diff options
Diffstat (limited to 'content/know/concept/material-derivative/index.pdc')
| -rw-r--r-- | content/know/concept/material-derivative/index.pdc | 8 |
1 files changed, 4 insertions, 4 deletions
diff --git a/content/know/concept/material-derivative/index.pdc b/content/know/concept/material-derivative/index.pdc index af65ca0..1c6bfdc 100644 --- a/content/know/concept/material-derivative/index.pdc +++ b/content/know/concept/material-derivative/index.pdc @@ -57,7 +57,7 @@ then we can rewrite this expression like so: $$\begin{aligned} \dv{t} f\big(x(t), y(t), z(t), t\big) - &= \pdv{f}{t} + \va{v} \cdot \nabla f + &= \pdv{f}{t} + (\va{v} \cdot \nabla) f \end{aligned}$$ Note that $\va{v} = \va{v}(\va{r}, t)$, @@ -73,7 +73,7 @@ and is known as the **material derivative** or **comoving derivative**: $$\begin{aligned} \boxed{ \frac{\mathrm{D}f}{\mathrm{D}t} - \equiv \pdv{f}{t} + \va{v} \cdot \nabla f + \equiv \pdv{f}{t} + (\va{v} \cdot \nabla) f } \end{aligned}$$ @@ -89,14 +89,14 @@ but in fact the definition also works for vector fields $\va{U}(\va{r}, t)$: $$\begin{aligned} \boxed{ \frac{\mathrm{D} \va{U}}{\mathrm{D}t} - \equiv \pdv{f}{t} + \va{v} \cdot \nabla \va{U} + \equiv \pdv{\va{U}}{t} + (\va{v} \cdot \nabla) \va{U} } \end{aligned}$$ Where the advective term is to be evaluated in the following way in Cartesian coordinates: $$\begin{aligned} - \va{v} \cdot \nabla \va{U} + (\va{v} \cdot \nabla) \va{U} = \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix} \cdot |