at ${T}_{c}$, while in the hydrodynamic region the sound frequency is characterized by $z=2\ensuremath{-}\frac{1}{2}\ensuremath{\eta}$ and the damping by yet another exponent. This breakdown of scaling is also reflected by the fact that different fixing conditions, i.e., different choices of the frequency, lead to different values of $z$. All of these apparently different transformations lead to the same modified dynamical scaling relations.

Gas-liquid mass transfer and hydrodynamic flow region in packed columns with cocurrent upward flow.

Liquid holdup and interfacial area were measured in columns packed with 1/2 and 1 in. ceramic spheres for upward cocurrent-flow mode, where the ratio of packing to column diameter, dp/dT, is 0.085, 0.128 and 0.169. The empirical equations of liquid holdup Φl for void fraction e are given as follows: Φl/e=κRehiRe-jgwhere κ=1.8, h=0.03, j=0.28 for Rel 695Reg-0.5. The empirical equations for interfacial area ap are also presented as follows: apdp/(1-Φl/e)=αRemlReno(dp/dT)-twhere the values of α, m, n and t are 16, 0.05, -0.4 and 0 for bubble (1), 2.2, 0.05, -0.2 and 0.5 for churn, 0.24. 0.27, 0 and 0.7 for pseudospray, 0.26, 2/3, -1/4 and 0.4 for bubble (11), 1.0×10-2, 2/3, 0 and 1.4 for pseudopulse and 2.9×10-4, 2/3, 0.2 and 2.5 for pulse flow, respectively. The equation for respective hydrodynamic flow boundary was found by combining two of these. The predicted mass-transfer coefficients for liquid phase given by the simple equation for downward flow mode agree with the literature data in a wide range of gas and liquid flow rates.

Critical dynamics of a stochastic n-vector model below T c

The dynamic properties of a stochastic n-vector model are investigated for T < T c in d=4− ϵ dimensions. Besides the non-conserved order parameter the model involves also the conserved densities of generators of the symmetry group O( n). We calculate the excitation spectra of those conserved densities and the transverse fluctuations of the order parameter to linear order in ϵ in the hydrodynamic region kξ⪡1. The propagating modes have linear dispersion and quadratic damping in accordance with the phenomenological theory. The relaxing modes, however, exhibit non-hydrodynamic wavenumber dependence with a relaxation rate ω k ∞ k d 2 .

Boundary of hydrodynamic flow region and gas-phase mass-transfer coefficient in packed column with cocurrent downward flow.

Boundary of hydrodynamic flow region and gas-phase mass-transfer coefficient in packed column with cocurrent downward flow.

Intrinsic angular momentum of3He-A and its hydrodynamics with unclamped normal fluid

It is shown that the stress tensor can be symmetrised by a proper redefinition of the (linear) momentum density, even if an intrinsic angular momentum is present. The difference between 'intrinsic' and 'external' angular momentum turns out to be a matter of definition. There is no independent hydrodynamic variable due to the angular momentum conservation law. Some expressions for the (linear) momentum density are discussed and the use of the symbol Cij for different quantities by various authors is clarified. The tensor Cij contains two independent parameters by Galilean invariance. Angular monentum conservation including the dipole-dipole interaction is discussed. The orbit wave spectrum shows a quadratic k dependence in the hydrodynamic region and different results are obtained only outside that region or by the use of non-hydrodynamic variables. In two appendices the mode spectrum (with unclamped normal fluid) is discussed and Graham's free energy is explicitly written down.

