Showing papers on "Velocity gradient published in 1972"

The determination of the bulk stress in a suspension of spherical particles to order c 2

Abstract: An exact formula is obtained for the term of order c2 in the expression for the bulk stress in a suspension of force-free spherical particles in Newtonian ambient fluid, where c is the volume fraction of the spheres and c [Lt ] 1. The particles may be of different sizes, and composed of either solid or fluid of arbitrary viscosity. The method of derivation circumvents the familiar obstacle, of non-absolutely convergent integrals representing the effect of all pair interactions in which one specified particle takes part, by the judicious use of a certain quantity which is affected by the presence of distant particles in a similar way and whose mean value is known exactly. The bulk stress is in general of non-Newtonian form and depends on the statistical properties of the suspension which in turn are dependent on the type of bulk flow.The formula contains two functions which are parameters of the flow field due to two spherical particles immersed in fluid in which the velocity gradient is uniform at infinity. One of them, p(r, t), represents the probability density for the vector r separating the centres of the two particles. The variation of p(r, t) for a moving material point in r-space due to hydrodynamic action is found in terms of a function q(r), and this gives p(r, t) explicitly over the whole of the region of r-space occupied by trajectories of one particle centre relative to another which come from infinity. In a region of closed trajectories, steady-state hydrodynamic action alone does not determine the relation between the values of p (r, t) for different material points. The function q(r) is singular when the spheres touch, and the contribution of nearly-touching spheres to the bulk stress is evidently important. Approximate numerical values of all the relevant functions are presented for the case of rigid spherical particles of uniform size.In the case of steady pure straining motion of the suspension, all trajectories in r-space come from infinity, the suspension has isotropic structure and the stress behaviour can be represented (to order c2) in terms of an effective viscosity . It is estimated from the available numerical data that for a suspension of identical rigid spherical particles \[ {\mathop\mu\limits^{*}}/\mu = 1 + 2.5c + 7.6c^2, \] the error bounds on the coefficient of c2 being about ∓ 0.8. In the important case of steady simple shearing motion, there is a region of closed trajectories of one sphere centre relative to another, of infinite volume. The stress system is here not of Newtonian form, and numerical results are not obtainable until the probability, density p(r, t) can be made determinate in the region of closed trajectories by the introduction of some additional physical process, such as three-sphere encounters or Brownian motion, or by the assumption of some particular initial state.In the analogous problem for an incompressible solid suspension it may be appropriate to assume that for many methods of manufacture p(r, t) is uniform over the accessible part of r-space, in which event the solid suspension has ‘Newtonian’ elastic behaviour and the ratio of the effective shear modulus to that of the matrix is estimated to be 1 + 2·5c + 5·2c2 for a suspension of identical rigid spheres.

Flow Differentiation in Igneous Dikes and Sills: Profiles of Velocity and Phenocryst Concentration

Abstract: Mechanical interactions between phenocrysts during magma flow give rise to a grain-dispersive pressure. During intrusion into a dike or sill, in the absence of forces other than those of grain interaction, the grain-dispersive pressure must be constant across the flow width. As a result, the concentration of phenocrysts must decrease toward the walls to offset the increase in the velocity gradient as the walls are approached. This mechanism has been offered as an explanation for the observed rapid but gradational increases in content from phenocryst-poor margins to a phenocryst-rich center, especially in picritic dikes and sills. A relation for the grain dispersive pressure is solved together with the momentum equation of fluid flow, utilizing an empirical relation for the apparent viscosity of the phenocryst suspension. Solutions for steady flow between parallel walls demonstrate pluglike velocity profiles as well as phenocryst-concentration increases toward the center away from the walls. The velocity is nearly the maximum value within the central half of the flow. Therefore very strong pseudoplastic non-Newtonian behavior of the magma need not be assumed to explain the observed phenocryst concentration variations.

Interfacial Organisms: Passive Ventilation in the Velocity Gradients near Surfaces

Abstract: A variety of animals, including certain sponges, tube-dwellinlg worms, tropical termites, and prairie dogs, either are themselves arranged or construct domiciles arranged to permit flow of fluid inside the system driven by a velocity gradient in an external stream of fluid.

A mathematical model to predict the growth and elimination of inclusions in liquid steel stirred by natural convection

Abstract: A model is developed to predict the rate of removal and the change in the size distribution of inclusions in a melt stirred by natural convection. Difficulties in obtaining an exact solution to the problem due to lack of adequate knowledge for the velocity fields in the melts are discussed. The model is based on Smoluchowski’s Theory of Gradient Collision to obtain the probability of collision between two inclusions under an arbitrarily chosen velocity gradient. Initial size distributions obtained in experimental heats are used as the input to the model. Various conditions are proposed by which inclusions are removed from the melt. The rates of removal are compared with the experimentally obtained rate of removal of oxides. It is observed that a boundary layer effect and the presence of a thin liquid metal film prevent rapid removal of inclusions from the stirred melts. Inclusion size distribution predicted by the model agrees qualitatively with the experimentally observed size distribution. It is postulated that the surface forces play a significant role in coalescence and assimilation of inclusions. Finally, the application of similar models to understand the removal of inclusions in such processes as argon sparging, solidification, degassing and electroslag remelting are advocated.

