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Diffstat (limited to 'source/know/concept/material-derivative/index.md')
| -rw-r--r-- | source/know/concept/material-derivative/index.md | 38 |
1 files changed, 19 insertions, 19 deletions
diff --git a/source/know/concept/material-derivative/index.md b/source/know/concept/material-derivative/index.md index 16c2c66..93e8ad0 100644 --- a/source/know/concept/material-derivative/index.md +++ b/source/know/concept/material-derivative/index.md @@ -11,22 +11,22 @@ layout: "concept" --- Inside a fluid (or any other continuum), we might be interested in -the time evolution of a certain intensive quantity $f$, +the time evolution of a certain intensive quantity $$f$$, e.g. the temperature or pressure, -represented by a scalar field $f(\va{r}, t)$. +represented by a scalar field $$f(\va{r}, t)$$. -If the fluid is static, the evolution of $f$ is simply $\ipdv{f}{t}$, +If the fluid is static, the evolution of $$f$$ is simply $$\ipdv{f}{t}$$, since each point of the fluid is motionless. However, if the fluid is moving, we have a problem: -the fluid molecules at position $\va{r} = \va{r}_0$ are not necessarily -the same ones at time $t = t_0$ and $t = t_1$. -Those molecules take $f$ with them as they move, +the fluid molecules at position $$\va{r} = \va{r}_0$$ are not necessarily +the same ones at time $$t = t_0$$ and $$t = t_1$$. +Those molecules take $$f$$ with them as they move, so we need to account for this transport somehow. To do so, we choose an infinitesimal "blob" or **parcel** of the fluid, which always contains the same specific molecules, -and track its position $\va{r}(t)$ through time as it moves and deforms. -The value of $f$ for this parcel is then given by: +and track its position $$\va{r}(t)$$ through time as it moves and deforms. +The value of $$f$$ for this parcel is then given by: $$\begin{aligned} f(\va{r}, t) @@ -34,9 +34,9 @@ $$\begin{aligned} = f\big(x(t), y(t), z(t), t\big) \end{aligned}$$ -In effect, we have simply made the coordinate $\va{r}$ dependent on time, +In effect, we have simply made the coordinate $$\va{r}$$ dependent on time, and have specifically chosen the time-dependence to track the parcel. -The net evolution of $f$ is then its "true" (i.e. non-partial) derivative with respect to $t$, +The net evolution of $$f$$ is then its "true" (i.e. non-partial) derivative with respect to $$t$$, allowing us to apply the chain rule: $$\begin{aligned} @@ -46,8 +46,8 @@ $$\begin{aligned} &= \pdv{f}{t} + v_x \pdv{f}{x} + v_y \pdv{f}{y} + v_z \pdv{f}{z} \end{aligned}$$ -Where $v_x$, $v_y$ and $v_z$ are the parcel's velocity components. -Let $\va{v} = (v_x, v_y, v_z)$ be the velocity vector field, +Where $$v_x$$, $$v_y$$ and $$v_z$$ are the parcel's velocity components. +Let $$\va{v} = (v_x, v_y, v_z)$$ be the velocity vector field, then we can rewrite this expression like so: $$\begin{aligned} @@ -55,12 +55,12 @@ $$\begin{aligned} &= \pdv{f}{t} + (\va{v} \cdot \nabla) f \end{aligned}$$ -Note that $\va{v} = \va{v}(\va{r}, t)$, -that is, the velocity can change with time ($t$-dependence), -and depends on which parcel we track ($\va{r}$-dependence). +Note that $$\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}$$-dependence). Of course, the parcel is in our imagination: -$\va{r}$ does not really depend on $t$; +$$\va{r}$$ does not really depend on $$t$$; after all, we are dealing with a continuum. Nevertheless, the right-hand side of the equation is very useful, and is known as the **material derivative** or **comoving derivative**: @@ -75,11 +75,11 @@ $$\begin{aligned} 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 $f$ +so that we can find the evolution of $$f$$ without needing to worry about the continuum's motion. -That was for a scalar field $f(\va{r}, t)$, -but in fact the definition also works for vector fields $\va{U}(\va{r}, t)$: +That was for a scalar field $$f(\va{r}, t)$$, +but in fact the definition also works for vector fields $$\va{U}(\va{r}, t)$$: $$\begin{aligned} \boxed{ |