Different form of the Navier--Stokes equations

Question

Normally I write the incompressible Newtonian isothermal flow Navier--Stokes equations as follows:

$$\displaystyle \frac{\partial v}{\partial t} -\nu\Delta v +\color{red}{(\nabla v)v} +\frac{1}{\rho}\nabla p = f_b $$

But I'm reading now the followwing

$$\displaystyle \frac{\partial v}{\partial t} -\nu\Delta v +\color{red}{\nabla\cdot( vv)} +\frac{1}{\rho}\nabla p = f_b $$

Are both equations the same? I can't prove that are the same, that is why I'm thinking that are different equations (and is not different notation only).

Answer 1

The operation in the second equation is the divergence of the dyadic product $vv$.The dyadic product of two vectors is a square matrix. In this case, a $3\times3$ matrix where $(vv)_{ij}=v_iv_j$.

The divergence of this product is given by $$\nabla\cdot(vv)=(\nabla\cdot v)v+v\cdot\nabla v$$ You can find the details of the vector identity here. The first term vanish by virtue of incompressibility constraint, $\nabla\cdot{v}=0$.

Answer 2

Tensorial notations are helpful here as an alternative--and an excellent way to prove-- vector calculus identities mentionned in other answers (if you can't remember them of the top of your head). The Navier-Stokes equations for the i-th component of $v$ are:

$$ \partial_t v_i + \sum_{1\le j \le 3}v_j \partial_j v_i = -\partial_i p + \sum_{1\le j \le 3}\partial_j\partial_j v_i$$

where I use the shorthand notations $\partial_t=\frac{\partial}{\partial t}$ and $\partial_i = \frac{\partial}{\partial x_i}$.

Incompressibility is: $$ \sum_{1\le j \le 3}\partial_j v_j = 0$$

Now let us use this notation to explicit the i-th component of the term in red: $$ \left\{\boldsymbol{\nabla \cdot }( \boldsymbol{vv}) \right\}_i =\boldsymbol{\nabla \cdot }(v_i \boldsymbol{v})= \sum_{1\le j \le 3} \partial_j\left( v_i v_j\right) $$

We can distribute the derivative, and use imcompressibility: $$ \left\{\boldsymbol{\nabla \cdot }( \boldsymbol{vv}) \right\}_i =v_i \underbrace{\sum_{1\le j \le 3} \partial_j v_j}_{=0} + \sum_{1\le j \le 3}v_j \partial_j v_i = \boldsymbol{v \cdot\nabla} v_i$$

This is how, for incompressible flows, advection can be written in these two different (but equivalent) forms. Written as a divergence, energy budgets over control volumes are then easily derived.