Small-angle critical neutron scattering from cobalt

Total and inelastic-neutron-scattering measurements have been made of the small-angle critical scattering from polycrystalline cobalt both above and below its Curie temperature ${T}_{C}$. Below ${T}_{C}$ the scattering is dominated by well-defined spin-wave modes which exhibit quadratic dispersion, $\ensuremath{\hbar}\ensuremath{\omega}(q)=D{q}^{2}$. Within the hydrodynamic region, the spin-wave stiffness constant is found to have the power-law temperature dependence $D \ensuremath{\propto} {(T\ensuremath{-}{T}_{C})}^{x}$ where $x=0.39\ifmmode\pm\else\textpm\fi{}0.05$. As ${T}_{C}$ is approached, the spin waves renormalize and broaden but no evidence of a central peak indicative of the longitudinal component of the susceptibility is observed. Above ${T}_{C}$, the exponents $\ensuremath{\gamma}=1.23\ifmmode\pm\else\textpm\fi{}0.05$ and $\ensuremath{\nu}=0.65\ifmmode\pm\else\textpm\fi{}0.04$, describing the power-law dependences of the static susceptibility and the inverse correlation range, respectively, have been obtained from the small-angle scans after taking full account of the inelasticity of the scattering. In addition, the critical scattering has been calibrated directly against the nuclear incoherent scattering and in this way the interaction range ${r}_{1}$, which appears in the classical and modified Ornstein-Zernike expressions for the asymptotic form of the spin pair correlation function, has also been determined. The linewidths of the quasielastic critical scattering have been measured over a range of wave vectors at temperatures up to 150\ifmmode^\circ\else\textdegree\fi{}C above ${T}_{C}$. At ${T}_{C}$ the linewidths vary with $q$ to the 2.4\ifmmode\pm\else\textpm\fi{}0.2 power. Above ${T}_{C}$, the linewidths are well described by a dynamical scaling function which, however, differs from that previously found for iron.

Microscopic boundary layer effects and rough sphere rotation

A calculation of the rough sphere angular velocity correlation function is presented which takes into account the presence of both the microscopic boundary layer and outer hydrodynamic regions around the test particle. The boundary layer region is approximately accounted for by the slip boundary condition while generalized hydrodynamic equations (frequency dependent transport coefficients plus fluid spin variables) are used to describe the outer region. The approximate calculation of the slip coefficient and the use of the slip coefficient to account for processes occurring in the boundary layer are discussed in the context of kinetic theory results. Since a description of both regions is incorporated, the present theory contains both microscopic and collective effects. The structure of the result is compared in some detail to a recent renormalized kinetic theory calculation on the same system and various levels of approximation are examined. The results are in good agreement with the molecular dynamics calculations.

Static and dynamic light scattering studies of critical phenomena. I. Upper and lower consolute points of 3-methylpyridine–D2O

To further probe the ’’universality’’ ideas of critical phenomena, the turbidity, light scattering intensity, and Rayleigh linewidth have been measured for the 3-methylpyridine–D2O binary liquid system in the region of both the entropy-driven lower (LCST) and the enthalpy-driven upper (UCST) critical solution temperatures. The LCST was found to be 38.240 °C at XD2O=0.923 and the UCST was 115.440 °C at XD2O=0.9005, both under a nitrogen atmosphere in a sealed cell. In addition to being the first complete light scattering study of two critical points in a single system, the UCST is the highest temperature attempted for such investigations and the LCST study is only the second complete LCST study that has been reported. For the LCST mixture, γ=1.27±0.03 and ν=0.65±0.03; for the UCST system, γ=1.27±0.03 and ν=0.61±0.02. Some deviations from OZ behavior were noted which are related to multiple scattering effects. The critical linewidth data, ranging from the hydrodynamic region (Kξ≪1) to the critical region (Kξ≫1), were compared to the original Kawasaki expression. Deviations similar to those previously reported for other critical systems were observed which resulted from the constant viscosity assumption. Kawasaki viscosities η* at given ΔT values were determined semiquantitatively to infer the different nature of the viscosity anomaly in the UCST and LCST regions. Analysis of the linewidth data also yielded the ’’apparent’’ exponent ν* describing the behavior of the diffusion coefficient. For the LCST system, ν* =0.61±02; for the UCST system, ν*=0.62±02.

