Help with Navier-Stokes Equation: Symbols & Meaning

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The Navier-Stokes equations describe fluid motion, incorporating various forces acting on a fluid. The equation presented includes terms for the total or convective derivative, which accounts for changes in velocity over time and space. The term "DP" likely refers to the pressure gradient, while "Dt" represents the divergence of the stress tensor, which includes pressure and shear stress components. The final "f" indicates external body forces like gravity. Overall, the discussion emphasizes the equation's role in expressing conservation of momentum for continuous materials.
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I was wondering if someone could help me this Navier-Stokes Equation.

f[(δv/δt) + v.Dv] = -DP + Dt + f

Could someone maybe explain the symbols and what it means.
I'm not sure but I think Navier-Stokes equations describe fluid motion.

(The P could be ρ. I'm not too sure)

Thanks
 
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You are correct that the equations describe fluid motion. I have a beautiful proof that solutions always exist in three dimensions, but unfortunately it is too big for this marginal comment.
 
iasc said:
I was wondering if someone could help me this Navier-Stokes Equation.

f[(δv/δt) + v.Dv] = -DP + Dt + f

Could someone maybe explain the symbols and what it means.
I'm not sure but I think Navier-Stokes equations describe fluid motion.

(The P could be ρ. I'm not too sure)

Thanks

I can't completely parse what you wrote, but some of it I can decipher:

The term in [] (not sure what that 'f' is doing there), is the "total" or "convective" derivative. It simply means that a spatial quantity (the velocity field vector 'v') is allowed to vary both in time and in space. I am assuming 'D' is a nabla (del) operator.

The term DP could be the gradient of pressure term, but the Dt term is normally the divergence of the stress tensor, so those two terms are a little ambiguous. The stress tensor consists of both an isotropic part (the pressure) and the off-diagonal antisymmetric components (the shear stress). The final 'f' is used if there is an external body force: gravity, centripetal forces, electromagnetic, etc. etc.

The Navier-Stokes equation you wrote is nothing more than ma=F for a continuous material. If ma(or []) = 0, then you have conservation of momentum.
 
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