Lubrication theory in highly compressible porous media: the mechanics of skiing, from red cells to humans

A generalized lubrication theory that is applicable to highly deformable po rous layers is developed using an effective-medium approach (Brinkman equat ion). This theory is valid in the limit where the structure is so compressi ble that the normal forces generated by elastic compression of the fibres c omprising the solid phase are negligible compared to the pressure forces ge nerated within the porous layer. We assume that the deformation of the soli d phase is primarily due to boundary compression as opposed to the motion o f the fluid phase. A generalized Reynolds equation is derived in which the spatial variation of the Darcy permeability parameter, alpha = H/rootK(p), due to the matrix compression is determined by new local hydrodynamic solut ions for the flow through a simplified periodic fibre model for the deforme d matrix. Here H is the undeformed layer thickness and K-p the Darcy permea bility. This simplified model assumes that the fibres compress linearly wit h the deformed gap height in the vertical direction, but the fibre spacing in the horizontal plane remains unchanged. The model is thus able to captur e the essential nonlinearity that results from large-amplitude deformations of the matrix layer. The new theory shows that there is an unexpected striking similarity betwee n the gliding motion of a red cell moving over the endothelial glycocalyx t hat lines our microvessels and a human skier or snowboarder skiing on compr essed powder. In both cases one observes an order-of-magnitude compression of the matrix layer when the motion is arrested and predicts values of alph a that are of order 100. In this large-alpha limit one finds that the press ure 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 c ell to explain the striking pop out phenomenon in which a red cell that sta rts from rest will quickly lift off the surface and then glide near the edg e of the glycocalyx and also for the unexpectedly large apparent viscosity measured by Pries et al. (1994) in vivo.