# How do the navier stoke equations model materials who "forget" their original form?

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:

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?

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.