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Skating on slippery ice

by J. M. J. van Leeuwen

Submission summary

As Contributors: J.M.J. van Leeuwen
Arxiv Link: (pdf)
Date accepted: 2017-12-12
Date submitted: 2017-09-22
Submitted by: van Leeuwen, J.M.J.
Submitted to: SciPost Physics
Discipline: Physics
Subject area: Fluid Dynamics
Approach: Theoretical


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.

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Friction Ice Skating

Published as SciPost Phys. 3, 042 (2017)

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 sub-hydrodynamical. 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 too-long
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 quasi-static 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 equal-length ice-cylinder
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 ill-posed. 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
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.

Reports on this Submission

Anonymous Report 1 on 2017-10-12 Invited Report

  • Cite as: Anonymous, Report on arXiv:1706.08278v3, delivered 2017-10-12, doi: 10.21468/SciPost.Report.260


I thank the author for clarifying the points that were raised and recommend the paper to be published.

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