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Skating on slippery ice
by J. M. J. van Leeuwen
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Submission summary
Authors (as registered SciPost users):  J.M.J. van Leeuwen 
Submission information  

Preprint Link:  http://arxiv.org/abs/1706.08278v3 (pdf) 
Date accepted:  20171212 
Date submitted:  20170922 02:00 
Submitted by:  van Leeuwen, J.M.J. 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The friction of a stationary moving skate on smooth ice is investigated, in particular in relation to the formation of a thin layer of water between skate and ice. It is found that the combination of ploughing and sliding gives a friction force that is rather insensitive for parameters such as velocity and temperature. The weak dependence originates from the pressure adjustment inside the water layer. For instance, high velocities, which would give rise to high friction, also lead to large pressures, which, in turn, decrease the contact zone and so lower the friction. The theory is a combination and completion of two existing but conflicting theories on the formation of the water layer.
Author comments upon resubmission
The constructive remarks of the referees are highly appreciated. With respect to the general points raised by the referees, I would like to reply the following.
 Comparison with experiments. Indeed there is regrettably little comparison with experiments. This is mainly due to the fact that few relevant experiments exist. The investigation started as an attempt to explain direct measurements of the thickness of the water layer. However the laboratory experiments could not reliably detect water layers. This conforms the theoretical calculations leading to water layers of the order of fractions of a mu m thick. Such water layers are, so far, too thin to see in the variation of the dielectric constant.
What if the water layer is absent? Friction generates heat, no matter the origin of the friction. Whether the heat leads to melting depends on the temperature of the ice. We find in Section 13 that at low temperatures and slow velocities, the heat available for melting becomes so small, that the water layer thickness becomes subhydrodynamical. Under these circumstances our calculation is not applicable. For velocities relevant for skating the water layer can be treated as a hydrodynamical system and the friction is then given by the friction in the water layer and independent of the surface properties of ice and skate.
If the formation of the water layer plays no role, the friction is determined by the properties of the surface of ice. This is demonstrated in measurements of B. Weber et al. (ref. 5) where a friction is found orders of magnitude higher than for skating conditions.
Role of the bite angle. The calculation is restricted to a horizontal skate, while a finite bite angle is indeed important for real skating, since it can not be avoided. It is likely that a finite bite angle can be discussed in the same context as the present paper (see the work of Le Berre and Pomeau, ref.~9), but it would have made the already toolong paper even longer, without adding to the essential mechanism of skating.
Thermal properties of the skates. The thermal properties of the skate are not ignored. In the appendix D, the heat flow inside the water layer is discussed, which is determined by the temperatures of ice surface and the skate. Since the skate will not be colder than the ice, the fraction of heat flowing towards the skate will not be larger than 50% of the produced heat. This is independent of the thermal properties of the skate.
Validity of the assumptions. I could not find any direct experimental arguments pro or contra the basic assumption Eq.~(6). Thus the proof or disproof of (6) has to come from measurements. It requires firstly a precision measurement of the hardness under quasistatic intrusion in order to determine the excess pressure. Next it requires intrusion rates under controlled speed and pressure. To the best of my knowledge these experiments, comparable to the intrusion rate of skates, are not available.
I have made in the introduction of the paper a number of modifications to illuminate the above mentioned points. A number of improvements suggested by the first referee are taken over without change i.e. the remarks 3,4,8,9,13,14,15 and 16. I am grateful for the scrutiny of this referee.
The other points give rise the following commentary.
Point 1. The calculation of the Hertzian contact between the skate, seen as a cylinder of curvature 22 m and length 1.1 mm, and an equallength icecylinder of infinite radius, is not relevant for the question whether the deformation of a skate is elastic or plastic. For that question the deformation sideways from the skate is most important. In particular, assuming that the skate has sharp edges, the deformation will always be plastic, since a sharp edge yields a kink in the ice surface, leading to a diverging counter pressure. In practise the edges are rounded off and that makes the question elastic vs plastic illposed. The counter pressure is maximal at the edges and therefore the degree of rounding off determines the limit of elastic deformations
Point 5. The rate of change of the through made by the skate is due to melting and ploughing. So it is the sum of the melting velocity v_m and the downward ploughing (or receding velocity) v_{ice}. I have made this more explicit in the text.
Point 6. The velocity comes in as the ratio of the derivative with respect to position and to time.
Point 10. What happens with the water layer, after the lowest point of the skate has passed, is irrelevant for the friction with the skate, since there is no more contact with the skate. I have stated this explicitly in the revised text.
Point 11. The water layer originates in melting. For V=0 no water is generated and only ploughing remains in the theory. The ploughing force is given by the hardness, provided the hardness is measured quasistatically. This would indeed yield a method to measure the hardness of ice.
Fig. 8. In the program that generates Fig. 8, it is hard to give the axis appropriate labels. The role of the axes is clarified in the caption.
Hopefully the changes made according to this reply have improved the paper.
Published as SciPost Phys. 3, 042 (2017)