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Sorry for the screenshot but I don't want to try to format this on latex:

We have this annotation of the Navier-Stokes equations: enter image description here

I am particularly puzzled by the viscosity/stress term. For an elastic material, say jelly, that term makes sense to me. When you apply light deformations to the jelly (don't rupture it) it will want to return back to its original form, so that stress term is telling you how locally "unhappy" a deformed point is.

But with something like water, stress forces are "forgotten" rather quickly, water does experience some "unhappiness" under local shearing at extremely small time scales, but clearly, once water has reconfigured itself it doesn't care what its original form was anymore. For example, if I grab an elastic material like foam and twist it in zero gravity it will go back to how it was prior to deformation under its internal forces. But if I do that to a bunch of water, the water will just remain under its new twisted configuration indefinitely.

But water has some viscosity, it is not 0, so if the viscosity term is computed from the original configuration, any simulation will eventually converge back to the original fluid form. This suggests to me that things like water must "forget" and "update" their deformation matrix over multiple time steps. Like the stress forces at time $t_i$ must be computed from those at time $t_{i-1}$ not from those at time $t_{0}$.

But then this suggests that any numerical approximation that discretizes in time will grow inaccurately as the time steps grow larger and larger since the distance between the time steps grows.

So how exactly do you discretize your computation of the deformation matrix properly? Do you use some kind of extrapolation based on your prior time steps?

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The key difference between fluids and solids is that in a fluid, the stress is $$ \nu \nabla \mathbf v, $$ (or some variation involving the symmetric gradient instead of the gradient) which involves the velocity. On the other hand, for solids, the stress is $$ C \nabla \mathbf u, $$ with the stress-strain tensor $C$ and involving the displacement instead of the velocity.

As a consequence, once you "deform" a fluid and let it come to rest again (velocity is zero), there is no stress any more in the same way as when you let a solid return to its original configuration (displacement is zero) there is no stress any more. But for the fluid, what matters is that the velocity is zero, no matter by how much you have deformed the original fluid from its previous state.

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    $\begingroup$ This made it click, thank you $\endgroup$
    – Makogan
    Apr 11, 2023 at 3:15
  • $\begingroup$ One could add here some words about amorphous materials (glasses, metallic glasses etc.) $\endgroup$ Apr 11, 2023 at 4:04
  • $\begingroup$ @MaximUmansky One could, but maybe we don't have to go the viscoelastic route today :-) $\endgroup$ Apr 11, 2023 at 16:04

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