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This question is asked from the viewpoint of continuum mechanics, its integral laws, and jump conditions.

Consider an object with a flat bottom, say a cubic block of concrete, moving with friction on a horizontal floor. We use a coordinate system where the floor is at rest. Let's say the constant, horizontal velocity of the block is $\pmb{v}$. There is friction, that is, a shear stress $\pmb{\tau}$ between the block and the floor, representing a flow of momentum between the two.

Take the imaginary horizontal control surface that separates the block and the floor. The surface normal is $\pmb{n}$.

Now, in situations where the velocities of the bodies or of the surface are orthogonal to the surface normal, and there are no discontinuities in velocity – so not our present case –, the energy flux through the surface can be written as

$$Q + \pmb{n}\cdot \pmb{\tau}\cdot \pmb{u}$$

where $Q$ is the heat flux and $\pmb{u}$ is the velocity of the body; the second term in the sum is simply the mechanical power transmitted through the surface. This formula applies for example to an imaginary surface along the (laminar) flow of some fluid, and gives the exchange of energy between the fluid above and below the surface owing to viscosity and to possible temperature gradients.

It seems that the formula above cannot be used in the case of the block moving on the floor, because of the discontinuity of the velocity – the block is moving, the floor is not.

So there surely are some jump conditions involved here. Can anyone give insights and literature references about them?

Cheers!

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The flux of mechanical energy, $\pmb{n}\cdot \pmb{\tau}\cdot \pmb{u}$, is still valid at any surface where all of the variables are well defined. So as you approach the discontinuity from one side you have a well defined flux of mechanical energy $\pmb{n}\cdot \pmb{\tau}\cdot \pmb{u_+}$, and as you approach the discontinuity from the other side you also have a well defined flux of mechanical energy $\pmb{n}\cdot \pmb{\tau}\cdot \pmb{u_-}$.

At the discontinuity the flux is not well defined, precisely because the velocity is discontinuous. But invoking the conservation of energy we can state that at the discontinuity energy is converted from mechanical to some other form at a rate of $\pmb{n}\cdot \pmb{\tau}\cdot (\pmb{u_+}-\pmb{u_-})$. Usually this would be heat generated at the discontinuity, but in principle it could be some other form of energy such as acoustic or electromagnetic.

In principle, it is important to distinguish between heat flux and the heat generated at the discontinuity. The velocity discontinuity would typically serve as the boundary of a thermodynamic analysis, with a heat source at the discontinuity driving heat flow through the continuum of both materials according to the standard thermodynamic laws.

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  • $\begingroup$ Fantastic, Thank you! So if I understand correctly the jump condition for the whole flux is $[[Q + \pmb{n}\cdot\pmb{\tau}\cdot\pmb{u}]] = 0$, but there are no separate conditions for the heating and power terms, and in this case the energy flux on one side could be completely in the form of mechanical power, and on the other in the form of heat? This is extremely interesting because if we choose a different coordinate system, in which the block is at rest and the floor moves, then the same situation would have an opposite interpretation. $\endgroup$
    – pglpm
    Commented Apr 7 at 14:02
  • $\begingroup$ I see how this coordinate dependence could lead to confusion regarding entropy flux, and your final remarks are referring to this if I understand correctly. Could you kindly share any literature that discusses these matters in depth? Cheers! $\endgroup$
    – pglpm
    Commented Apr 7 at 14:04
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    $\begingroup$ @pglm unfortunately I don’t have any literature on this topic. It is stuff I have had to cobble together when I was looking at this. But you can verify these principles with simple systems like blocks on a floor or levers. I suspect a good mechanical engineering text may have what you want, but I don’t own any to recommend. $\endgroup$
    – Dale
    Commented Apr 7 at 15:12
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    $\begingroup$ @pglpm regarding what happens in other reference frames, interestingly you are correct that the mechanical power can reverse direction and change amount. However, interestingly the amount of energy converted from mechanical to other forms is invariant. All frames agree on the power converted at the discontinuity regardless of the differing mechanical power flow $\endgroup$
    – Dale
    Commented Apr 7 at 15:15

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