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The Navier-Stokes equation can be defined as follows (taken from wikipedia):

$$\rho\frac{D\mathbf{u}}{Dt} = - \nabla p + \nabla \cdot \boldsymbol \tau + \rho\,\mathbf{g}$$ , where $$\frac{D}{Dt} = \frac{\partial}{\partial t} + \mathbf{u}\cdot\nabla$$

If we expand the $\frac{D}{Dt}$ in the left-hand side we get: $$ \rho \frac{D\mathbf{u}}{Dt} = \rho \left(\frac{\partial\mathbf{u}}{\partial t} + \mathbf{u}\cdot\left(\mathbf{u} \cdot \nabla \right) \right) $$

But the continuity equation tells us that $ \mathbf{u}\cdot\nabla = 0 $, so what the meaning of this component? Why is it there if we know it should be 0?

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  • $\begingroup$ Your 2nd term is written incorrectly. It should read $\mathbf{u}\centerdot \nabla \mathbf{u}$ $\endgroup$ – Chet Miller Mar 6 '20 at 15:29
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I'm not sure what second equation you are referring to exactly, but I suspect you are talking about the conservation of mass equation. The conservation of mass equation for an incompressible flow reads:

$$ \nabla \cdot \vec{u} = \frac{\partial u_i}{\partial x_i} = 0 $$

This is not the same thing as saying $\vec{u} \cdot \nabla = u_i \frac{\partial}{\partial x_i} = 0$ however! Because $\nabla$ is really an operator, you can't transpose the dot product.

Additionally, your expansion of $D\vec{u}/Dt$ is incorrect. Remember it is the operator $D/Dt$ applied to $\vec{u}$, not a multiplication. It should read:

$$ \frac{D\vec{u}}{D t} = \left( \frac{\partial}{\partial t} + \vec{u} \cdot \nabla \right) \vec{u} = \frac{\partial \vec{u}}{\partial t} + \vec{u} \cdot \nabla \vec{u} = \frac{\partial u_i}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} $$

and $\nabla \vec{u} = \frac{\partial u_i}{\partial x_j} \neq 0$ in general.

I've written them out both in vector form and in Einstein summation form because I think the summation form makes it more clear why the dot product of $\vec{u}$ and $\nabla$ don't transpose.

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  • $\begingroup$ Nice, I always thought divergence is just regular dot product, which is commutative. Thank you, it's clear now $\endgroup$ – Maciej Dziuban Mar 6 '20 at 15:45

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