Universal Electromagnetic Waves in Dielectric

The dielectric susceptibility of a wide class of dielectric materials follows, over extended frequency ranges, a fractional power-law frequency dependence that is called the "universal" response. The electromagnetic fields in such dielectric media are described by fractional differential equations with time derivatives of non-integer order. An exact solution of the fractional equations for a magnetic field is derived. The electromagnetic fields in the dielectric materials demonstrate fractional damping. The typical features of "universal" electromagnetic waves in dielectric are common to a wide class of materials, regardless of the type of physical structure, chemical composition, or of the nature of the polarizing species, whether dipoles, electrons or ions.Comment: 19 pages, LaTe

Computation of incompressible viscous flows through turbopump components

A finite-difference, three-dimensional, incompressible Navier-Stokes formulation for calculating the flow through turbopump components is presented. The solution method is based on the pseudocompressibility approach and uses an implicit-upwind differencing scheme together with the Gauss-Seidel line-relaxation method. Both steady and unsteady flow calculations can be performed using the presented algorithm. In this paper, the equations are solved in steadily rotating reference frames by using the steady-state formulation in order to simulate the flow through a turbopump inducer. Eddy viscosity is computed by using the Baldwin-Lomax model. Numerical results are compared with experimental measurements and good agreement is found between the two. Time-accurate calculations will be reported in future publications

Power-law rheology in the bulk and at the interface: quasi-properties and fractional constitutive equations

Consumer products, such as foods, contain numerous polymeric and particulate additives that play critical roles in maintaining their stability, quality and function. The resulting materials exhibit complex bulk and interfacial rheological responses, and often display a distinctive power-law response under standard rheometric deformations. These power laws are not conveniently described using conventional rheological models, without the introduction of a large number of relaxation modes. We present a constitutive framework using fractional derivatives to model the power-law responses often observed experimentally. We first revisit the concept of quasi-properties and their connection to the fractional Maxwell model (FMM). Using Scott-Blair's original data, we demonstrate the ability of the FMM to capture the power-law response of ‘highly anomalous’ materials. We extend the FMM to describe the viscoelastic interfaces formed by bovine serum albumin and solutions of a common food stabilizer, Acacia gum. Fractional calculus allows us to model and compactly describe the measured frequency response of these interfaces in terms of their quasi-properties. Finally, we demonstrate the predictive ability of the FMM to quantitatively capture the behaviour of complex viscoelastic interfaces by combining the measured quasi-properties with the equation of motion for a complex fluid interface to describe the damped inertio-elastic oscillations that are observed experimentally.United States. National Aeronautics and Space Administration (Microgravity Fluid Sciences (Code UG) for support of this research under grant no. NNX09AV99G

Microscopic dynamics and failure precursors of a gel under mechanical load

Material failure is ubiquitous, with implications from geology to everyday life and material science. It often involves sudden, unpredictable events, with little or no macroscopically detectable precursors. A deeper understanding of the microscopic mechanisms eventually leading to failure is clearly required, but experiments remain scarce. Here, we show that the microscopic dynamics of a colloidal gel, a model network-forming system, exhibit dramatic changes that precede its macroscopic failure by thousands of seconds. Using an original setup coupling light scattering and rheology, we simultaneously measure the macroscopic deformation and the microscopic dynamics of the gel, while applying a constant shear stress. We show that the network failure is preceded by qualitative and quantitative changes of the dynamics, from reversible particle displacements to a burst of irreversible plastic rearrangements

Wave Propagation in a Fractional Viscoelastic Tissue Model: Application to Transluminal Procedures

In this article, a wave propagation model is presented as the first step in the development of a new type of transluminal procedure for performing elastography. Elastography is a medical imaging modality for mapping the elastic properties of soft tissue. The wave propagation model is based on a Kelvin Voigt Fractional Derivative (KVFD) viscoelastic wave equation, and is numerically solved using a Finite Difference Time Domain (FDTD) method. Fractional rheological models, such as the KVFD, are particularly well suited to model the viscoelastic response of soft tissue in elastography. The transluminal procedure is based on the transmission and detection of shear waves through the luminal wall. Shear waves travelling through the tissue are perturbed after encountering areas of altered elasticity. These perturbations carry information of medical interest that can be extracted by solving the inverse problem. Scattering from prostate tumours is used as an example application to test the model. In silico results demonstrate that shear waves are satisfactorily transmitted through the luminal wall and that echoes, coming from reflected energy at the edges of an area of altered elasticity, which are feasibly detectable by using the transluminal approach. The model here presented provides a useful tool to establish the feasibility of transluminal procedures based on wave propagation and its interaction with the mechanical properties of the tissue outside the lumen.University College London, United KingdomTalentia scholarship (grant C2012H-75146405T-1) from the regional government of Andalusia, Spainthe Ministry of Education and Science, Spain, grants DPI2017-83859-R, EQC2018-004508-P and UNGR15-CE3664Andalusia, Spain, grants SOMM17/6109/UGR, B-TEP-026-UGR18, IE2017-5537 and P18-RT-165

