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9 files changed, 389 insertions, 4 deletions

diff --git a/content/know/concept/capillary-action/index.pdc b/content/know/concept/capillary-action/index.pdc
index e76b88a..7b6d7fb 100644
--- a/content/know/concept/capillary-action/index.pdc
+++ b/content/know/concept/capillary-action/index.pdc
@@ -5,6 +5,8 @@ publishDate: 2021-03-29
 categories:
 - Physics
 - Fluid mechanics
+- Fluid statics
+- Surface tension
 
 date: 2021-03-07T20:42:28+01:00
 draft: false
diff --git a/content/know/concept/cauchy-stress-tensor/index.pdc b/content/know/concept/cauchy-stress-tensor/index.pdc
new file mode 100644
index 0000000..6c7c97a
--- /dev/null
+++ b/content/know/concept/cauchy-stress-tensor/index.pdc
@@ -0,0 +1,245 @@
+---
+title: "Cauchy stress tensor"
+firstLetter: "C"
+publishDate: 2021-03-31
+categories:
+- Physics
+- Continuum physics
+
+date: 2021-03-31T09:43:33+02:00
+draft: false
+markup: pandoc
+---
+
+# Cauchy stress tensor
+
+Roughly speaking, **stress** is the solid equivalent of fluid pressure:
+it describes the net force acting on an imaginary partition surface inside a solid.
+However, unlike fluids at rest,
+where the pressure is always perpendicular to such a surface,
+solid stress is usually much more complicated.
+
+Formally, the concept of stress can be applied to any continuum
+(not just solids), including fluids,
+but it is arguably most intuitive for solids.
+
+
+## Definition
+
+In the solid, imagine an infinitesimal cube
+whose sides, $\dd{S}_x$, $\dd{S}_y$ and $\dd{S}_z$,
+are orthogonal to the $x$, $y$ and $z$ axes, respectively.
+There is a force $\dd{\va{F}}_1$ acting on $\dd{S}_x$,
+$\dd{\va{F}}_2$ on $\dd{S}_y$, and $\dd{\va{F}}_3$ on $\dd{S}_z$.
+Then we can decompose each of these forces, for example:
+
+$$\begin{aligned}
+    \dd{\va{F}}_1
+    = \va{e}_x F_{x1} + \va{e}_y F_{y1} + \va{e}_z F_{z1}
+\end{aligned}$$
+
+Where $\va{e}_x$, $\va{e}_y$ and $\va{e}_z$ are the basis unit vectors.
+If we divide each of the force components by the area $\dd{S}_x$
+(like in a fluid, in order to get the pressure),
+we find the stresses $\sigma_{xx}$, $\sigma_{yx}$ and $\sigma_{zx}$
+that are being "felt" by the $x$ surface element $\dd{S}_x$:
+
+$$\begin{aligned}
+    \dd{\va{F}}_1
+    = \big( \va{e}_x \sigma_{xx}  + \va{e}_y \sigma_{yx} + \va{e}_z \sigma_{zx} \big) \dd{S}_x
+\end{aligned}$$
+
+The perpendicular component $\sigma_{xx}$ is called a **tensile stress**,
+and its sign is always chosen so that a positive value corresponds to a tension,
+i.e. the $x$-side is pulled away from the rest of the cube.
+The tangential components $\sigma_{yx}$ and $\sigma_{zx}$
+are called **shear stresses**.
+
+Evidently, the other two forces $\dd{\va{F}}_2$ and $\dd{\va{F}}_3$
+can be decomposed in the exact same way,
+yielding nine stress components in total:
+
+$$\begin{aligned}
+    \dd{\va{F}}_2
+    &= \va{e}_x F_{x2} + \va{e}_y F_{y2} + \va{e}_z F_{z2}