Measurements of the dynamic structure factor near the lambda temperature in liquid helium

The dynamic structure factor $S(q,\ensuremath{\omega})$ was measured by Brillouin scattering in liquid $^{4}\mathrm{He}$ at $q=1.79\ifmmode\times\else\texttimes\fi{}{10}^{5}$ ${\mathrm{cm}}^{\ensuremath{-}1}$. Results were obtained at two densities: $\ensuremath{\rho}=0.175$ g/${\mathrm{cm}}^{3}$ for which ${P}_{\ensuremath{\lambda}}=23.1$ bars and $\ensuremath{\rho}=0.179$ g/${\mathrm{cm}}^{3}$ for which ${P}_{\ensuremath{\lambda}}=28.5$ bars. Values of the reduced temperature $\ensuremath{\epsilon}\ensuremath{\equiv}\frac{(T\ensuremath{-}{T}_{\ensuremath{\lambda}})}{{T}_{\ensuremath{\lambda}}}$ varied from -5 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}2}$ to -5 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}6}$ and from 1.3 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}5}$ to 5 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}2}$. This range involves both the hydrodynamic and the critical regions; values of $q\ensuremath{\xi}$ fall between 5 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}2}$ and 21 below ${T}_{\ensuremath{\lambda}}$ and between 4.5 and 2 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}2}$ above ${T}_{\ensuremath{\lambda}}$. This article presents the data on the critical mode, which is second sound below the transition and heat diffusion above. The most striking result is that the damping of the critical mode is essentially independent of $\ensuremath{\epsilon}$ over the entire range of temperatures studied. This behavior is qualitatively different from the strong temperature dependence, consistent with dynamic scaling, which is observed at low frequencies. On the other hand, any dispersion in the velocity of high frequency (\ensuremath{\sim} several MHz) second sound, if present, is less than 2%. $S(q,\ensuremath{\omega})$ retains a two-peaked structure well into the critical region below ${T}_{\ensuremath{\lambda}}$; it still appears in the measured spectra (which contain the instrumental width as well) at $q\ensuremath{\xi}\ensuremath{\sim}4$. At ${T}_{\ensuremath{\lambda}}$ and in the critical region just above, there is evidence for a non-Lorentzian shape of $S(q,\ensuremath{\omega})$. The critical mode contribution to $S(q,\ensuremath{\omega})$ is compared with a theoretical calculation of Hohenberg, Siggia, and Halperin based on the planar-spin model of $^{4}\mathrm{He}$. The theory matches both the overall width and the general features of the shape of $S(q,\ensuremath{\omega})$ to within our spectral resolution at ${T}_{\ensuremath{\lambda}}$. However, closer to $q\ensuremath{\xi}=1$ in the critical region, and in the hydrodynamic regions both above and below ${T}_{\ensuremath{\lambda}}$, the theory predicts linewidths which are too small.

Dynamic critical properties of a stochastic n-vector model

A stochastic model is introduced with a non-conserved order parameter of n real components coupled to the density of generators for the rotation of the order parameter. As microscopic background it is based on a lattice dynamic model with O(n) symmetry. In the symmetry breaking phase a rearranged perturbation scheme is introduced in which the response and correlation functions are expressed in terms of new blocks free from the singularities of the self-energies. It is shown that the transverse fluctuations of the order parameter and the fluctuations of that component of the generator field which rotates the order parameter around its equilibrium position have common excitation spectra. The dynamic renormalization group is applied to the model in a form somewhat different from those introduced earlier. A detailed analysis is carried out to O(ϵ). Though the stable fixed point for d < 4 depends on n it leads to the dynamic critical exponent z = 12d for all values of n. In the hydrodynamic region above Tc the excitation spectra turn out to be independent of n to O(ϵ). Consequences of the energy conservation and anisotropy are discussed.

