# Navier-Stokes Equations in Einstein Notation and its relation to Poisson's Equation

The Navier-Stokes equations are: $$\partial_t v + v \cdot \nabla v = - \nabla p + \nu\nabla^2 v, \quad v \in \mathbb{R}^3\\ \nabla \cdot v = 0$$

I have seen that, using Einstein notation (which I am new to), the above can be written as $$\partial_t v_i + v_j\partial_j v_i = - \partial_i p + \nu \partial_{jj}v_i, \quad v \in \mathbb{R}^3\\ \partial_iv_i = 0.$$

Note that here I am using the notation that $$\partial_j = \partial/\partial x_j$$.

Using this notation, I would like to take the divergence of the N-S equations to eliminate the pressure term. I am not sure if this is correct, but my work is as follows: $$\partial_j (\partial_t v_i + v_j\partial_j v_i + \partial_i p - \nu \partial_{jj}v_i) \\= \partial_j\partial_tv_i + \partial_jv_j\cdot\partial_jv_i + v_j\cdot \partial_{jj}v_i + \partial_{ji} p - \nu\partial_{jjj}v_i.$$ What is confusing me is when to use an index of $$i$$ and when to use an index of $$j$$, is there some general rule for this? Also, why is there subscript on $$p$$?

Once I figure this out I assume the rest is just a trivial application of the divergence-free condition and from there we recover Poisson's equation.

Note that I have already read the following posts: Index notation with Navier-Stokes equations and Questions about Navier-Stokes equations, Einstein notation, tensor rank but unfortunately to no avail.

You are taking the inner product of $$\nabla$$ and $$\mathbf v$$, so you need to make sure they both have the same index: $$\partial_i\left[\partial_tv_i+v_j\partial_jv_i\right]=\partial_i\left[-\partial_ip+\nu\partial_{jj}v_i\right]$$ Your first term should drop to zero due to the divergence condition (after interchanging partial derivatives), then you can work on sums for the remaining terms to more clearly see the Poisson equation, $$\partial_{ii}p=f\left(\cdots\right)$$
• In Frisch's text on turbulence he claims that with periodic boundary conditions, we can eliminate the pressure by taking the divergence and using the divergence-free condition of $v$, resulting in a Poisson equation. By solving this Poisson equation he claims we have eliminated the pressure term. Regardless though, for pedagogical purposes, how would the divergence of the equation look like in Einstein notation? Jul 28, 2022 at 20:59
• Yes that is why I chose three js, i.e. $\partial^3/\partial x_{j}^3$ Jul 28, 2022 at 21:07