TransWikia.com

Navier-Stokes Convective term divergence of tensor is a row vector

Physics Asked on April 1, 2021

Consider the convective term from the Navier-Sokes equation (e.g. Navier–Stokes momentum equation (conservation form) from https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations):
$$
nabla cdot (rho vec u otimes vec u ).
$$

Writing the outer product for three dimensional Eucledian space:
$$
vec u otimes vec u = rho
begin{pmatrix}
uu & vu & wu
uv & vv & wv
uw & vw & ww
end{pmatrix}
$$

we will arrive at
$$
nabla cdot (rho vec u otimes vec u ) =
begin{pmatrix}
dfrac{partial}{partial x}
dfrac{partial}{partial y}
dfrac{partial}{partial z}
end{pmatrix}
cdot
begin{pmatrix}
rho uu & rho vu & rho wu
rho uv & rho vv & rho wv
rho uw & rho vw & rho ww
end{pmatrix}
=
begin{pmatrix}
dfrac{partial}{partial x} &dfrac{partial}{partial y} &dfrac{partial}{partial z}
end{pmatrix}
begin{pmatrix}
rho uu & rho vu & rho wu
rho uv & rho vv & rho wv
rho uw & rho vw & rho ww
end{pmatrix}
=
begin{pmatrix}
dfrac{partial rho uu}{partial x}
+ dfrac{partial rho uv}{partial y}
+ dfrac{partial rho uw}{partial z},
& dfrac{partial rho vu}{partial x}
+ dfrac{partial rho vv}{partial y}
+ dfrac{partial rho vw}{partial z},
& dfrac{partial rho wu}{partial x}
+ dfrac{partial rho wv}{partial y}
+ dfrac{partial rho ww}{partial z}
end{pmatrix}
$$

My confusion is that the result is a row vector (1 x 3) not a column one (3 x 1) which is the case for all other terms in the equation. Shouldn’t the convective term be
$$
left( nabla cdot (rho vec u otimes vec u ) right)^T ~?
$$

If we consider the NS equations in convective form (see Navier–Stokes momentum equation (convective form) from the link above) the convective term:
$$
vec u cdot nabla vec u
$$

is in my opinion ambiguous. We can evaluate it like this $vec u cdot (nabla vec u)$ or like this $(vec u cdot nabla) vec u$. The former delivers a row vector and the latter a column vector. Some authors use explicitly the second form (see Incompressible Navier–Stokes equations (convective form) on the Wiki page)

Why is this term incosistent (row vector) with other terms of NS equation? Doesn’t it matter and I should ignore this fact? Do I miss something?

I found several questions about the convective term (here or here) but none of these addresses my issue.

One Answer

It should be written as $rho ({bf v}cdot nabla){bf v}$, or in index notation as $rho v_i partial_i v_j$.

It's best to avoid the "row vector" and "column vector" language when working with tensor products. Indeed, in my opinion, the use of the tensor product notation in the Wikipedia is unnecessarily confusing for the likely reader of the article. I would write the momentum conservation equation as $$ partial_t(rho v_i) +partial_j (rho v_i v_j +delta_{ij}p)=hbox{viscosity term}, $$
which, when combined with mass conservation $$ partial_t rho+ partial_i (rho v_i)=0, $$ leads to the Euler form of the NS equation $$ rho( partial_t v_i +v_j partial_j v_i)= - partial_ip + hbox{viscosity term}. $$

Answered by mike stone on April 1, 2021

Add your own answers!

Ask a Question

Get help from others!

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP