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Rapid flow of granular materials
Published in M. Oda, K. Iwashita, Mechanics of Granular Materials, 2020
depends on the nine parameters on the right. In turn, they are, the strain, non-dimensional strain-rate, the Savage number (Savage, 1984), the Bagnold number (Bagnold, 1954), the density ratio, the non-dimensional particle fluctuation velocity, the particle friction coefficient, restitution coefficient, and solid concentration. Among these, the non-dimensional strain rate is the ratio of relative mean velocity between neighboring particles and the sound speed. The Savage number is the ratio of the mean particle inertia and its weight. The Bagnold number is the ratio of the mean particle inertia and the viscous force. The density ratio represents the relative importance of the solid stress to fluid stress. The non-dimensional particle fluctuation velocity is the ratio of particle’s random motion to its mean relative motion with respect to neighboring particles.
B
Published in Carl W. Hall, Laws and Models, 2018
where N = number present at t t = time k = constant Keywords: bacteria, growth, microbiology, pathology Source: Brown, S. B. and Brown, L. B. 1972. BAER LAW (BIOLOGY); VON BAER LAW; LAW OF VON BAER (1837) The more general features that are common to all members of a group of animals are developed in the embryo earlier than the special features that distinguish the various members of the group. The development of an organism proceeds from the generalized (homogeneous) to the specialized (heterogeneous) condition, so that the earliest embryonic stages of related organisms are identical and distinguishing features develop later. This law was the predecessor of the theory of recapitulation. Keywords: animals, development, embryo, features BAER, Karl Ernst Ritter von, 1792-1876, Estonian naturalist, geologist, and embryologist Sources: Critchley, M. 1978; Friel, J. P. 1974; Landau, S. I. 1986. See also EMBRYOGENESIS; HAECKEL; RECAPITULATION BAER LAW (GEOLOGY); VON BAER LAW The principle according to which the rotation of the Earth causes asymmetrical, lateral erosion of stream beds. Keywords: erosion, rotation, stream beds BAER, Karl Ernst Ritter von, 1792-1876, Estonian naturalist, geologist, and embryologist Source: Gary, M. et al. 1973. See also STREAM BANK BAGNOLD NUMBER, Ba OR NBa A dimensionless group that represents the drag force related to the gravitational force for saltation: NBa = 3 co g V2/4 d g p where co = a constant g = gas density p = particle density V = fluid velocity d = diameter g = gravitation Keywords: drag, gravitational, saltation BAGNOLD, Ralph A., 1896-1990, British physicist Sources: Bolz, R. E. and Tuve, G. L. 1970; Parker, S. P. 1992; Pelletier, R. A. 1994. BAIRSTON NUMBER--SEE MACH BALFOUR LAW The speed with which the ovum forms segments is approximately proportional to the concentration of the protoplasm in the ovum. The size of the segments is inversely proportional to the concentration of protoplasm in the ovum.
Sediment and fluid properties
Published in Arved J. Raudkivi, Loose Boundary Hydraulics, 2020
Bagnold presented the results, Figure 2.7, as G2=τ*ρsd2λµ2andpyρsd2λµ2 versus B=λ2ρsd2(dudy)2λ3/2µdudy=λ1/2ρsd2dudyµ where G has the form of a Reynolds number and the Bagnold number B is the ratio of shear stress from grain inertia to viscous shear, with the proportionality to λ3/2 inserted from observations. The ratio τ*/py was found to range from 0.32 to about 0.75, Figure 2.8. The transition region covers approximately 450 > N > 40 or 3000 > G2 > 100.
A review of lahars; past deposits, historic events and present-day simulations from Mt. Ruapehu and Mt. Taranaki, New Zealand
Published in New Zealand Journal of Geology and Geophysics, 2021
Jonathan Procter, Anke Zernack, Stuart Mead, Michael Morgan, Shane Cronin
Hungr (1995) divided grain-flow modelling into two main categories, either (1) lumped mass models that idealise the sliding body as a single point and are unable to account for internal deformation, or (2) continuum models based on rheological formulae (e.g. Newtonian, Bingham, Coulomb). Continuum numerical mass-flow models have developed from describing two major non-dimensional regimes using either the Bagnold number or the Savage number (Bagnold 1954). The Bagnold number approach describes the interstitial fluid and the interaction of particles, particle density, packing, strain rate and viscosity of fluid; this results in the classification of flow as being either macro-viscous or inertial. The Savage number focuses on the transfer of momentum between particles and the interparticle friction that develops; flows are either in a collisional or frictional regime. The modelling of granular flows has hence focused on determination of Mohr-Coulomb stresses acting on piles of grains.
Investigation of the effects of particle size on the performance of classical gravity concentration equipment
Published in Mineral Processing and Extractive Metallurgy Review, 2022
Damla Izerdem, S. Levent Ergun
The Bagnold number, a dimensionless shear rate group, is given by Equation (15), and the linear concentration (λ) is given by Equation (15) (Holtham 1992):
Effect of particle size distribution on rheology of high concentration limestone–water slurry for economic pipeline transportation
Published in Particulate Science and Technology, 2019
Sambit Senapati, Jayant Kumar Pothal, Akash Mohanty
In case of concentrated suspensions, many different types of particle interaction depending upon the particle size, shear rate, solid concentration, and temperature have been discussed by some investigators (Bagnold 1954; Coussot and Ancey 1999; Hunt et al. 2002). The linear dependence of shear and normal stresses on shear rate in the macro-viscous regime in a concentric cylinder rheometer was identified by Bagnold (1954). The shear stress varied linearly with fluid viscosity and influenced by the solids fraction, ϕ. The linear concentration, ε was then defined as the ratio of particle diameter to the mean radial separation and was correlated to the solid fraction (ϕ) by the following equation:where ϕm is the maximum possible concentration. The Bagnold number N derived by scaling the stresses in lower and higher shear rate regions can be written as:where ρ (kg/m3) is the particle density, d (m) is the particle size ε is the linear concentration, ỳ (s−1) is the shear rate, and ηp (Pas) is the Bingham viscosity. The computed values of ε and N for the five limestone samples in the studied range of slurry concentrations are given in Table 3. Interestingly, the calculated values of N obtained for the five limestone samples were in the range of 0.0009–7.43 which were less than 40 indicating the linear macro-viscous regime for dense particle flows (Coussot and Ancey 1999). Thus, the particle interactions due to collision impulses and collision rates influenced shear stress values for the limestone slurry samples. Furthermore, the linear concentration (ε) values obtained for sample S-1 and S-4 were found to be maximum and minimum, respectively, in the investigated range of concentrations. Therefore, the rheological data collected for the limestone samples were mostly affected by viscous and contact forces (lubrication, solid friction, collision) during our experimental measurements in concentric cylinder rheometer.