Impulse acoustics based ejection of ferrofluid grains from a ferrofluid: the blueprint of a concept for a nozzle-free in
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New Wave Dynamics in Granular State Vitali F. Nesterenko Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093, U.S.A.
ABSTRACT The unusual feature of granular state is the negligible linear range of the interaction force between neighboring particles resulting in zero sound speed in uncompressed material. This makes linear and weakly nonlinear continuum approach based on Korteveg - de Vries equation invalid. This paper describes a results based on more general approach. INTRODUCTION Granular materials are highly nonlinear materials according to a few physically different reasons. For example Hertz law for compression of two elastic granules (or more general powerlaw) has no linear part even for relatively small displacements in the vicinity of zero compression force. Nonlinearity can be caused by other principally different reason - structural rearrangements under applied load. This paper addresses nonlinearities connected with interaction law between particles. Even this part of nonlinear behavior is able to produce a qualitatively different mode of wave propagation. It places granular materials in a special class according to wave dynamics. This was a reason for the introduction of the concept of “sonic vacuum” - the medium where the traditional wave equation is not a basic equation for wave dynamics [1]. The synthesis of all components of highly nonlinear behavior is a very exciting area for future research. Hopefully this development will also extend to other newly designed materials, which will poses highly nonlinear properties desirable for applications.
LONG-WAVE EQUATION FOR “STRONGLY COMPRESSED” HERTZ CHAIN Wave propagation in one-dimensional granular material is considered taking into account that particles interact with each other according to the Hertz law [2]. It is assumed that a onedimensional chain of identical spherical granules is subjected to constant compression forces F applied to both ends and securing the initial displacement δ0 between particle centers. The particle equation of motion becomes [1]: ui = A δ 0 – ui + ui – 1 m = 4 πR 3ρ 0, 3
A=
32
– A δ 0 – ui + 1 + ui
E 2R
32
, N – 1 ≥ i ≥ 2,
12
3 1 – ν2 m
,
(1)
BB3.1.1 Downloaded from https:/www.cambridge.org/core. University of Arizona, on 27 Apr 2017 at 13:28:32, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1557/PROC-627-BB3.1
here m is the mass of the particle, E and ρ0 are the Young’s modules and density of particle material, R is the granule radii, ν is the Poisson coefficient. It is assumed that the distance between the particle centers does not exceed 2R if particles are spherical. After anharmonic and long-wave approximations, Eq. 1 can be transformed into nonlinear Boussinesq equation [1]. It can be further transformed [1,3] inside the same approximation into Korteweg-de Vries equation [4] which describes propagation of disturbances in one direction ξ t + c 0ξ x + γξ xxx + σ ξξ x = 0, 2c 0
ξ = – ux
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