On the formulation of hereditary cohesive-zone models

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The thesis presents novel formulations of hereditary cohesive zone models able to capture rate-dependent crack propagation along a defined interface. The formulations rely on the assumption that the measured fracture energy is the sum of an intrinsic fracture energy, related to the rupture of primary bonds at the atomic or molecular level, and an additional dissipation caused by any irreversible mechanisms present in the material and occurring simultaneously to fracture. The first contribution can be accounted for by introducing damage-type internal variables, which are to be driven by a rateindependent evolution law in order to be coherent with the definition as intrinsic energy. It is then proposed that the additional dissipation can be satisfactorily characterised by the same continuum-type material constitutive law obeyed by the interface material considered as a continuum: it is postulated that the dimensional reduction whereby a three-dimensional thin layer is idealized as a surface does not qualitatively alter the functional description of the free energy. The specific application considered is mode-I crack propagation along a rubber interface. After focusing on viscoelasticity as a suitable candidate to reproduce rubber’s behaviour, firstly the most common relaxation function, namely a single exponential term, is considerd after which the attention is turned to the use of fractional calculus and the related fractional integral kernel. A comparison with experimental results is presented. A shortcoming of the proposed approach is then noted, in that certain features of experimentally measured responses (i.e.the non-monotonicity of the critical energy-release rate with respect to crack speed) will be shown to be out of reach for the described modelling paradigm. A novel micromechanical formulation is then sketched in an attempt to qualitatively understand the phenomenon. An additional interface damaging mode is introduced, physically inspired by the desire to reproduce the formation of fibrils in a neighbourhood of the crack tip. Fibril formation is then driven by a variational argument applied to the whole of the interface, yielding its non-local character. Upon the introduction of an anisotropic fracture energy, motivated by experimental considerations, it is noted how the model can predict a non-monotonic energy-release rate vs crack speed behaviour, at least for a simple loading mode.Dunlop Oil & Marine Ltd and EPSR

Time-Domain Methods for Diffusive Transport in Soft Matter

Passive microrheology [12] utilizes measurements of noisy, entropic fluctuations (i.e., diffusive properties) of micron-scale spheres in soft matter to infer bulk frequency-dependent loss and storage moduli. Here, we are concerned exclusively with diffusion of Brownian particles in viscoelastic media, for which the Mason-Weitz theoretical-experimental protocol is ideal, and the more challenging inference of bulk viscoelastic moduli is decoupled. The diffusive theory begins with a generalized Langevin equation (GLE) with a memory drag law specified by a kernel [7, 16, 22, 23]. We start with a discrete formulation of the GLE as an autoregressive stochastic process governing microbead paths measured by particle tracking. For the inverse problem (recovery of the memory kernel from experimental data) we apply time series analysis (maximum likelihood estimators via the Kalman filter) directly to bead position data, an alternative to formulas based on mean-squared displacement statistics in frequency space. For direct modeling, we present statistically exact GLE algorithms for individual particle paths as well as statistical correlations for displacement and velocity. Our time-domain methods rest upon a generalization of well-known results for a single-mode exponential kernel [1, 7, 22, 23] to an arbitrary M-mode exponential series, for which the GLE is transformed to a vector Ornstein-Uhlenbeck process

Pair Production in Low Luminosity Galactic Nuclei

Electron-positron pairs may be produced near accreting black holes by a variety of physical processes, and the resulting pair plasma may be accelerated and collimated into a relativistic jet. Here we use a self-consistent dynamical and radiative model to investigate pair production by \gamma\gamma collisions in weakly radiative accretion flows around a black hole of mass M and accretion rate \dot{M}. Our flow model is drawn from general relativistic magnetohydrodynamic simulations, and our radiation field is computed by a Monte Carlo transport scheme assuming the electron distribution function is thermal. We argue that the pair production rate scales as r^{-6} M^{-1} \dot{M}^{6}. We confirm this numerically and calibrate the scaling relation. This relation is self-consistent in a wedge in M, \dot{M} parameter space. If \dot{M} is too low the implied pair density over the poles of the black hole is below the Goldreich-Julian density and \gamma\gamma pair production is relatively unimportant; if \dot{M} is too high the models are radiatively efficient. We also argue that for a power-law spectrum the pair production rate should scale with the observables L_X \equiv X-ray luminosity and M as L_X^2 M^{-4}. We confirm this numerically and argue that this relation likely holds even for radiatively efficient flows. The pair production rates are sensitive to black hole spin and to the ion-electron temperature ratio which are fixed in this exploratory calculation. We finish with a brief discussion of the implications for Sgr A* and M87.Comment: 21 pages, 10 figures, 1 table. Accepted for publication in Ap