Anisotropic Energy Distribution in Three-Dimensional Vibrofluidized Granular Systems

We examine the energy distribution in a three-dimensional model granular system contained in an open cylinder under the influence of gravity. Energy is supplied to the system by a vibrating base. We introduce spatially resolved, partial particle-particle ``dissipations'' for directions parallel and perpendicular to the energy input, respectively. Energy balances show that the total (integrated) ``dissipation'' is less than zero in the parallel direction while greater than zero in the perpendicular directions. The energy supplied to the perpendicular directions is dissipated by particle-wall collisions. We further define a fractional energy transfer, which in the steady state represents the fraction of the power supplied by the vibrating base that is dissipated at the wall. We examine the dependence of the fractional energy transfer on the number of particles, the velocity of the vibrating base, the particle-particle restitution coefficient, and the particle-wall restitution coefficient. We also explore the influence of the system parameters on the spatially dependent partial dissipations.Comment: 10 pages, 10 figures, RevTeX forma

Exact solution of a one-dimensional Boltzmann equation for a granular tracer particle

We consider a one-dimensional system consisting of a granular tracer particle of mass $M$ in a bath of thermalized particles each of mass $m$. When the mass ratio, $M/m$, is equal to the coefficient of restitution, $\alpha$, the system maps to a a one-dimensional elastic gas. In this case, Boltzmann equation can be solved exactly. We also obtain expressions for the velocity autocorrelation function and the diffusion coefficient. Numerical simulations of the Boltzmann equation are performed for $M/m\neq \alpha$ where no analytical solution is available. It appears that the dynamical features remain qualitatively similar to those found in the exactly solvable case.Comment: 17 pages, 3 figures, Accepted in Physica

Thermalization of an anisotropic granular particle

We investigate the dynamics of a needle in a two-dimensional bath composed of thermalized point particles. Collisions between the needle and points are inelastic and characterized by a normal restitution coefficient $\alpha<1$. By using the Enskog-Boltzmann equation, we obtain analytical expressions for the translational and rotational granular temperatures of the needle and show that these are, in general, different from the bath temperature. The translational temperature always exceeds the rotational one, though the difference decreases with increasing moment of inertia. The predictions of the theory are in very good agreement with numerical simulations of the model.Comment: 7 pages, 6 Figures, submitted to PRE. Revised version (Fig1, Fig5 and Fig6 corrected + minor typos

Studies of Mass and Size Effects in Three-Dimensional Vibrofluidized Granular Mixtures

We examine the steady state properties of binary systems of driven inelastic hard spheres. The spheres, which move under the influence of gravity, are contained in a vertical cylinder with a vibrating base. We computed the trajectories of the spheres using an event-driven molecular dynamics algorithm. In the first part of the study, we chose simulation parameters that match those of experiments performed by Wildman and Parker. Various properties computed from the simulation including the density profile, granular temperature and circulation pattern are in good qualitative agreement with the experiments. We then studied the effect of varying the mass ratio and the size ratio independently while holding the other parameters constant. The mass and size ratio are shown to affect the distribution of the energy. The changes in the energy distributions affect the packing fraction and temperature of each component. The temperature of the heavier component has a non-linear dependence on the mass of the lighter component, while the temperature of the lighter component is approximately proportional to its mass. The temperature of both components is inversely dependent on the size of the smaller component.Comment: 14 Pages, 12 Figures, RevTeX

Diffusion of impurities in a granular gas

Diffusion of impurities in a granular gas undergoing homogeneous cooling state is studied. The results are obtained by solving the Boltzmann--Lorentz equation by means of the Chapman--Enskog method. In the first order in the density gradient of impurities, the diffusion coefficient $D$ is determined as the solution of a linear integral equation which is approximately solved by making an expansion in Sonine polynomials. In this paper, we evaluate $D$ up to the second order in the Sonine expansion and get explicit expressions for $D$ in terms of the restitution coefficients for the impurity--gas and gas--gas collisions as well as the ratios of mass and particle sizes. To check the reliability of the Sonine polynomial solution, analytical results are compared with those obtained from numerical solutions of the Boltzmann equation by means of the direct simulation Monte Carlo (DSMC) method. In the simulations, the diffusion coefficient is measured via the mean square displacement of impurities. The comparison between theory and simulation shows in general an excellent agreement, except for the cases in which the gas particles are much heavier and/or much larger than impurities. In theses cases, the second Sonine approximation to $D$ improves significantly the qualitative predictions made from the first Sonine approximation. A discussion on the convergence of the Sonine polynomial expansion is also carried out.Comment: 9 figures. to appear in Phys. Rev.

