Search results for "equation"
showing 10 items of 4219 documents
A smeared seismicity constitutive model
2004
The classical application of rate and state dependent frictional constitutive laws has involved the instabilities developed between two sliding surfaces. In such a situation, the behaviour and evolution of asperities is the controlling mechanism of velocity weakening. However, most faults have a substantial thickness and it would appear that it is the bulk behaviour of the fault gouge, at whatever scale, that is important. The purpose of this paper is to explore how bulk frictional sliding behaviour may be described. We explore here the consequences of applying the rate and state framework initially developed to describe the frictional behaviour at the interface between two interacting slid…
Qualitative characterisation of effective interactions of charged spheres on different levels of organisation using Alexander’s renormalised charge a…
2005
Abstract Effective interactions are conveniently determined from experimental or numerical data by fitting a Debye–Huckel potential with an effective charge Z ∗ and an effective electrolyte concentration c ∗ as free parameters. In this contribution we numerically solved the Poisson–Boltzmann equation to obtain the so-called renormalised charge Z PBC ∗ . For sufficiently large bare charge Z one finds a saturation of Z ∗ which scales as Z ∗ = A a / λ B , where a is the particle radius, λ B the Bjerrum length and A a proportionality factor of order (8–10). The saturation value increases with increased total micro-ion concentration and shows a shallow minimum as a function of packing fraction. …
Shear behaviour of undiluted polyisobutylenes
1979
Some new data in shear flow are presented for two commercial polyisobutylene samples, namely Vistanex LMMH and L 100. In particular beyond a few steady state results, the tangential stress build-up after a sudden imposition of a shear rate and the decay after cessation of steady shear flow have been collected. The data are used to further test a constitutive equation already advanced by some of the authors. The comparison seems to confirm the validity of the proposed model, whose single adjustable parameter is shown to be independent of molecular weight.
Shock formation in the dispersionless Kadomtsev-Petviashvili equation
2016
The dispersionless Kadomtsev-Petviashvili (dKP) equation $(u_t+uu_x)_x=u_{yy}$ is one of the simplest nonlinear wave equations describing two-dimensional shocks. To solve the dKP equation we use a coordinate transformation inspired by the method of characteristics for the one-dimensional Hopf equation $u_t+uu_x=0$. We show numerically that the solutions to the transformed equation do not develop shocks. This permits us to extend the dKP solution as the graph of a multivalued function beyond the critical time when the gradients blow up. This overturned solution is multivalued in a lip shape region in the $(x,y)$ plane, where the solution of the dKP equation exists in a weak sense only, and a…
Numerical study of the Kadomtsev–Petviashvili equation and dispersive shock waves
2018
A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev-Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schrodinger equation in the semiclassical limit.
Capturing blast waves in granular flow
2007
Abstract In this paper we continue the analysis of compressible Euler equations for inelastic granular gases described by a granular equation of state due to Goldshtein and Shapiro [Goldshtein A, Shapiro M. Mechanics of collisional motion of granular materials. Part 1: General hydrodynamic equations. J Fluid Mech 1995;282:75–114], and an energy loss term accounting for inelastic collisions. We study the hydrodynamics of blast waves in granular gases by means of a fifth-order accurate scheme that resolves the evolution under different restitution coefficients. We have observed and analyzed the formation of a cluster region near the contact wave using the one-dimensional and two-dimensional v…
Long-term FRII jet evolution: Clues from three-dimensional simulations
2018
We present a long-term numerical three-dimensional simulation of a relativistic outflow designed to be compared with previous results from axisymmetric, two-dimensional simulations, with existing analytical models and state-of-art observations. We follow the jet evolution from 1~kpc to 200~kpc, using a relativistic gas equation of state and a galactic profile for the ambient medium. We also show results from smaller scale simulations aimed to test convergence and different three-dimensional effects. We conclude that jet propagation can be faster than expected from axisymmetric simulations, covering tens of kiloparsecs in a few million years, until the dentist drill effect produced by the gr…
Capturing shock waves in inelastic granular gases
2005
Shock waves in granular gases generated by hitting an obstacle at rest are treated by means of a shock capturing scheme that approximates the Euler equations of granular gas dynamics with an equation of state (EOS), introduced by Goldshtein and Shapiro [J. Fluid Mech. 282 (1995) 75-114], that takes into account the inelastic collisions of granules. We include a sink term in the energy balance to account for dissipation of the granular motion by collisional inelasticity, proposed by Haff [J. Fluid Mech. 134 (1983) 401-430], and the gravity field added as source terms. We have computed the approximate solution to a one-dimensional granular gas falling on a plate under the acceleration of grav…
A flux-split algorithm applied to conservative models for multicomponent compressible flows
2003
In this paper we consider a conservative extension of the Euler equations for gas dynamics to describe a two-component compressible flow in Cartesian coordinates. It is well known that classical shock-capturing schemes applied to conservative models are oscillatory near the interface between the two gases. Several authors have addressed this problem proposing either a primitive consistent algorithm [J. Comput. Phys. 112 (1994) 31] or Lagrangian ingredients (Ghost Fluid Method by Fedkiw et al. [J. Comput. Phys. 152 (1999) 452] and [J. Comput. Phys. 169 (2001) 594]). We solve directly this conservative model by a flux-split algorithm, due to the first author (see [J. Comput. Phys. 125 (1996) …
Dissipative shock waves in all-normal-dispersion mode-locked fiber lasers
2014
4 pags.; 4 figs.; OCIS codes: (140.4050) Mode-locked lasers; (140.3510) Lasers, fiber.