0
$\begingroup$

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.

$\endgroup$

2 Answers 2

1
$\begingroup$

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, $$\partial_{ii}p=f\left(\cdots\right) $$

$\endgroup$
1
  • $\begingroup$ That makes a lot of sense now, thanks! $\endgroup$
    – CBBAM
    Jul 29 at 0:19
-2
$\begingroup$

You can only repeat an index twice: that means sum over it. Your equation with three js is meaningless. You need to take a curl in order to eliminate the grad p term, not a div. That will involve the alternating tensor which you may not yet know about. You are not restricted to i and j, you can use any suffix you like for summed indices but the single ones must match in every term.

$\endgroup$
4
  • $\begingroup$ 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? $\endgroup$
    – CBBAM
    Jul 28 at 20:59
  • $\begingroup$ I used three js to indicate a third order derivative, is there an alternate way to denote this in Einstein notation? Also, how come there is no subscript on the pressure term? $\endgroup$
    – CBBAM
    Jul 28 at 21:00
  • $\begingroup$ You still need to indicate which x you are differentiating with respect to. So jjk would be twice wrt xj and once wrt xk $\endgroup$
    – CWPP
    Jul 28 at 21:05
  • $\begingroup$ Yes that is why I chose three js, i.e. $\partial^3/\partial x_{j}^3$ $\endgroup$
    – CBBAM
    Jul 28 at 21:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.