+    = \big( \va{e}_x \sigma_{xy}  + \va{e}_y \sigma_{yy} + \va{e}_z \sigma_{zy} \big) \dd{S}_y
+    \\
+    \dd{\va{F}}_3
+    &= \va{e}_x F_{x3} + \va{e}_y F_{y3} + \va{e}_z F_{z3}
+    = \big( \va{e}_x \sigma_{xz}  + \va{e}_y \sigma_{yz} + \va{e}_z \sigma_{zz} \big) \dd{S}_z
+\end{aligned}$$
+
+The total force $\dd{\va{F}}$ on the entire infinitesimal cube
+is simply the sum of the previous three:
+
+$$\begin{aligned}
+    \dd{\va{F}}
+    = \dd{\va{F}}_1 + \dd{\va{F}}_2 + \dd{\va{F}}_3
+\end{aligned}$$
+
+We can then decompose $\dd{\va{F}}$ into its net components
+along the $x$, $y$ and $z$ axes:
+
+$$\begin{aligned}
+    \dd{\va{F}}
+    = \va{e}_x \dd{F}_x + \va{e}_y \dd{F}_y + \va{e}_z \dd{F}_z
+\end{aligned}$$
+
+From the preceding equations, we find that these components are given by:
+
+$$\begin{aligned}
+    \dd{F}_x
+    &= \sigma_{xx} \dd{S}_x + \sigma_{xy} \dd{S}_y + \sigma_{xz} \dd{S}_z
+    \\
+    \dd{F}_y
+    &= \sigma_{yx} \dd{S}_x + \sigma_{yy} \dd{S}_y + \sigma_{yz} \dd{S}_z
+    \\
+    \dd{F}_z
+    &= \sigma_{zx} \dd{S}_x + \sigma_{zy} \dd{S}_y + \sigma_{zz} \dd{S}_z
+\end{aligned}$$
+
+We can write this much more compactly using index notation,
+where $i, j \in \{x, y, z\}$:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{F}_i
+        = \sum_{j} \sigma_{ij} \dd{S}_j
+    }
+\end{aligned}$$
+
+The stress components $\sigma_{ij}$ can be written as a second-order tensor
+(i.e. a matrix that transforms in a certain way),
+called the **Cauchy stress tensor** $\hat{\sigma}$:
+
+$$\begin{aligned}
+    \boxed{
+        \hat{\sigma} =
+        \{ \sigma_{ij} \} =
+        \begin{pmatrix}
+            \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
+            \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
+            \sigma_{zx} & \sigma_{zy} & \sigma_{zz}
+        \end{pmatrix}
+    }
+\end{aligned}$$
+
+Then $\dd{\va{F}}$ is written even more compactly
+using the dot product, with $\dd{\va{S}} = (\dd{S}_x, \dd{S}_y, \dd{S}_z)$:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{\va{F}}
+        = \hat{\sigma} \cdot \dd{\va{S}}
+    }
+\end{aligned}$$
+
+All forces on the cube's sides can be written in this form.
+**Cauchy's stress theorem** states that the force on *any*
+surface element inside the solid can be written like this,
+simply by projecting it onto the $x$, $y$ and $z$ zero-planes
+to get the areas $\dd{S}_x$, $\dd{S}_y$ and $\dd{S}_z$.
+
+Note that for fluids, the pressure $p$ was defined
+such that $\dd{\va{F}} = - p \dd{\va{S}}$.
+If we wanted to define $p$ for solids in the same way,
+we would need $\hat{\sigma}$ to be diagonal *and*
+all of its diagonal elements to be identical.
+Since this is almost never the case,
+the scalar pressure is ill-defined in solids.
+
+
+## Equilibrium
+
+The total force $\va{F}$ acting on a (non-infinitesimal) volume $V$ of the solid
+is given by the sum of the total body force $\va{F}_b$ and total surface force $\va{F}_s$,
+where $\vec{f}$ is the body force density:
