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Diffstat (limited to 'source/know/concept/material-derivative')
| -rw-r--r-- | source/know/concept/material-derivative/index.md | 10 |
1 files changed, 3 insertions, 7 deletions
diff --git a/source/know/concept/material-derivative/index.md b/source/know/concept/material-derivative/index.md index 6bb83c5..4eb43e9 100644 --- a/source/know/concept/material-derivative/index.md +++ b/source/know/concept/material-derivative/index.md @@ -36,7 +36,7 @@ $$\begin{aligned} In effect, we have simply made the coordinate $$\va{r}$$ dependent on time, and have specifically chosen the time-dependence to track the parcel. -The net evolution of $$f$$ is then its "true" (i.e. non-partial) derivative with respect to $$t$$, +The evolution of $$f$$ is then its derivative with respect to $$t$$, allowing us to apply the chain rule: $$\begin{aligned} @@ -58,11 +58,7 @@ $$\begin{aligned} Note that $$\va{v} = \va{v}(\va{r}, t)$$, that is, the velocity can change with time ($$t$$-dependence), and depends on which parcel we track ($$\va{r}$$-dependence). - -Of course, the parcel is in our imagination: -$$\va{r}$$ does not really depend on $$t$$; -after all, we are dealing with a continuum. -Nevertheless, the right-hand side of the equation is very useful, +This result is very useful for fluid dynamics, and is known as the **material derivative** or **comoving derivative**: $$\begin{aligned} @@ -76,7 +72,7 @@ The first term is called the **local rate of change**, and the second is the **advective rate of change**. In effect, the latter moves the frame of reference along with the material, so that we can find the evolution of $$f$$ -without needing to worry about the continuum's motion. +without needing to explicitly account for the continuum's motion. That was for a scalar field $$f(\va{r}, t)$$, but in fact the definition also works for vector fields $$\va{U}(\va{r}, t)$$: |