Why is velocity divergence contained in the Navier-Stokes equation even if it's 0?

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?

• Your 2nd term is written incorrectly. It should read $\mathbf{u}\centerdot \nabla \mathbf{u}$ Mar 6, 2020 at 15:29

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.

• Nice, I always thought divergence is just regular dot product, which is commutative. Thank you, it's clear now Mar 6, 2020 at 15:45