Abstract
A generalized lubrication theory that is applicable to highly deformable porous layers
is developed using an effective-medium approach (Brinkman equation). This theory
is valid in the limit where the structure is so compressible that the normal forces
generated by elastic compression of the fibres comprising the solid phase are negligible
compared to the pressure forces generated within the porous layer. We assume that
the deformation of the solid phase is primarily due to boundary compression as
opposed to the motion of the fluid phase. A generalized Reynolds equation is derived
in which the spatial variation of the Darcy permeability parameter,
α = H/√Kp, due
to the matrix compression is determined by new local hydrodynamic solutions for the
flow through a simplified periodic fibre model for the deformed matrix. Here H is
the undeformed layer thickness and Kp the Darcy permeability. This simplified model
assumes that the fibres compress linearly with the deformed gap height in the vertical
direction, but the fibre spacing in the horizontal plane remains unchanged. The model
is thus able to capture the essential nonlinearity that results from large-amplitude
deformations of the matrix layer.The new theory shows that there is an unexpected striking similarity between the
gliding motion of a red cell moving over the endothelial glycocalyx that lines our
microvessels and a human skier or snowboarder skiing on compressed powder. In
both cases one observes an order-of-magnitude compression of the matrix layer when
the motion is arrested and predicts values of α that are of order 100. In this large-α
limit one finds that the pressure and lift forces generated within the compressed
matrix are four orders-of-magnitude greater than classical lubrication theory. In the
case of the red cell these repulsive forces may explain why red cells do not experience
constant adhesive molecular interactions with the endothelial plasmalemma, whereas
in the case of the skier or snowboarder the theory explains why a 70 kg human can
glide through compressed powder without sinking to the base as would occur if the
motion is arrested. The principal difference between the tightly fitting red cell and
the snowboarder is the lateral leakage of the excess pressure at the edges of the
snowboard which greatly diminishes the lift force. A simplified axisymmetric model is
presented for the red cell to explain the striking pop out phenomenon in which a red
cell that starts from rest will quickly lift off the surface and then glide near the edge
of the glycocalyx and also for the unexpectedly large apparent viscosity measured by
Pries et al. (1994) in vivo.
Publisher
Cambridge University Press (CUP)
Subject
Mechanical Engineering,Mechanics of Materials,Condensed Matter Physics
Cited by
132 articles.
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