+
+$$\begin{aligned}
+    \va{F}
+    = \va{F}_b + \va{F}_s
+    = \int_V \va{f} \dd{V} + \oint_S \hat{\sigma} \cdot \dd{\va{S}}
+\end{aligned}$$
+
+We can rewrite the surface term using the divergence theorem,
+where $\top$ is the transpose:
+
+$$\begin{aligned}
+    \va{F}_s
+    = \oint_S \hat{\sigma} \cdot \dd{\va{S}}
+    = \int_V \nabla \cdot \hat{\sigma}^{\top} \dd{V}
+\end{aligned}$$
+
+For some people, this equation may be more enlightening in index notation,
+where $\vec{\sigma}_i$ is the $i$th row of the tensor (implicitly transposed):
+
+$$\begin{aligned}
+    F_{s, i}
+    = \oint_S \sum_j \sigma_{ij} \dd{S_j}
+    = \int_V \sum_{j} \nabla_{\!j} \sigma_{ij} \dd{V}
+    = \int_V \nabla \cdot \vec{\sigma}_i \dd{V}
+\end{aligned}$$
+
+In any case, the total force $\va{F}$ can then be expressed
+as a single volume integral over $V$:
+
+$$\begin{aligned}
+    \va{F}
+    = \int_V \va{f} \dd{V} + \int_V \nabla \cdot \hat{\sigma}^{\top} \dd{V}
+    = \int_V \va{f^*} \dd{V}
+\end{aligned}$$
+
+Where we have defined the **effective force density** $\va{f^*}$ as follows:
+
+$$\begin{aligned}
+    \boxed{
+        \va{f^*}
+        = \va{f} + \nabla \cdot \hat{\sigma}^{\top}
+    }
+\end{aligned}$$
+
+The volume $V$ is in **mechanical equilibrium** if the net force acting on it amounts to zero:
+
+$$\begin{aligned}
+    \va{F}
+    = 0
+\end{aligned}$$
+
+However, because $V$ is abritrary, the equilibrium condition for the whole solid is in fact:
+
+$$\begin{aligned}
+    \boxed{
+        \va{f^*}
+        = 0
+    }
+\end{aligned}$$
+
+This is reminiscent of the equilibrium condition of a fluid
+(see [hydrostatic pressure](/know/concept/hydrostatic-pressure/)).
+Note that it is a set of coupled differential equations,
+which needs boundary conditions at the object's surface.
+Newton's third law states that the two sides of the boundary
+exert opposite forces on each other,
+so the boundary condition is continuity of the **stress vector**
+$\hat{\sigma} \cdot \va{n}$:
+
+$$\begin{aligned}
+    \boxed{
+        \hat{\sigma}_{\mathrm{outer}} \cdot \va{n}
+        = - \hat{\sigma}_{\mathrm{inner}} \cdot \va{n}
+    }
+\end{aligned}$$
+
+Where the normal of the outer surface is $\va{n}$,
+and the normal of the inner surface is $-\va{n}$.
+Note that the above equation does *not* mean
+that $-\hat{\sigma}_{\mathrm{inner}}$ equals $\hat{\sigma}_{\mathrm{outer}}$:
+the tensors are allowed to be very different,
+as long as the stress vector's three components are equal.
+
+
+## References
+1.  B. Lautrup,
+    *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
+    CRC Press.
+
diff --git a/content/know/concept/hydrostatic-pressure/index.pdc b/content/know/concept/hydrostatic-pressure/index.pdc
index 3bbb0af..90e57ce 100644
--- a/content/know/concept/hydrostatic-pressure/index.pdc
+++ b/content/know/concept/hydrostatic-pressure/index.pdc
@@ -5,6 +5,7 @@ publishDate: 2021-03-12
 categories:
 - Physics
 - Fluid mechanics
+- Fluid statics
 
