Skating on slippery ice

1. Detailed treatment based on hydrodynamic approach for clarifying the origin of slippery ice surface
2. Combining the theory of Le Berre and Pomeau [9], which only accounts for the e ects of melting and the theory of Lozowski and Szilder, which equals the pressure in the water layer to the hardness of ice to overcome the phenomena of slippery ice surface

1. The assumptions the author uses are bit far from the knowledge obtained from microscopic measurement and molecular dynamics simulation

As is written as the strength of the papers, I enjoy reading the manuscript, as it provides insight into the long-lived question - why the ice surface slippery - based on the hydrodynamics approaches. The analysis is consistent with the well-known phenomena, meaning that the theory presented here could be a solid hypothesis. However, I am not convinced whether this theory can explain the ice friction behavior beyond the skating condition. It seems very important to clarify in which conditions the current theory is valid and in which condition the assumption is broken down. We can control the weight and velocity as well as temperature in a much wider range than the ranges discussed here. It is important to comment on how this theory is applicable for such wider ranges of the physical condition.

For the microscopic measurement on the ice surface - in particular the liquid water on the top of the ice surface, I would suggest the author to read several literature.
- http://pubs.acs.org/doi/abs/10.1021/acs.chemrev.6b00045 (Fig. 16)
- http://aip.scitation.org/doi/abs/10.1063/1.2940195
These may be contradictory with the author's assumption, but it seems quite important to comment on the consistency/inconsistency of the author's assumption with these measurement/computational analysis.

1. In Figure 8, the titles of the axes should be given.

Unifies two earlier, contradicting skating theories.

Gives very reasonable results.

Very interesting topic.

The origin of the lubricating water film is the heat generated in the lubricating water film, so what if there is no lubricating water film?

Only very little comparison to experiment.

Bite Angle under which the skate contacts the ice surface is ignored.

Thermal properties of the skate are ignored.

Skating on ice is usually assumed to be possible due to a lubricating water film on the ice surface. This water layer however, has never been observed in situ. Van Leeuwen presents a theory of ice skating in which the friction force consists of two contributions: at low velocities the skate ploughs through the ice, leaving a track that results mainly from the plastic deformation of the ice surface. At higher speeds, the pressure within a water film that separates the skate from the ice builds up. The heat dissipated into this film during sliding then drives the melting of surface ice. The combined effect of ploughing through the ice and shearing the water film leads to a friction force that is only weakly dependent on velocity and roughly a factor 2 lower than that measured experimentally by de Koning et al. The theory unifies two existing ice skating theories that assume instant deformation of the ice surface when the pressure exceeds the hardness of the ice (Lozowski and Szilder), or no plastic deformation of the ice surface (Le Berre and Pomeau).

As such, the manuscript forms a significant improvement to existing ice skating theories. I therefore recommend the manuscript to be published if the following changes/questions are satisfactorily addressed by the author:

1. If the skate on ice contact is modelled as a Hertzian contact between a cylinder with radius 22 m, length 1.1 mm and a second cylinder with equal length and infinite radius, the contact pressure does not exceed 3 MPa which should be below the penetration hardness of the ice. Should the conclusion not be that the contact is elastic?

2. The angle between the skate and the ice (bite angle) is assumed to be 0. The bite angle will impact, for instance, the stationary length of contact, $l_0$, strongly. The experiments from de Koning et al suggest that the friction force is not too sensitive to the bite angle. Is this also expected for the present theory? I suppose that if the water layer can fill the gap between skate and ice caused by the bite angle the effect on friction will be limited?

3. Page 4 below eq. 4: perhaps 'in contact with the skate' instead of 'in touch with the skate'.

4. Page 6 below eq. 8 'mayor' should be 'major'

5. I do not understand equation 9: I would expect the speed with which the ice recedes, $v_{ice}$, to be the sum of the speed with which the skate recedes, $v_{sk}$, and the speed with which the water layer grows,$v_m$.

6. Where does the speed, $V$, in equation 10 come from?

7. Why is the thermal conductivity of the skate no parameter in the theory? I find it surprising that more heat flows into the ice than into the skate although the skate can in principle transport heat faster.

8. It may be good to mention that the viscosity of water is not very sensitive to the typical ice skating pressures.

9. page 8 above eq. 21: 'Eq. (21) dominates' should be 'Eq. (20) dominates'

10. For the calculation of the friction force that results from shearing the water film; does the -l<x<0 part of the skate not also shear a water film?

11. If the heat source for melting ice is sheared water, where does that water come from (it is not present at V=0)?

12. The data from reference 13 (FIG 3) should allow for an experimental estimate of $\gamma$ and show whether or not the Bingham model is appropriate for ice. It may also be interesting to compare the penetration hardness from ref 13 to that reported in the paper.

13. Page 16 bottom 'treat it hydrodynamic' should be 'treat it as a hydrodynamic system'

14. Bottom page 17 'not very emcouraging' should be 'not very encouraging'

15. Appendix A under eq. 58 it says 'skate blade z = h(x)' this should be d(x)

16. Appendix C refers to expressions 66-68 above equation 82. Should this be 7?