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Jun
7
comment Incompressible Navier-Stokes pressure solve in simulations
Your second equation is right (I expanded my answer).
Jun
7
revised Incompressible Navier-Stokes pressure solve in simulations
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Jun
6
comment Incompressible Navier-Stokes pressure solve in simulations
Not sure what you mean by "what to do". Obviously, hundreds of text books have been written about analytical solutions to these equations, and about methods for solving them numerically. Many codes can be downloaded, optimized for all sorts of conditions.
Jun
6
comment Incompressible Navier-Stokes pressure solve in simulations
Sorry, I fixed the index structure. This is just the divergence of the NS equation, $\partial_t u_i+u_j\nabla_ju_i=-1/\rho\nabla_iP +\nu\nabla^2u_i$, using the fact that $\nabla_i u_i=0$.
Jun
6
revised Incompressible Navier-Stokes pressure solve in simulations
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Jun
5
answered Incompressible Navier-Stokes pressure solve in simulations
Jun
4
answered Feynman-Bethe Critical Mass Formula
May
31
comment In a fluid, why are the shear stresses $\tau_{xy}$ and $\tau_{yx}$ equal?
I tried to be a little more explicit.
May
31
revised In a fluid, why are the shear stresses $\tau_{xy}$ and $\tau_{yx}$ equal?
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May
31
answered In a fluid, why are the shear stresses $\tau_{xy}$ and $\tau_{yx}$ equal?
May
30
answered Quark-gluon plasma: status
May
30
comment Gauge potential for locomotion at low Reynolds number
The paper by Shapere and Wilczek has a number of worked examples. Do those not answer your question?
May
29
answered How to compute scattering amplitude $\gamma\pi^+\to\pi^+\pi^0$
May
28
awarded  Supporter
May
28
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May
28
comment Seeking Reference on Transport of Momentum by Diffusion of Mass
I have not looked at the book, but note that in the energy/entropy the leading term is of second order in diffusive fluxes (this follows from the second law of thermodynamics), so corrections beyond Navier-Stokes start at third order.
May
28
answered Seeking Reference on Transport of Momentum by Diffusion of Mass
May
27
comment Perfect fluid and Cauchy momentum equation
1) The perfect fluid is a systematic approximation (gradient corrections can be made arbitrarily small by considering smooth flows, and can be accounted for order by order in the gradient expansion). 2) $\epsilon=0$ is not a systematic approximation. 3) The only thing I can imagine is that $\epsilon=0$ corresponds to neglecting internal energy compared to rest mass energy density, which is reasonable for a non-relativistic fluid. There is still no godd reason for dropping $\rho\epsilon$, but keeping $P$.
May
27
revised Perfect fluid and Cauchy momentum equation
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May
27
comment Perfect fluid and Cauchy momentum equation
I think this is not just terminology. $T_{\mu\nu}=(\rho+P)u_\mu u_\nu+Pg_{\mu\nu}$ and $j_\mu=\rho u_\mu$ cannot both be correct. You can consider $\epsilon=0$ (in your notation) but i) there is no such fluid known to man (finite pressure but zero internal energy), ii) the original question is pointless, because we are in the non-relativistic limit from the start.