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Propagation and attenuation of sound through a layered phononic crystal with viscous constituents is theoretically studied. The Navier–Stokes equation with appropriate boundary conditions is solved and the dispersion relation for sound is obtained for a periodic layered heterogeneous structure where at least one of the constituents is a viscous fluid. Simplified dispersion equations are obtained when the other component of the unit is either elastic solid, viscous fluid, or ideal fluid. The limit of low frequencies when periodic structure homogenizes and the frequencies close to the band edge when propagating Bloch wave becomes a standing wave are considered and enhanced viscous dissipation is calculated. Angular dependence of the attenuation coefficient is analyzed. It is shown that transition from dissipation in the bulk to dissipation in a narrow boundary layer occurs in the region of angles close to normal incidence. Enormously high dissipation is predicted for solid–fluid structure in the region of angles where transmission practically vanishes due to appearance of so-called “transmission zeros,” according to El Hassouani, El Boudouti, Djafari-Rouhani, and Aynaou [Phys. Rev. B 78, 174306 (2008)]. For the case when the unit cell contains a narrow layer of high viscosity fluid, the anomaly related to acoustic manifestation of Borrmann effect is explained.
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Finite boundary effects on the spherical Rayleigh–Taylor instability between viscous fluids
For the Rayleigh–Taylor unstable arrangement of a viscous fluid sphere embedded in a finite viscous fluid spherical shell with a rigid boundary and a radially directed acceleration, a dispersion relation is developed from a linear stability analysis using the method of normal modes. aR1 is the radially directed acceleration at the interface. ρi denotes the density, μi is the viscosity, and Ri is the radius, where i = 1 is the inner sphere and i = 2 is the outer sphere. The dispersion relation is a function of the following dimensionless variables: viscosity ratio s=μ1μ2, density ratio d=ρ1ρ2, spherical harmonic mode n, B=R1aR1ρ22μ221/3, H=R2R1, and the dimensionless growth rate α=σμ2aR12ρ21/3, where σ is the exponential growth rate. We show that the boundedness provided by the outer spherical shell has a strong influence on the instability behavior, which is reflected not only in the modulation of the growth rate but also in the selection of the most unstable modes that are physically possible. This outer boundary effect is quantified by the relative magnitude of the radius ratio H. We find that when H is close to unity, lower order harmonics are excluded from becoming the most unstable within a vast region of the parameter space. In other words, the effect of H has precedence over the other controlling parameters d, B, and a wide range of s in establishing what the lowest most unstable mode can be. When H ∼ 1, low order harmonics can become the most unstable only for s ≫ 1. However, in the limit when s → ∞, we show that the most unstable mode is n = 1 and derive the dispersion relation in this limit. The exclusion of most unstable low order harmonics caused by a finite outer boundary is not realized when the outer boundary extends beyond a certain threshold length-scale in which case all modes are equally possible depending on the value of B.
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- Award ID(s):
- 1748883
- PAR ID:
- 10597334
- Publisher / Repository:
- American Institute of Physics
- Date Published:
- Journal Name:
- AIP Advances
- Volume:
- 12
- Issue:
- 4
- ISSN:
- 2158-3226
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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