Radial diffusion and entrance effects in porous flow-through electrodes

Radial diffusion has two effects on the performance of a flow-through electrode: (a) it may limit the maximum experimentally attainable degree of conversion, and/or (b) the polarization at a certain current output may be lower for smaller than for larger pores. A dimensionless groupwas developed as a criterion of reactant conversion under certain conditions of flow rate, pore diameter and diffusion coefficient. The lengths of both diffusional and hydrodynamic entrance regions are calculated. The ratio between the length of the diffusional entrance region and the electrode thickness is related to the criterion of reactant conversion . It was shown that in all conditions of fully developed laminar flow inside the pores, radial diffusion must be fast with respect to axial convection: hence complete reactant conversion is possible. The effect of radial hydrodynamic dispersion is briefly outlined. It increases the rate of radial mass transfer, hence the degree of conversion and decreases the length of the diffusional entrance region. The length of the hydrodynamic entrance region would normally be much shorter than the diffusional entrance length. They become comparable only when the radial dispersion coefficient is of the same order as the kinematic viscosity of the electrolyte.

Molecular dynamics in liquid cyclopropane - I. — Self-diffusion Measurements by Quasielastic Neutron Scattering and N.M.R. Spin Echo

The translational diffusion of liquid cyclopropane was studied by quasielastic neutron scattering (Q.N.S.) and by N.M.R. spin-echo in the temperature range 155-297 K. The values of the self-diffusion coefficient derived from the two techniques was the same within experimental accuracy (D = 0.78 x 10-5 cm2 s-1 at 155 K, and D = 9.2 x 10-5 cm2 s-1 at 297 K). This shows that high-resolution Q.N.S. measurements at small momentum transfer give accurate information in the hydrodynamic region.

Liquid-phase volumetric and mass-transfer coefficient, and boundary of hydrodynamic flow region in packed column with cocurrent downward flow.

Liquid-phase volumetric and mass-transfer coefficient, and boundary of hydrodynamic flow region in packed column with cocurrent downward flow.

Interfacial area and boundary of hydrodynamic flow region in packed column with cocurrent downward flow.

Liquid holdup and interfacial areas were measured in packed columns with cocurrent downward flow. An empirical equation of liquid holdup Φt is presented in terms of Reynolds numbers of gas Reg(=dsGg/μg) and liquid Rel, surface shape factor of packing φ, void fraction e and ratio of packing to column diameter dp/T, where the ratio is smaller than 0.13. This equation is different for the dispersed bubble flow and other flow regions. The empirical equation of interfacial area ap in the respective flow regions varies as follows: apdp/(1-Φl/e)=ωΦ-mRenlReqg(dp/T)-twhere ω=7.5×10-5, m=Q.2, n=Q.15, q=2/3, t=2.5 for spray flow; ω=2.2×10-4, m=0.3, n=2/3, q=0.2, t=2.5 for pulse flow; ω=3.9×10-3, m=0.1, n=0.4, q=p. t=2 for trickle flow; ω=2.8×10-7, m=0.9, n=1.8, q=0, t=3.3 for dispersed bubble flow. The equation of the boundary in the respective flow regions was found by equating the two of them. The predicted boundaries are in excellent agreement with the literature data given from the analysis of liquid pulse frequencies. The predictions for interfacial areas also agree well with the literature data.

Effect of surfactants on the flow of falling liquid films

The effect of surfactants on the velocity profiles in falling non-rippling liquid films has been examined. An appropriate solution of the Navier—Stok Our own experimental data and published data on the increase in film thickness and on the decrease of surface velocity in the presence of surfactants a The existence of three separate hydrodynamic regions along the film length is indicated by our experimental data.

Deposition of aerosols in the entrance of a tube

Deposition of aerosols in the hydrodynamic entrance region of a circular tube was analyzed with an integral method. It was found that for flow at low Schmidt numbers ( v/ D < 0.01) the rate of deposition approaches that of uniform velocity distribution and at high Schmidt numbers ( v/ D > 100) the rate of deposition approaches that of parabolic velocity distribution.