 date: 2021-03-12T14:37:56+01:00
 draft: false
@@ -167,9 +168,16 @@ $$\begin{aligned}
     = \nabla \Big( \Phi + w(p) \Big)
 \end{aligned}$$
 
-By defining the **effective gravitational potential** $\Phi^* = \Phi + w(p)$,
-we get the cleanest form yet of the equilibrium condition,
-which states that $\Phi^*$ must be a constant:
+From this, let us now define the
+**effective gravitational potential** $\Phi^*$ as follows:
+
+$$\begin{aligned}
+    \boxed{
+        \Phi^* = \Phi + w(p)
+    }
+\end{aligned}$$
+
+This results in the cleanest form yet of the equilibrium condition, namely:
 
 $$\begin{aligned}
     \boxed{
diff --git a/content/know/concept/material-derivative/index.pdc b/content/know/concept/material-derivative/index.pdc
new file mode 100644
index 0000000..36113cc
--- /dev/null
+++ b/content/know/concept/material-derivative/index.pdc
@@ -0,0 +1,121 @@
+---
+title: "Material derivative"
+firstLetter: "M"
+publishDate: 2021-03-30
+categories:
+- Physics
+- Fluid mechanics
+- Fluid dynamics
+- Continuum physics
+
+date: 2021-03-30T19:39:28+02:00
+draft: false
+markup: pandoc
+---
+
+# Material derivative
+
+Inside a fluid (or any other continuum), we might be interested in
+the time evolution of a certain intensive quantity $f$,
+e.g. the temperature or pressure,
+represented by a scalar field $f(\va{r}, t)$.
+
+If the fluid is static, the evolution of $f$ is simply $\pdv*{f}{t}$,
+since each point of the fluid is motionless.
+However, if the fluid is moving, we have a problem:
+the fluid molecules at position $\va{r} = \va{r}_0$ are not necessarily
+the same ones at time $t = t_0$ and $t = t_1$.
+Those molecules take $f$ with them as they move,
+so we need to account for this transport somehow.
+
+To do so, we choose an infinitesimal "blob" or **parcel** of the fluid,
+which always contains the same specific molecules,
+and track its position $\va{r}(t)$ through time as it moves and deforms.
+The value of $f$ for this parcel is then given by:
+
+$$\begin{aligned}
+    f(\va{r}, t)
+    = f(\va{r}(t), t)
+    = f\big(x(t), y(t), z(t), t\big)
+\end{aligned}$$
+
+In effect, we have simply made the coordinate $\va{r}$ dependent on time,
+and have specifically chosen the time-dependence to track the parcel.
+The net evolution of $f$ is then its "true" (i.e. non-partial) derivative with respect to $t$,
+allowing us to apply the chain rule:
+
+$$\begin{aligned}
+    \dv{t} f\big(x(t), y(t), z(t), t\big)
+    &= \pdv{f}{t} + \pdv{f}{x} \dv{x}{t} + \pdv{f}{y} \dv{y}{t} + \pdv{f}{z} \dv{z}{t}
+    \\
+    &= \pdv{f}{t} + v_x \pdv{f}{x} + v_y \pdv{f}{y} + v_z \pdv{f}{z}
+\end{aligned}$$
+
+Where $v_x$, $v_y$ and $v_z$ are the parcel's velocity components.
+Let $\va{v} = (v_x, v_y, v_z)$ be the velocity vector field,
+then we can rewrite this expression like so:
+
+$$\begin{aligned}
+    \dv{t} f\big(x(t), y(t), z(t), t\big)
+    &= \pdv{f}{t} + \va{v} \cdot \nabla f
+\end{aligned}$$
+
+Note that $\va{v} = \va{v}(\va{r}, t)$,
+that is, the velocity can change with time ($t$-dependence),
+and depends on which parcel we track ($\va{r}$-dependence).
+
+Of course, the parcel is in our imagination:
+$\va{r}$ does not really depend on $t$;
+after all, we are dealing with a continuum.
+Nevertheless, the right-hand side of the equation is very useful,
+and is known as the **material derivative** or **comoving derivative**:
+
+$$\begin{aligned}
+    \boxed{
+        \frac{\mathrm{D}f}{\mathrm{D}t}
+        \equiv \pdv{f}{t} + \va{v} \cdot \nabla f
+    }
+\end{aligned}$$
+
+The first term is called the **local rate of change**,
+and the second is the **advective rate of change**.
+In effect, the latter moves the frame of reference along with the material,
+so that we can find the evolution of $f$
+without needing to worry about the continuum's motion.
+
+That was for a scalar field $f(\va{r}, t)$,
+but in fact the definition also works for vector fields $\va{U}(\va{r}, t)$:
+
+$$\begin{aligned}
+    \boxed{
+        \frac{\mathrm{D} \va{U}}{\mathrm{D}t}
+        \equiv \pdv{f}{t} + \va{v} \cdot \nabla \va{U}
+    }
+\end{aligned}$$
+
+Where the advective term is to be evaluated in the following way:
+
+$$\begin{aligned}
+    \va{v} \cdot \nabla \va{U}
+    =
+    \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}
+    \cdot
+    \begin{bmatrix}
+        \displaystyle\pdv{U_x}{x} & \displaystyle\pdv{U_x}{y} & \displaystyle\pdv{U_x}{z} \\
+        \displaystyle\pdv{U_y}{x} & \displaystyle\pdv{U_y}{y} & \displaystyle\pdv{U_y}{z} \\
+        \displaystyle\pdv{U_z}{x} & \displaystyle\pdv{U_z}{y} & \displaystyle\pdv{U_z}{z}
+    \end{bmatrix}
+    =
+    \begin{bmatrix}
+        v_x \displaystyle\pdv{U_x}{x} & v_y \displaystyle\pdv{U_x}{y} & v_z \displaystyle\pdv{U_x}{z} \\
+        v_x \displaystyle\pdv{U_y}{x} & v_y \displaystyle\pdv{U_y}{y} & v_z \displaystyle\pdv{U_y}{z} \\
+        v_x \displaystyle\pdv{U_z}{x} & v_y \displaystyle\pdv{U_z}{y} & v_z \displaystyle\pdv{U_z}{z}
+    \end{bmatrix}
+\end{aligned}$$
+
+
+
+## References
+1.  B. Lautrup,
+    *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
+    CRC Press.
diff --git a/content/know/concept/meniscus/index.pdc b/content/know/concept/meniscus/index.pdc
index c3d1c6a..04e5daf 100644
--- a/content/know/concept/meniscus/index.pdc
+++ b/content/know/concept/meniscus/index.pdc
@@ -5,6 +5,8 @@ publishDate: 2021-03-11
 categories:
 - Physics
 - Fluid mechanics
+- Fluid statics
+- Surface tension
 
