B. Svendsen et al., ON THE ROLE OF MECHANICAL INTERACTIONS IN THE STEADY-STATE GRAVITY FLOW OF A 2-CONSTITUENT MIXTURE DOWN AN INCLINED PLANE, Proceedings - Royal Society. Mathematical, physical and engineering sciences, 452(1948), 1996, pp. 1189-1205
In this work, we investigate the isothermal gravity-driven Stokes how
of a mixture of two constant true density viscous fluids which are ove
rlain by a (single-constituent) constant density viscous fluid down an
inclined-plane. The continuum thermody namical theory for such a syst
em implies that, in the simplest case, the constituents of such a mixt
ure interact mechanically with each other because of (1) friction or d
rag between the constituents, and (2) the non-uniform (volume) distrib
ution of constituents, in the mixture. The former interaction is propo
rtional to the relative velocity of the two constituents, and the latt
er to the gradient of the volume fraction. The coefficient of the volu
me fraction gradient in this latter interaction has the dimensions of
pressure, and is usually interpreted as the fluid pressure p in the ca
se of a fluid-solid mixture. More generally, however, this pressure re
presents that maintaining saturation in the mixture. In this work, we
formulate a model for a saturated mixture in which this coefficient ta
kes a slightly more general form, i.e. delta p, where delta is a dimen
sionless constant varying between 0 and 1. In particular, in the conte
xt of the thin-layer approximation, analytical solutions of the lowest
-order non-dimensionalized constituent momentum balances, under the us
ual assumption delta = 1, yield only pure constituent-1 or pure consti
tuent-2 'mixtures'. On the other hand, numerical solution of these mom
entum balances for delta not equal 1 yield non-trivial volume fraction
variations with depth in the layer, and hence represent true mixture
solutions. Applying this model to the case of a sediment-ice mixture,
such as that found in a glacier or ice sheet, one obtains good qualita
tive agreement with observations on the variation of sediment in these
bodies with depth for delta greater than or equal to 0.95, i.e. in th
is case the sediment remains concentrated at the bottom of the layer.