An application of contact melting theory to skates sliding on ice

The problem of a skate blade sliding over ice is a complex and classic problem, with an early form considered by Reynolds over a century ago. The problem is revisited herein: a thin layer of water in between the skate blade and the ice surface is assumed to exist, and acts as a lubricant for the sliding motion of the skate blade. The existence of the melt layer is caused by viscous friction in the liquid film itself, instead of pressure melting. Governing equations are considered for a Newtonian and inviscid fluid of constant density. These equations are reduced by considering some scaling analysis to determine the negligible terms, and a simpler planar flow is considered. Through some straightforward manipulations of the governing equations, the viscous stress on the surface of the skate blade is analytically expressed as a function of the depth of the melt layer. Other results are used to posit an approximate expression for the non-constant depth of the melt layer, and this is used to calculate the frictional force. The results are compared to others in the area, and limitations on the modelling are discussed.