Rates of change of flutter Mach number and flutter frequency.

Abstract: free" region which he perceives to begin at about 10-15D. This characterization seems a misnomer, in that the shear behaves in a continuous fashion over all x/D studied, as can be seen from Fig. 1. What does happen is that the velocity defect—and hence the mean-flow velocity gradient du/dr—decays to a small value very rapidly. Since all eddy viscosity models are utilized to predict the shear in conjunction with a gradient transport model [see Eq. (2)], one can then question the applicability of any such model when the velocity gradient becomes small. However, the present model was developed based upon the axial behavior of the shear and the velocity defect, both of which show no discernible change in behavior over the complete range of the data. More precisely, the slope of the distribution of these quantities (i.e., the decay rate) shows no change in value over the range of the experiment. Thus, no distinct regimes can be logically identified. Further, the numerical prediction is in roughly equivalent agreement with the data over the complete range in x/D studied, so that clearly the region of applicability of the present model extends at least that far downstream.

Theory of Second‐Order Stress in a Dense Gas

Abstract: The phenomenological coefficients associated with stress tensor terms proportional to the square of the velocity gradient have been computed for dense gases composed of rigid spheres and of particles interacting with a square‐well potential. Most of the calculations reported here were obtained from a generalization of the moment method which is appropriate to a dense gas. A detailed comparison between theory and experiment leads to the following conclusions: (i) Reiner's torsional flow observations for gases and for liquid toluene are not attributable to second‐order stress. At low densities kinetic theory predicts an effect with the opposite algebraic sign from that observed experimentally. At high densities the theory predicts a reversal of sign. At all densities the magnitudes of the theoretical and experimental effects differ by several orders of magnitude. (ii) Oliver and MacSporran's measurements of jet thrust due to liquids issuing from circular orifices differ by many orders of magnitude from the predictions of kinetic theory. However, at high densities there is agreement with regard to the algebraic sign of the effect. (iii) Oliver and MacSporran also studied jet thrust from a slot, but were unable to observe any effect with gases. Kinetic theory is in agreement with this.

[13] Flow birefringence

Abstract: Publisher Summary Theoretical considerations indicate that if the solute macromolecules are geometrically and therefore optically asymmetric, the double refraction of the system appears as a result of the equilibrium between the orienting effect of the applied field and the disordering effect of the Brownian motion. The flow-birefringence apparatus is simple, and it can be easily constructed in the laboratory. The method of measurements is also simple. The measurement of the extinction angle in a certain range of the velocity gradient can give quantitative information on the size and shape of rigid solute particles by the determination of the rotary diffusion constant. In some cases, the change of the shape with environmental conditions can be detected by the extinction angle change. On the other hand, information on the optical anisotropy of solute particles obtained by flow birefringence is mostly qualitative. However, the theoretical foundation is not sufficient for quantitative analyses.

A method to separate delayed elasticity from viscous flow specially applied to borate glasses

Abstract: Shear under stress in alkali borate glasses has been studied with the aid of a Pochettino viscosimeter. A new method of applying the stress, as compared with those of previous experiments in the literature, has been used. With this method the delayed elastic effects can be separated satisfactorily from the viscous flow. Consequently, this method has yielded a measuring procedure to determine the velocity gradient for purely viscous flow. This method consists in measuring the velocity gradients at a certain deformation time at two different stress levels and taking their average. Rheological properties (viscosity and Bingham yield value) calculated from the average velocity gradients appear to be practically independent of time. Measurements performed in this way are considered sufficiently reliable to permit conclusions about the rheological properties. Finally, the observed delayed elastic effects have been explained in terms of the structure of alkali borate glasses.

Concentration-stress convection and the properties of slow flows of mixtures of gases

Abstract: In the general case, changes in the temperature of a gas bring about motion of the gas, i.e., thermostrcss convection [1, 2]. It is shown below that in mixtures of gases there is an analogous phenomenon, i.e., concentration-stress convection, resulting from concentration gradients; examples of the latter are given. Some of the results of [1, 2] are also correlated. In [1, 2] investigations were made of the basic properties of a class of slow flows, “nonrarefied” (Knudsen number K→0) of a monoatomic gas with Reynolds numbers R∼1 and with relative temperature drops in the flow θ=T*−1Δ T≲1, which, with adhesive boundary conditions, cannot be described by the Navier-Stokes equations (T* is the characteristic temperature). Under these circumstances, it is necessary to take account of some of the Barnett terms of the momentum equation, due to temperature gradients, and to slipping of the temperature. In the case of slow movements of a mixture of gases, there are analogous effects, due to concentration gradients, if the relative concentration drops Nα=yα*−1Δyα are of the order of magnitude of unity [2]. Here, yα-nα/n; nα is the number of particles of the α-th component of the mixture in unit volume, n=Σnα. Maxwell [3] was the first to investigate the question of the stresses in rarefied gases, due to nonhomogeneity of the temperature. In this case, he neglected terms of the tensor of the temperature stresses containing products of the first derivatives with respect to T along the coordinates. Then, the Barnett temperature terms entering into the momentum equation have a “gradient” form and can be combined with the pressure. Therefore, Maxwell drew the conclusion that temperature stresses do not bring about motion, and that motion can arise only as the result of slipping of the temperature; his investigation reduced to a consideration of the redistribution of the pressure due to temperature stresses in a quiescent gas. After this, the above question was touched upon in the book [4], in which the following note is made: “for velocity gradients on the order of 1 sec−1 and values of [∂2T/∂xi∂xj] on the order of 1 deg/cm2, it is impossible to completely neglect the temperature stresses in comparison with the ordinary viscous stresses, even at ordinary pressures, and they can play a role in experiments aimed at determination of the viscosity, in which inequality of the temperatures is allowed.” As far as is known to the authors, this exhausts the existing literature discussion of the effect of Barnett stresses (both temperature and concentration) on gas flows with K→0. In concluding our review, we note still another phenomenon, for a description of which, with K≪1, we must bring in the Barnett approximation: in the case of slow flows of a mixture of gases, some of the Barnett (due to velocity gradients) terms in the expression for the diffusional velocity are of the same order as the usual barodifferential term of this expression [4, 5]. There is demonstrated below the “symmetry” of some of the flow properties of a one-component gas in a mixture of gases, in the presence, respectively, of Barnett temperature and concentration stresses. The article considers conditions with the absence of convection due to the corresponding Barnett stresses. It gives examples of concentration-stress flows and discusses the properties of flows due to weak temperature and diffusion slipping.

Note on the No-stress Boundary Condition at the Edge of the Ice Pack

Abstract: The theoretical modelling of the large-scale motion of the arctic ice pack is receiving increasing attention as the economic importance of the region increases. One of the most widely used types of model is the so-called "viscous fluid" model. ... The boundary condition at the edge of the ice pack is an important feature of most such models. In some cases a no-slip condition seems appropriate, but in others, when the ice near the boundary has a low compactness (fraction of ice coverage) or the boundary occurs away from a coast, some other condition may be more appropriate. One that is often suggested is a no-stress condition, which is often assumed to imply that there is no velocity gradient perpendicular to the boundary. When the edge occurs away from a coast, the latter assumption is wrong. It suffices for present purposes to assume that we are dealing with an incompressible two-dimensional fluid. In this case the viscous force per unit of area (corresponding to volume in three dimensions) is del·(A del v), where A is an isotropic but possibly variable coefficient of eddy viscosity, and v, the large-scale averaged horizontal ice velocity, has components u and v in the x and y directions respectively. The notation del v ... is equivalent to the tensor [partial derivative of vi with respect to xj], where i and j vary independently over all coordinate directions, and (del·del v)i = Sum over the index j of (partial derivative with respect to xj of the partial derivative of vi with respect to xj). ... Since the viscous force is the divergence of the stress, the quantity A del v is often thought of as the eddy stress (or "internal ice stress"). That this is not true is easily seen by noting that the tensor A del v, to be referred to here as the "pseudo-stress" tensor, is not symmetrical. The non-diagonal elements of the stress tensor, which must be equal, are ½A(partial derivative of v with respect to x + partial derivative of u with respect to y). The distinction made here is irrelevant in determining the viscous forces, since the stress tensor and the pseudo-stress tensor differ by a tensor of zero divergence .... In large-scale ocean models which employ eddy viscosity, the stress itself is often required in connection with boundary conditions, particularly at the sea surface, or naviface .... Here, however, those who use the pseudo-stress are saved both by scale considerations and by the fact that w=0 (where w is the vertical or z-component of velocity), hence the (partial derivative of w with respect to x)=0 and the (partial derivative of w with respect to y)=0, at the naviface, so that the stress components there reduce to A(partial derivative of u with respect to z) and A(partial derivative of v with respect to z). In the "viscous liquid" model of an ice pack bounded by open water, we at last have a case in which the distinction between real stress and pseudo-stress assumes geophysical importance. ... Assuming (without loss of generality) that the edge is oriented with its outward normal in the first quadrant at an angle of theta to the x-axis, we have for the direction cosines: n1 = +cos(theta), n2 = +sin(theta), t1 = +sine(theta), t2= -cos(theta). The appropriate expression of the condition that there be no tangential stress at the boundary becomes: sin(2theta)(partial derivative of u with respect to x - partial derivative of v with respect to y) - cos(2theta)(partial derivative of v with respect to x - partial derivative of u with respect to y) = 0. ... If the boundary is oriented along a coordinate axis this reduces to (partial derivative of v with respect to x) + (partial derivative of u with respect to y) = 0, which qualitatively means that shears at the boundary are permitted, provided that they are part of a locally uniform rotation and do not produce deformation of the ice field. If one also wishes to assume zero normal stress at the boundary, there is an additional condition given by: (partial derivative of u with respect to x)cos² (theta) + (partial derivative of v with respect to y)sin²(theta) + ½(partial derivative of v with respect to x + partial derivative of u with respect to y)sin(2theta) = 0. These are purely mathematical deductions; the appropriateness of the physical conditions is a more difficult question which can only be answered experimentally. The physical condition of zero tangential stress qualitatively means that no deformation of the ice field can take place at the boundary. Techniques for measuring the deformation of the ice fields are now under development. It is suggested that it would be interesting to measure the deformation of ice fields near the boundary, even though a measurement of non-zero deformation (which the author suspects would be found, since external driving forces will in general tend to produce deformation) would not distinguish critically between the correctness of the boundary condition and the basic validity of the "viscous liquid" type of model.

