Abstract
A fluid mechanics model for the slow deformations of layered geophysical materials is adopted. The problem consists of the slow viscous flow of two horizantally stratified fluid layers of different viscosities as being laterally squeezed between two frictionless planes. For solution a perturbation scheme is applied and the resulting approximation equations are solved numerically. The interface and free surface shapes, and the streamline patterns within the fluid layers are given.
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Abbreviations
- A kj, Bkj, Ckj Dk, Ekj, Fkj Gkj, Kkj, Rk, Skj :
-
perturbation coefficients, defined in (47) through (52)
- H a, H b :
-
dimensionless constant elevation of the interface and free surface, H′ a/L′ i, H′ b/L′ i, respectively.
- H′ a, H′ b :
-
constant elevation of the interface and free surface, respectively
- k :
-
constant indicating the first - (k=1) or second-order (k=2) approximations
- L :
-
dimensionless independent variable; distance from x=0 to the vertical plate, L′/L′ i
- L′ :
-
independent variable; distance from x′=0 to the vertical plate
- p :
-
dimensionless pressure, p′L′ i/μ′U′ i
- p′ :
-
pressure
- Re :
-
Reynolds number, ρ′U′ i L′ i/μ′
- t′ :
-
time
- U′ :
-
speed of the moving wall
- x :
-
dimensionless coordinate parallel to the stationary plate, x′/L′ i
- x′ :
-
coordinate parallel to the stationary plate
- y :
-
dimensionless coordinate vertical to the stationary plate, y′/L′ i
- y′ :
-
coordinate vertical to the stationary plate
- ε :
-
perturbation parameter, defined in (28)
- η a, η b :
-
dimensionless elevation of the interface and free surface, η′ a/L′ i, η′ b/L′ i, respectively
- η′ a, η′ b :
-
elevation of the interface and free surface, respectively
- μ′ :
-
viscosity
- μ r :
-
ratio of the viscosity of the lower fluid to the viscosity of the higher fluid layer, μ′ a/μ′ b
- ρ′ :
-
density
- Φ :
-
dummy variable indicating one of the dependent variables, defined in (28)
- ψ :
-
dimensionless stream function, ψ′/U′ i L′ i
- ψ′ :
-
stream function
- a:
-
lower fluid layer
- b:
-
higher fluid layer
- i:
-
at initial time
- k :
-
first- (k=1) or second-order (k=2) solution
- 0, 1, 2, ...:
-
zeroth-, first-, second-, ... order solutions
- a:
-
evaluation at y=η a(x, L)
- b:
-
evaluation at y=η b(x, L)
References
Orowan, E., Science 146 (1964) 1003.
Gordon, R. B., J. Geophys. Res. 70 (1965) 2413.
Turcotte, D. L., K. E. Torrance, and A. T. Hunt, Methods in Computational Physics, edited by B. A. Bolt, Academic Press, New York, 1973.
Artyushkov, E. V., J. Geophys. Res. 77 (1972) 6454.
Thompson, E. G., L. R. Mack, and F. S. Lin, Developments in Mechanics, Vol. 5, Proc. 11th Midwestern Mechanics Conference, Iowa State University Press, Ames, 1969.
Dieterich, J. H. and N. L. Carter, Amer. J. Sci. 267 (1969) 129.
Dieterich, J. H. and E. T. Onat, J. Geophys. Res. 74 (1969) 2081.
Mack, L. R. and E. G. Thompson, Quart. J. Mech. and Appl. Math. 25 (1972) 105.
Thompson, E. G. and L. R. Mack, J. Geophys. Res. 75 (1970) 1625.
Wehausen, J. V. and E. V. Laitone, Surface Waves, Encyclopedia of Physics 9, Springer-Verlag, Berlin, 1960.
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Meric, R.A. Lateral squeezing of horizontally stratified viscous layers. Appl. Sci. Res. 31, 241–252 (1975). https://doi.org/10.1007/BF00389217
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DOI: https://doi.org/10.1007/BF00389217