Showing papers on "Velocity gradient published in 1979"

P and S waves diffracted around the core and the velocity structure at the base of the mantle

Abstract: Summary. Decay spectra and dT/dΔ on 13 great circle paths for P and seven for SH, are used in a joint interpretation of P and SH diffraction data, in terms of a transition zone with linear velocity profile above the core—mantle boundary (CMB). Error analysis reveals that the effect of large-scale lateral variations at the base of the mantle can be significant, and this effect is accounted for in the standard errors as associated with the data and used in the inversion procedure. Employing a full wave theoretical approach discussed previously, synthetic decay spectra and dT/dΔ are calculated for every individual path, to be compared to the real data. Following recommendations of a previous paper, it is illustrated here by examples that thickness zt and velocity gradients dα/dz, dβ/dz of the transition zone have a characteristic effect on the decay spectra of P and SH, and these effects are taken into account in searching the model space for the best fitting model (in the least-squares sense). Subject to the constraint of a linear velocity profile, the combined P and SH data set allows a thickness zt of 50–100 km and the P and S velocities both decrease with depth. The preferred model, PEMC-L01, has zt= 75 km and dα/dz=dβ/dz 2–0.0019; the S velocity gradient is close to critical and the ‘index of inhomogeneities’η∼ 4.5. There is a suggestion of an additional low-Q at the base of the mantle, but this is not absolutely required due to the large standard errors associated with the relevant data. The models are determined primarily from the decay spectra. They also fit the dT/dA data reasonably well, but these data do not discriminate between several different models considered here. Allowing for the possibility of lateral variations, previously published data can be reconciled with the present model, possibly with some modifications warranted by short-period information. The physical implications are an anomalous density gradient and/or superadiabatic temperature gradient above CMB. The numerical results with regard to density and/or temperature must be treated with caution, since the seismic parameters dφ/u'z and η are very sensitive to apparently small changes in the model.

Nonlinear least-squares inversion of traveltime data for a linear velocity-depth relationship

Abstract: Nonlinear least‐squares inversion of traveltime data is applied to the problem of determining thicknesses, velocities, and velocity gradients in laterally homogeneous, horizontally layered structures. The parametric forms of the traveltime equations are used for the calculations. Results of inversions on randomly inaccurate synthetic data show that the method will not determine the gradient consistently when using reflection traveltimes. Good results are obtained, however, when using traveltimes of energy refracted in a layer by the velocity gradient. Thicknesses and average velocity in the case of reflections, or velocity at the top of the layer in the case of refractions, are also determined. Partial derivatives determined during the course of the least‐squares inversion can be used to place limits on errors in the determined parameters.

Velocity distributions in the flow of solid particles in an inclined open channel

Abstract: The velocity of solid particles flowing in an inclined open channel of which the bottom plate was covered with very rough sandpaper was measured for three kinds of particle. Except for the region near the bottom plate, the velocity distribution normal to the bottom plate appeared to be linear. The velocity gradient in that region was almost independent of the thickness of the particle layer and increased as the slope of the channel became steep. These observed velocity distributions were analyzed based on the variational principle, by which the velocity distribution could be obtained as the solution which minimized a certain integral consisting of several energy terms. It was found that such analysis could explain the main feature of the particle flow in an inclined channel, and the following relation between stress and rate of deformation was obtained. τyz=kτy-kμy(dvz/dy) It was also found that the critical inclination angle of the channel, which was anticipated by the analysis, corresponded to the angle of repose.

Oceanic upper crustal structure from variable angle seismic reflection – refraction profiles

Abstract: Summary. The travel times of reflected and refracted arrivals observed on variable angle seismic profiles can be inverted to produce a first approximation to the variation of seismic velocity with depth. A method is discussed for deducing the interval velocity above a reflector by finding the best match between the observed variable angle reflection travel times and those derived by ray tracing through a series of trial models. Further constraints can be placed on the rate of change of velocity with depth by consideration of the amplitudes of both reflections and refractions. Synthetic seismograms calculated by the reflectivity method demonstrate that different kinds of velocity structure in the oceanic volcanic basement layer produce characteristically different variations of amplitude with range. In particular, the exact nature of the shear wave velocity transition at the top of the basement has a marked effect on the compressional wave reflection amplitudes. Pre-critical reflection amplitudes are seen to be generally more sensitive to the velocity structure at the sediment-basement interface than are post-critical reflections, whilst head wave amplitudes are strongly dependent on the velocity gradient in the upper part of the basement. The second part of the paper presents an analysis of variable angle seismic profiles obtained over the Madeira abyssal plain in the North Atlantic, using the travel time and amplitude interpretation techniques discussed earlier. It is concluded that the velocity structure of this area of typical oceanic crust comprises a stack of distinct layers within each of which the velocity increases smoothly with depth. An upper basement layer 600 to 700 m thick is a remnant of an originally lower velocity seismic layer 2A which is found on younger crust. Beneath this upper layer the compressional wave velocity increases with a gradient of less than 0.7/s to the base of seismic layer 2. The velocity gradient within the underlying seismic layer 3 is constrained by refraction amplitudes as less than 0.1/s.

