diff options
Diffstat (limited to 'content/know/concept/reynolds-number')
| -rw-r--r-- | content/know/concept/reynolds-number/index.pdc | 11 |
1 files changed, 9 insertions, 2 deletions
diff --git a/content/know/concept/reynolds-number/index.pdc b/content/know/concept/reynolds-number/index.pdc index bd18f2f..ff5e793 100644 --- a/content/know/concept/reynolds-number/index.pdc +++ b/content/know/concept/reynolds-number/index.pdc @@ -25,6 +25,12 @@ $$\begin{aligned} = - \frac{\nabla p}{\rho} + \nu \nabla^2 \va{v} \end{aligned}$$ +In this case, the gravity term $\va{g}$ +has been absorbed into the pressure term: +$p \to p\!+\!\rho \Phi$, +where $\Phi$ is the gravitational scalar potential, +i.e. $\va{g} = - \nabla \Phi$. + Let us introduce the dimensionless variables $\va{v}'$, $\va{r}'$, $t'$ and $p'$, where $U$ and $L$ are respectively a characteristic velocity and length of the system at hand: @@ -108,7 +114,7 @@ such that redimensionalizing yields: $$\begin{aligned} \pdv{\va{v}}{t} + (\va{v} \cdot \nabla) \va{v} - = - \nabla p + = - \frac{\nabla p}{\rho} \end{aligned}$$ Which is simply the main [Euler equation](/know/concept/euler-equations/) @@ -133,7 +139,8 @@ $$\begin{aligned} This equation is called the **unsteady Stokes equation**. Usually, however, such flows are assumed to be steady -(i.e. time-invariant), leading to the **steady Stokes equation**: +(i.e. time-invariant), leading to the **steady Stokes equation**, +with $\eta = \rho \nu$: $$\begin{aligned} \boxed{ |