Measuring the velocity in a boundary layer with a total-head pitot tube

Abstract: Results are shown of an experimental study concerning the effects of viscosity, of the velocity gradient, and of the wall proximity on the readings of a total-head Pitot tube during the measurement of the velocity distribution in the viscous sublayer region of a turbulent boundary layer.

Flow and Mixing of Powder in the Vertical-Cylinder-Type Mixer

Abstract: The flow pattern and the mixing process of powder in the vertical-cylinder-type mixer were investigated using photographic observation and measurements of the concentration profile of a tracer. Furthermore, using white glass particles as a tracer, the flow pattern in the mixer was studied by the model mixer, in which glass beads were made transparent by methylbenzoate, having same refractive index with glass beads.The flows of powder varied according to the local position in the mixer. Those characteristics were shown in term of the local rate coefficient of mixing.Desirable mixing states were observed under the condition of strong shear force and large rectangular velocity gradient in powder at the front edge of the paddle. In addition, many flowing circuits and mixing zones were found at the other regions.

Direct Measurement of the Velocity Gradient in a Fluid Flow

Abstract: Conclusions Results of the present experiments show increasing plateau pressure and decreasing length of the interaction region with decreasing Reynolds number and decreasing wall-to-recovery temperature ratio Values of plateau pressure coefficient were proportional to the square root of the skin-friction coefficient of the undisturbed boundary layer, a scaling law derived from Chapman's formulation of free interaction, and maximum static pressure gradients were found to agree reasonably well with Zukoski's correlation of adiabatic data

The influence of coanda effect on blow-off limit

Abstract: The behavior of a premixed acetylene-air flame on a slot burner adjacent to a wall is described. The so-called wall reattachment or Coanda effect is seen to occur. The blow-off flow rate is shown to be greater than that obtained with an ordinary slot burner. The results obtained for various slot sizes are demonstrated to correlate approximately when plotted as critical average velocity gradient versus mixture ratio.

Mass transfer in a laminar boundary layer with moving interface

Abstract: Mass transfer coefficient is numerically obtained for a laminar boundary layer flow with tangentially moving interface. It is implied that the flow is similar and the dimensionless stream function is known. Dimensionless mass transfer coefficient can be approximately expressed as a function of a single parameter containing the velocity and the velocity gradient at the interface. The functional relation is analogous with that by Beek and Bakker and applicable to all similar boundary layer flows. It is also numerically shown that the general correlation gives a good estimate even in the case of small Schmidt number for uniform flow over a flat interface.

Method of investigating the extension of high-elastic polymers

Abstract: A method has been developed for studying the deformation of polymers in a field with a longitudinal velocity gradient. The changes in the longitudinal viscosity and modulus of high elasticity of polyisobutylene in uniaxial tension at constant extension and strain rates are compared.

Some Problems of the Use of P-Wave Dynamics to Find the Velocity Profile of the Upper Mantle

Abstract: This paper presents a numerical study of effect on the amplitude-distance curve of body waves caused by a discontinuity in the velocity gradient at a certain depth. The study was made with an elementary model; two layers with different gradients. The results can be used for more complicated models.