A Family of Turbulence Models for Three-Dimensional Boundary Layers

Abstract: The process of turbulent shear-stress generation is studied on the basis of the transport equations, as derived from the Navier-Stokes equations. The experimental observation that the vector of the shear stress is in general not parallel to the vector of the mean velocity gradient, is attributed mainly to the fluctuating pressure term. A new simple approximation to the part of the pressure-strain term that depends on the mean rate of strain is proposed. It is used as a closure assumption for a family of turbulence models the most complicated of which consists of the transport equations for the shear stresses and the turbulent energy equation. Dropping the transport terms, three-dimensional versions of the eddy viscosity relationships and mixing. length formulas are obtained, in which the eddy viscosity is nonisotropic. The relations describe typical features of turbulent boundary layers like the decrease of both the ratio of shear stress to turbulent energy and the conventional mixing length with the angle between the vectors of mean velocity gradient and mean velocity.

Resistance to shock‐front propagation in solids

Abstract: The structure of a strong shock front is modeled using a wall of dislocations. Motion of these dislocations changes the density of the solid medium and simultaneously relaxes the shear strains that would otherwise exist behind the front. The dislocation motion is highly dissipative because it requires atoms in intimate contact to slide over one another. This transports momentum down the velocity gradient. It is shown that the energy dissipation rate associated with dislocation motion is larger than for any other general mechanism in the system, including electron‐electron, electron‐phonon, and phonon‐phonon interactions. The reason for this is that electrons are not effective in transporting momentum, and atomic thermal velocities are substantially smaller than the velocities of the dislocations which move with the shock front. The viscosity coefficient associated with the dislocations is propertional to the shock velocity, so the viscous drag stress is proportional to the square of the shock velocity. This can lead to large drag pressures.

Calculation of Turbulent-Inviscid Flow Interactions with Large Normal Pressure Gradients

Abstract: A simple and economical iterative scheme is presented for calculating turbulent shear layers with significant pressure gradients normal to the plane of the layer, such as occur on highly-curved surfaces or near the trailing edges of lifting airfoils, and for matching the shear-layer calculations to calculations of the inviscid external flow. The iteration required to solve the elliptic equations describing the shear layer is combined with the iteration needed for the matching, and the finite-difference solution of the normal-component momentum equation is a simple quadrature at each iteration. Therefore computing time is little greater than in conventional displacement-surface calculations that ignore normal pressure gradients. OUNDARY-layer (thin-shear-layer) equations are derived from the Navier-Stokes equations by assuming that streamwise (x) gradients of velocity are small compared to velocity gradients normal to the surface (y). The resulting simplifications1 include the disappearance of the pressure gradient normal to the surface, from which follows the smallness of the velocity gradient dU/dy in the "inviscid" flow just outside the shear layer. This in turn leads to the concept of a displacement thickness to represent the displacement, nominally independent of y, of the "inviscid" flow streamlines near the shear layer. It is important to note that once the basic assumption fails, all the simplifications disappear. If the shear layer changes rapidly in the x direction, so that dV/dx is large, normal pressure gradients are im- portant both within the layer and outside it; not only does the displacement thickness fail to represent the displacement of the external flow, but its definition, even as a mere shear-layer parameter, becomes ambiguous. It will be seen that correc- tions based on the surface radius of curvature may be highly inaccurate. All but the most rapidly changing viscous or turbulent flows will still be recognizable as fairly thin shear layers, and the terms in the Navier-Stokes equations that are neglected in the boundary-layer equations will still be v fairly small. Therefore, thin-shear-layer concepts are still useful in analytic and computational work. In particular, the effects of the normal pressure gradient on the shear layer will be small enough to be included by iterative improvement of a con- ventional marching calculation rather than by a fully elliptic calculation, and this more economical approach has been adopted in the present work.

