The Navier-Stokes equations describe fluid flows in continuum mechanics.

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How to integrate twice of this viscous term?

I am reading a paper, and I do not understand why the author said the following term when integrated twice will become, $\int\limits_\Omega {{\rm{d}}\Omega {{\bf{\psi }}^{\bf{u}}}\cdot\nabla ...
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Convective and Diffusive terms in Navier Stokes Equations

My question has 2 parts: I just followed the derivation of Navier Stokes (for Control Volume CFD analysis) and was able to understand most parts. However, the book I use (by Versteeg) does not ...
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Reynolds number with hyper-viscosity

Is it possible to evaluate a Reynolds number when viscosity operator is substituted by hyper-viscosity operator at the power H (Laplacien to the power H) in the incompressible Navier-Stokes equations ...
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Occurrence of turbulences in Fluid Dynamics from the equations of motion?

How can it be shown that turbulences occur in Fluid Dynamics? I think poeple imply that they develope because of the $\text{rot}$ terms in the equations of motion, i.e. the Navier-Stokes equations, ...
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Free boundary conditions

I am trying to simulate liquid film evaporation with free boundary conditions (in cartesian coordinates) and my boundary conditions are thus: $$ \frac{\partial h}{\partial x} = 0, \qquad (1) $$ $$ ...
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Lagrangian for Euler Equations in general relativity

The stress energy tensor for relativistic dust $$ T_{\mu\nu} = \rho v_\mu v_\nu $$ follows from the action $$ S_M = -\int \rho c \sqrt{v_\mu v^\mu} \sqrt{ -g } d^4 x = -\int c \sqrt{p_\mu ...
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Would a solution to the Navier-Stokes Millennium Problem have any practical consequences?

I know the problem is especially of interest to mathematicians, but I was wondering if a solution to the problem would have any practical consequences. Upon request: this is the official problem ...
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Friction term in Navier-Stokes equation

The friction term in Navier-Stokes equation assumes that the viscosity coefficients are the same for the longitudinal and transverse directions. This doesn't seem intuitive, because the former is ...
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How to derive the Karman-Howarth-Monin relation for anisotropic turbulence?

I find the derivation of the Karman-Howarth-Monin relation in the book Turbulence from Frisch (1995) a bit to short. Can someone point me to a more detailed derivation of that relation, if possible in ...
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How to calculate the upper limit on the number of days weather can be forecast reliably?

To put it bluntly, weather is described by the Navier-Stokes equation, which in turn exhibits turbulence, so eventually predictions will become unreliable. I am interested in a derivation of the ...