Navier-Stokes transport coefficients of dd-dimensional granular binary mixtures at low density

The Navier-Stokes transport coefficients for binary mixtures of smooth inelastic hard disks or spheres under gravity are determined from the Boltzmann kinetic theory by application of the Chapman-Enskog method for states near the local homogeneous cooling state. It is shown that the Navier-Stokes transport coefficients are not affected by the presence of gravity. As in the elastic case, the transport coefficients of the mixture verify a set of coupled linear integral equations that are approximately solved by using the leading terms in a Sonine polynomial expansion. The results reported here extend previous calculations [V. Garz\'o and J. W. Dufty, Phys. Fluids {\bf 14}, 1476 (2002)] to an arbitrary number of dimensions. To check the accuracy of the Chapman-Enskog results, the inelastic Boltzmann equation is also numerically solved by means of the direct simulation Monte Carlo method to evaluate the diffusion and shear viscosity coefficients for hard disks. The comparison shows a good agreement over a wide range of values of the coefficients of restitution and the parameters of the mixture (masses and sizes).Comment: 6 figures, to be published in J. Stat. Phy

Transport coefficients for inelastic Maxwell mixtures

The Boltzmann equation for inelastic Maxwell models is used to determine the Navier-Stokes transport coefficients of a granular binary mixture in $d$ dimensions. The Chapman-Enskog method is applied to solve the Boltzmann equation for states near the (local) homogeneous cooling state. The mass, heat, and momentum fluxes are obtained to first order in the spatial gradients of the hydrodynamic fields, and the corresponding transport coefficients are identified. There are seven relevant transport coefficients: the mutual diffusion, the pressure diffusion, the thermal diffusion, the shear viscosity, the Dufour coefficient, the pressure energy coefficient, and the thermal conductivity. All these coefficients are {\em exactly} obtained in terms of the coefficients of restitution and the ratios of mass, concentration, and particle sizes. The results are compared with known transport coefficients of inelastic hard spheres obtained analytically in the leading Sonine approximation and by means of Monte Carlo simulations. The comparison shows a reasonably good agreement between both interaction models for not too strong dissipation, especially in the case of the transport coefficients associated with the mass flux.Comment: 9 figures, to be published in J. Stat. Phy

Do quantitative and qualitative shear wave elastography have a role in evaluating musculoskeletal soft tissue masses?

Objectives: To determine if quantitative and qualitative shear wave elastography have roles in evaluating musculoskeletal masses. Methods: 105 consecutive patients, prospectively referred for biopsy within a specialist sarcoma centre, underwent B-mode, quantitative (m/s) and qualitative (colour map) shear wave elastography. Reference was histology from subsequent biopsy or excision where possible. Statistical modelling was performed to test elastography data and/or B-mode imaging in predicting malignancy. Results: Of 105 masses, 39 were malignant and 6 had no histology but benign characteristics at 12 months. Radiologist agreement for B-mode and elastography was moderate to excellent Kw 0.52-0.64; PABAKw 0.85-0.90). B-Mode imaging had 78.8% specificity, 76.9% sensitivity for malignancy. Quantitatively, adjusting for age, B-mode and lesion volume there was no statistically significant association between longitudinal velocity and malignancy (OR [95% CI] 0.40[0.10, 1.60], p=0.193), but some evidence that higher transverse velocity was associated with decreased odds of malignancy (0.28[0.06, 1.28], p=0.101). Qualitatively malignant masses tended to be towards the blue spectrum (lower velocities); 39.5% (17/43) of predominantly blue masses were malignant, compared to 14.3% (1/7) of red lesions. Conclusions: Quantitatively and qualitatively there is no statistically significant association between shear wave velocity and malignancy. There is no clear additional role to B-mode imaging currently. Key Points: • Correlation between shear wave velocity and soft tissue malignancy was statistically insignificant• B-mode ultrasound is 76.9 % sensitive and 78.8 % specific• Statistical models show elastography does not significantly add to lesion assessmen

Microscale characterization of prostate biopsies tissues using optical coherence elastography and second harmonic generation imaging

© 2018 USCAP, Inc All rights reserved. Photonics, especially optical coherence elastography (OCE) and second harmonic generation (SHG) imaging are novel high-resolution imaging modalities for characterization of biological tissues. Following our preliminary experience, we hypothesized that OCE and SHG imaging would delineate the microstructure of prostate tissue and aid in distinguishing cancer from the normal benign prostatic tissue. Furthermore, these approaches may assist in characterization of the grade of cancer, as well. In this study, we confirmed a high diagnostic accuracy of OCE and SHG imaging in the detection and characterization of prostate cancer for a large set of biopsy tissues obtained from men suspected to have prostate cancer using transrectal ultrasound (TRUS). The two techniques and methods described here are complementary, one depicts the stiffness of tissues and the other illustrates the orientation of collagen structure around the cancerous lesions. The results showed that stiffness of cancer tissue was ∼57.63% higher than that of benign tissue (Young's modulus of 698.43±125.29 kPa for cancerous tissue vs 443.07±88.95 kPa for benign tissue with OCE. Using histology as a reference standard and 600 kPa as a cut-off threshold, the data analysis showed sensitivity and specificity of 89.6 and 99.8%, respectively. Corresponding positive and negative predictive values were 99.5 and 94.6%, respectively. There was a significant difference noticed in terms of Young's modulus for different Gleason scores estimated by OCE (P-value<0.05). For SHG, distinct patterns of collagen distribution were seen for different Gleason grade disease with computed quantification employing a ratio of anisotropic to isotropic (A:I ratio) and this correlated with disease aggressiveness