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.

References

References
1.
Paillet
,
F. L.
, and
White
,
J. E.
, 1982, “
Acoustic Modes of Propagation in the Borehole and Their Relationship to Rock Properties
,”
Geophys.
,
47
(
8
), pp.
1215
1228
.
2.
Chouet
,
B.
, 1986, “
Dynamics of a Fluid-Driven Crack in Three Dimensions by the Finite Difference Method
,”
J. Geophys. Res.
,
91
(
B14
), pp
13,967
13, 992
.
3.
Ferrazzini
,
V.
, and
Aki
,
K.
, 1987, “
Slow Waves Trapped in a Fluid-Filled Infinite Crack: Implications for Volcanic Tremor
,”
J. Geophys. Res.
,
92
(
B9
), pp.
9215
9223
.
4.
Korneev
,
V.
, 2008, “
Slow Waves in Fractures Filled with Viscous Fluid
,”
Geophys.
,
73
(
1
), pp.
N1
N7
.
5.
Julian
,
B.
, 1994, “
Volcanic Tremor: Nonlinear Excitation by Fluid Flow
,”
J. Geophys. Res.
,
99
(
B6
), pp
11,859
11, 877
.
6.
Balmforth
,
N. J.
,
Craster
,
R. V.
, and
Rust
,
A. C.
, 2005, “
Instability in Flow Through Elastic Conduits and Volcanic Tremor
,”
J. Fluid Mech.
,
527
, pp.
353
377
.
7.
Tsai
,
V. C.
, and
Rice
,
J. R.
, 2010, “
A Model for Turbulent Hydraulic Fracture and Application to Crack Propagation at Glacier Beds
,”
J. Geophys. Res.
,
115
.
8.
Geubelle
,
P. H.
, and
Rice
,
J. R.
, 1995, “
A Spectral Method for Three-Dimensional Elastodynamic Fracture Problems
,”
J. Mech. Phys. Solids
,
43
, pp.
1791
1824
.
9.
Pedley
,
T. J.
, 1980,
The Fluid Mechanics of Large Blood Vessels
,
Cambridge University Press
,
Cambridge
.
10.
Rahman
,
M.
, and
Barber
,
J. R.
, 1995, “
Exact Expressions for the Roots of the Secular Equation for Rayleigh Waves
,”
ASME J. Appl. Mech.
,
62
(
1
), pp.
250
252
.
11.
Winberry
,
J. P.
,
Anandakrishnan
,
S.
, and
Alley
,
R. B.
, 2009, “
Seismic Observations of Transient Subglacial Water-Flow Beneath MacAyeal Ice Stream, West Antarctica
,”
Geophys. Res. Lett.
,
36
.
You do not currently have access to this content.