 date: 2021-03-11T14:39:56+01:00
 draft: false
diff --git a/content/know/concept/rayleigh-plateau-instability/index.pdc b/content/know/concept/rayleigh-plateau-instability/index.pdc
index ae0a21d..df3d6ab 100644
--- a/content/know/concept/rayleigh-plateau-instability/index.pdc
+++ b/content/know/concept/rayleigh-plateau-instability/index.pdc
@@ -6,6 +6,7 @@ categories:
 - Physics
 - Fluid mechanics
 - Perturbation
+- Surface tension
 
 date: 2021-03-10T09:13:22+01:00
 draft: false
@@ -110,7 +111,8 @@ $$\begin{aligned}
 
 Before solving this, we need boundary conditions.
 The radial fluid velocity $u_r$ (the $r$-component of $\vec{u}$)
-at the column surface $r\!=\!R$ is the *material derivative* of $R_\epsilon$:
+at the column surface $r\!=\!R$ is the
+[material derivative](/know/concept/material-derivative/) of $R_\epsilon$:
 
 $$\begin{aligned}
     u_r(r\!=\!R)
diff --git a/content/know/concept/wetting/index.pdc b/content/know/concept/wetting/index.pdc
index e5bbadf..2cb7c08 100644
--- a/content/know/concept/wetting/index.pdc
+++ b/content/know/concept/wetting/index.pdc
@@ -5,6 +5,8 @@ publishDate: 2021-03-29
 categories:
 - Physics
 - Fluid mechanics
+- Fluid statics
+- Surface tension
 
 date: 2021-03-29T16:20:44+02:00
 draft: false
diff --git a/content/know/concept/young-dupre-relation/index.pdc b/content/know/concept/young-dupre-relation/index.pdc
index 6b6d89a..d3f36cb 100644
--- a/content/know/concept/young-dupre-relation/index.pdc
+++ b/content/know/concept/young-dupre-relation/index.pdc
@@ -5,6 +5,8 @@ publishDate: 2021-03-07
 categories:
 - Physics
 - Fluid mechanics
+- Fluid statics
+- Surface tension
 
 date: 2021-03-07T15:05:50+01:00
 draft: false
diff --git a/content/know/concept/young-laplace-law/index.pdc b/content/know/concept/young-laplace-law/index.pdc
index 505125e..20912ab 100644
--- a/content/know/concept/young-laplace-law/index.pdc
+++ b/content/know/concept/young-laplace-law/index.pdc
@@ -5,6 +5,7 @@ publishDate: 2021-03-11
 categories:
 - Physics
 - Fluid mechanics
+- Surface tension
 
 date: 2021-03-07T14:54:41+01:00
 draft: false