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Research Papers

Guided Waves Along Fluid-Filled Cracks in Elastic Solids and Instability at High Flow Rates

[+] Author and Article Information
Eric M. Dunham1

Assistant ProfessorDepartment of Geophysics and Institute for Computational and Applied Mathematics,  Stanford University, Stanford, CA 94305edunham@stanford.edu

Darcy E. Ogden

Assistant Professor  Scripps Institution of Oceanography,  University of California, San Diego, La Jolla, CA 92093dogden@ucsd.edu

1

Corresponding author.

J. Appl. Mech 79(3), 031020 (Apr 05, 2012) (7 pages) doi:10.1115/1.4005961 History: Received August 21, 2011; Revised January 13, 2012; Posted February 13, 2012; Published April 04, 2012; Online April 05, 2012

We characterize wave propagation along an infinitely long crack or conduit in an elastic solid containing a compressible, viscous fluid. Fluid flow is described by quasi-one-dimensional mass and momentum balance equations with a barotropic equation of state, and the wall shear stress is written as a general function of width-averaged velocity, density, and conduit width. Our analysis focuses on small perturbations about steady flow, through a constant width conduit, at an unperturbed velocity determined by balancing the pressure gradient with drag from the walls. Short wavelength disturbances propagate relative to the fluid as sound waves with negligible changes in conduit width. The elastic walls become more compliant at longer wavelengths since strains induced by opening or closing the conduit are smaller, and the fluid compressibility becomes negligible. As wavelength increases, the sound waves transition to crack waves propagating relative to the fluid at a slower phase velocity that is inversely proportional to the square-root of wavelength. Associated with the waves are density, velocity, pressure, and width perturbations that alter drag. At sufficiently fast flow rates, crack waves propagating in the flow direction are destabilized when drag reduction from opening the conduit exceeds the increase in drag from increased fluid velocity. This instability may explain the occurrence of self-excited oscillations in fluid-filled cracks.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

(a) Crack in unbounded elastic solid with opening 2w(x, t) containing fluid flowing in +x direction with width-averaged velocity u(x, t). (b) Perturbations in pressure (δp) and velocity (δu) carried by sound/crack waves with negligible damping. Waves propagating in the ±x direction have δu in phase with and of the same sign as ±δp. With the linearized equation of state, Eq. 10, density perturbations (δρ) have the same sign as δp. For Fourier mode perturbations with phase velocities less than the Rayleigh speed, conduit width perturbations (δw) have the same sign as δp.

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Figure 2

Plot of 1/F (Eq. 17) and H (Eq. 25) as a function of phase velocity c = −Im(s)/k for purely imaginary s, shown for ν = 1/4. For propagating waves, the ratio of conduit compressibility to fluid compressibility is F(k, s)Λ, where Λ is the quasi-static compressibility ratio defined in Eq. 20. Note that F is singular at the Rayleigh speed (c = cR ), making the conduit extremely compliant for waves propagating near this speed.

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Figure 3

Contours of |D(S)| in complex S plane, with D(S) defined in Eq. 26. Blue lines are contours from 0.1 to 1, plotted every 0.1; red lines continue from 1 up, with spacing of 1. Shown for cs /c0  = 2, ζ = 0.1, and ν = 1/4. (a) For Λ=0 and M0  = 0, solutions of D(S) = 0 are Rayleigh waves and damped sound waves. (b) Contours plotted for Λ=10 and M0  = 0. Black curves show trajectories of crack wave solutions as M0 is increased from 0 to 0.6 for viscous drag law (Table 1, m = 4, n = 0). The wave propagating with the flow becomes unstable beyond a critical M0 .

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Figure 4

Critical Mach number, M0 ≡ u0 /c0 , for instability, shown for viscous drag law (Table 1, m = 4, n = 0). Red curve is for quasi-static elasticity (Eq. 32) and blue curves show corresponding values with full elastodynamic response for several values of cs /c0 . Black curves show phase velocity c at neutral stability, which approaches but never exceeds the Rayleigh speed cR .

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Figure 5

Phase velocity of solutions propagating with flow (red and blue curves, with terminology in quotes), as a function of cs /c0 . Dashed black lines labeled c = … show limiting speeds. Shown for viscous drag law (Table 1) with M0  = 0.1 and Λ=0.1. Solution trajectories through transition region cs /c0  ∼ 1 differ for (a) ζ = 0.1 and (b) ζ = 0.3.

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