A computational fluid dynamics example solves the equations below. I'm trying to learn the software and a bit of fluid dynamics concurrently. I believe that this is an integral form of the Navier-Stokes equation, conserving quantities within fixed volume elements.
My skills understanding notation are "underdeveloped".
- In the first integral, what is the meaning of the colon?
- after the comma, what is the "upside-down A"-like symbol? Do I read that as for all $\mathbf{v}$?
From SfePy example navier_stokes/navier_stokes2d.py:
Making my own comments an answer.
Concerning the first question: as I already mentioned the notation $\textbf A:\textbf B$ most likely refers to the Double Dot Product. What exactly happens here (more precise what is multplied here) is, at least to me, not entirely clear since I am not that experienced within this field of mathematics and especially I have never encountered the Double Dot Product, as representative of products of dyadics, before. However, within the Wikipedia article of the Navier Stokes equations, subsection Compressible Flow (and it's used within Incompressible Flow too) it is also used as Double Dot Product from were I can conclude that is then which it refers to here. Additionally, the arcticle you linked calls it "double dot".
Concerning the second question: this one is one of the two commonly used quantifiers. On the one hand we got the exitential quantifier $\exists$ (
\existsin Tex, read as "there exists...") and one the other hand the universal quantifier $\forall$ (\forallin Tex, read as "for all..."). Here we read $\forall\underline v$ as "for all vectors $v$" and similiar $\forall q$ as "for all scalars $q$" (the information of vectors and scalars is not included within the quantifiers but within the remaining notation).