Comparison of laboratory and in situ compressional-wave velocity measurements on sediment cores from the western North Atlantic

Abstract: Laboratory and in situ velocity measurements have been made on six piston cores taken in the western North Atlantic Ocean. Sediments from the southwestern Bermuda Rise and Greater Antilles Outer Ridge are clays having velocities ranging mostly from 1500 to 1530 m/s and velocity gradients near 1 s−1. In cores from the Nares Abyssal Plain, the clayey sediments have comparable velocities, but interbedded silty turbidites exhibit much higher values (up to 1690 m/s). Velocity gradients are slightly higher in the abyssal-plain cores. After the laboratory measurements are corrected to in situ conditions, they show reasonable agreement in average velocity and velocity gradient with in situ measurements, although the in situ velocities average 10–12 m/s higher in the clayey cores and 15–20 m/s higher in the turbidites. This difference may be caused by reduction in the dynamic frame bulk modulus and/or the dynamic shear modulus due to visually undetected coring disturbance. The profilometer used to obtain the in situ measurements does not record the fine-scale variations in velocity that were measured in the laboratory, but it accurately determines average velocities and velocity gradient. Where cores were closely spaced (2–12 km apart), inter-core correlations in lithology, velocity, and bulk properties are possible. Fluctuations in the latter two parameters are very similar in position and magnitude from core to core, suggesting either that effects of coring disturbance are small or that they are uniform in a given kind of sedimentary bed. Inter-core comparison also shows that some beds are laterally discontinuous as a result of local (less than a few kilometers) patterns of seafloor erosion and deposition.

Optical extinction of randomly orientated and shear flow orientated colloidal kaolinite particles

Abstract: The optical transmission of a colloidal kaolinite dispersion subject to laminar shear flow is shown experimentally to depend on the magnitude of the velocity gradient, the particle size, the measurement wavelength and the colloid concentration. With the light direction parallel to that of the velocity gradient, the transmitted intensity change on shear was found to either increase or decrease compared with the dispersion at rest. The sign of the change for a given dispersion depends on the particle size in the dispersion and the measurement wavelength employed.The anomalous diffraction theory of optical scattering cross-section is applied to the case of the plate-like kaolinite particles. It is shown to account for the magnitude and the particle size dependence of the turbidity of dispersions at rest, where the particles are randomly orientated with respect to the incident light. It also explains why the transmitted intensity on shear increases or decreases depending on the particle size and wavelength, when the light is incident mostly on the particle faces.

Observation of Pretransitional Divergence of Shear Viscosity near a Smectic-A-Smectic-B Phase Transition

Abstract: Strong pretransitional divergence of shear viscosity was observed for the first time near a smectic-$A$\char22{}smectic-$B$ phase transition by capillary shear flow with the layer normal perpendicular to both the flow velocity and velocity gradient. The results are inconsistent with a single power-law type of critical exponent and are suggestive of an essential singularity. Possible implications of the results for recently proposed models of the smectic-$B$ phase and mechanisms for smectic-$A$\char22{}smectic-$B$ phase transitions are discussed.

Flow turbidity in colloidal kaolinite dispersions

Abstract: The optical transmission and turbidity of a colloidal dispersion of non-spherical particles depends on the number of particles per unit volume and the average extinction cross-section per particle. Laminar shear flow of the dispersion causes preferential orientation of the non-spherical particles and changes the average optical extinction cross-section per particle. The result is a velocity gradient dependence of the optical turbidity.This paper describes apparatus designed to make accurate flow turbidity measurements with the velocity gradient direction either parallel or at right-angles to the incident light direction.Results are presented for colloidal kaolinite clay particles dispersed in water and in a range of more viscous water + glycerol mixtures. A semi-empirical expression is developed which relates the turbidity to the velocity gradient by means of a time T and dimensionless parameter p. Values of T derived from the experimental results suggest that T is a relaxation time of an individual kaolinite particle averaged over the distribution of particle sizes in any particular clay fraction. The parameter p is found to be dependent on the polydispersity of the fraction, decreasing with increasing fraction breadth. The linear dependence of T on continuous phase viscosity indicates that the clay particles interact hydrodynamically with the continuous phase. The cubic dependence of T on the equivalent Stokes diameter (ESD) suggests that ESD is proportional to the face diameter of the plate-like kaolinite particles. The use of flow turbidity in colloidal particle size measurement is discussed, and the technique compared with that of flow birefringence.

Sewage aeration system - with cylindrical element revolving on vertical shaft above bubble releasing ring

Abstract: System is based on conducting ascending bubbles along a revolving vertical element which produces a thin laminar flow with a high radial velocity gradient. The laminar flow continue over the full height of the element. A revolving cylinder on a vertical shaft has a cone at the bottom which guides the air bubbles along its surface. Its top can be closed or open. The bubbles are produced in perforations of a ring to which compressed air is supplied through a pipe. The velocity gradient produces an inverse pressure gradient so that the bubbles adhere to the element as they rise to the surface. Their path is thereby lengthened and they are broken up into finer bubbles.

Hartree self-consistent field approach towards dilute polymer solutions

Abstract: The presence of elastic forces in the spring and bead model of polymer solutions suggests a discussion in terms of modes rather than of beads. The transition from a many-mode picture to a one-mode picture is effected by a factorization of the many-mode probability density into one-mode probability densities. These obey a system of coupled integro-differential equations of the Hartree type. Working in the Rouse mode representation and beginning with the longmolecule free-draining approximation good one-mode probability densities were derived which take into account Oseen's hydrodynamic interactions in an average fashion and form the starting basis for the well-known Hartree self-consistent procedure which goes beyond the approximation of the averaged Oseen tensor. This procedure can be used to improve the values of the viscosity and normal stress functions which we have calculated for the case of simple shear flow. The shear stress and the two relevant normal stress differences are non-zero and depend on the velocity gradient in a non-linear manner. The calculation of explicit numerical results requires computer work of moderate extent for which all required formulae are given in the appendix.

A note on momentum transfer above very rough surfaces

Abstract: Data from wind tunnel experiments on the deviations from unity of the non-dimensional velocity gradient, ΦM = (kz/u*)(δU/δz), close to very rough surfaces are reviewed and compared with field data discussed by J. R. Garratt. It is found for z/z0 1 in the wind tunnel experiments. This difference is discussed tentatively in terms of roughness element flexibility and porosity.

Optical extinction of randomly orientated and shear flow orientated colloidal kaolinite particles

Abstract: The optical transmission of a colloidal kaolinite dispersion subject to laminar shear flow is shown experimentally to depend on the magnitude of the velocity gradient, the particle size, the measurement wavelength and the colloid concentration. With the light direction parallel to that of the velocity gradient, the transmitted intensity change on shear was found to either increase or decrease compared with the dispersion at rest. The sign of the change for a given dispersion depends on the particle size in the dispersion and the measurement wavelength employed.The anomalous diffraction theory of optical scattering cross-section is applied to the case of the plate-like kaolinite particles. It is shown to account for the magnitude and the particle size dependence of the turbidity of dispersions at rest, where the particles are randomly orientated with respect to the incident light. It also explains why the transmitted intensity on shear increases or decreases depending on the particle size and wavelength, when the light is incident mostly on the particle faces.

Flow turbidity in colloidal kaolinite dispersions

Abstract: The optical transmission and turbidity of a colloidal dispersion of non-spherical particles depends on the number of particles per unit volume and the average extinction cross-section per particle. Laminar shear flow of the dispersion causes preferential orientation of the non-spherical particles and changes the average optical extinction cross-section per particle. The result is a velocity gradient dependence of the optical turbidity.This paper describes apparatus designed to make accurate flow turbidity measurements with the velocity gradient direction either parallel or at right-angles to the incident light direction.Results are presented for colloidal kaolinite clay particles dispersed in water and in a range of more viscous water + glycerol mixtures. A semi-empirical expression is developed which relates the turbidity to the velocity gradient by means of a time T and dimensionless parameter p. Values of T derived from the experimental results suggest that T is a relaxation time of an individual kaolinite particle averaged over the distribution of particle sizes in any particular clay fraction. The parameter p is found to be dependent on the polydispersity of the fraction, decreasing with increasing fraction breadth. The linear dependence of T on continuous phase viscosity indicates that the clay particles interact hydrodynamically with the continuous phase. The cubic dependence of T on the equivalent Stokes diameter (ESD) suggests that ESD is proportional to the face diameter of the plate-like kaolinite particles. The use of flow turbidity in colloidal particle size measurement is discussed, and the technique compared with that of flow birefringence.

Rheology of Colloid Systems

Abstract: Consider a unit volume of fluid consisting of an infinite number of layers, the lower surface of the bottom layer of which is fixed. This base is indicated in Figure 9.1 as the “reference plane”. A tangential force is applied at the top surface of the unit volume, which results in movement of the top layer in the direction of the applied force. The stacked layers slide one above the other by equal relative amounts to each other. The result is that a velocity gradient D = (dv/dy) is established perpendicular to the plane in which the top layer moves, wherein v is the velocity of the sliding layer and y is the distance of the layer from the “reference plane”. This velocity gradient is called rate of shear (or simply shear), and is measured in reciprocal seconds (m s-1 · m-1 = s-1). The applied tangential force is called shearing stress (or simply stress) and is indicated by τ. The stress is measured in dynes/cm2 or in newtons/m2.

Observations of velocity fields in WN and Of stars

Abstract: Systematic wavelength shifts of series of spectral line centers observed in many early type stars,generally interpreted as due to large scale motions,can give us information about the velocity gradients in stellar atmospheres.However,it should be borne in mind that the velocity gradients inferred from the observed displacements of spectral lines may not correspond to a unique alternative (e.g.see Karp 1978).Also,and especially when we are dealing with stars which have emission lines in their spectra,the structure of the velocity field depends on the assumed temperature structure of the atmosphere,i.e.in which atmospheric region do the lines originate.