From 07a63237de774b3a57a0975e03cf2c6b68f165b5 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Wed, 31 Mar 2021 19:57:54 +0200 Subject: Expand knowledge base --- content/know/category/continuum-physics.md | 9 + content/know/category/fluid-dynamics.md | 9 + content/know/category/fluid-statics.md | 9 + content/know/category/surface-tension.md | 9 + content/know/concept/capillary-action/index.pdc | 2 + .../know/concept/cauchy-stress-tensor/index.pdc | 245 +++++++++++++++++++++ .../know/concept/hydrostatic-pressure/index.pdc | 14 +- content/know/concept/material-derivative/index.pdc | 121 ++++++++++ content/know/concept/meniscus/index.pdc | 2 + .../concept/rayleigh-plateau-instability/index.pdc | 4 +- content/know/concept/wetting/index.pdc | 2 + .../know/concept/young-dupre-relation/index.pdc | 2 + content/know/concept/young-laplace-law/index.pdc | 1 + 13 files changed, 425 insertions(+), 4 deletions(-) create mode 100644 content/know/category/continuum-physics.md create mode 100644 content/know/category/fluid-dynamics.md create mode 100644 content/know/category/fluid-statics.md create mode 100644 content/know/category/surface-tension.md create mode 100644 content/know/concept/cauchy-stress-tensor/index.pdc create mode 100644 content/know/concept/material-derivative/index.pdc diff --git a/content/know/category/continuum-physics.md b/content/know/category/continuum-physics.md new file mode 100644 index 0000000..dbc04ec --- /dev/null +++ b/content/know/category/continuum-physics.md @@ -0,0 +1,9 @@ +--- +title: "Continuum physics" +firstLetter: "C" +date: 2021-03-31T09:48:05+02:00 +draft: false +layout: "category" +--- + +This page will fill itself. diff --git a/content/know/category/fluid-dynamics.md b/content/know/category/fluid-dynamics.md new file mode 100644 index 0000000..ed920c8 --- /dev/null +++ b/content/know/category/fluid-dynamics.md @@ -0,0 +1,9 @@ +--- +title: "Fluid dynamics" +firstLetter: "F" +date: 2021-03-31T09:48:16+02:00 +draft: false +layout: "category" +--- + +This page will fill itself. diff --git a/content/know/category/fluid-statics.md b/content/know/category/fluid-statics.md new file mode 100644 index 0000000..0ecaa84 --- /dev/null +++ b/content/know/category/fluid-statics.md @@ -0,0 +1,9 @@ +--- +title: "Fluid statics" +firstLetter: "F" +date: 2021-03-31T09:48:12+02:00 +draft: false +layout: "category" +--- + +This page will fill itself. diff --git a/content/know/category/surface-tension.md b/content/know/category/surface-tension.md new file mode 100644 index 0000000..8506028 --- /dev/null +++ b/content/know/category/surface-tension.md @@ -0,0 +1,9 @@ +--- +title: "Surface tension" +firstLetter: "S" +date: 2021-03-31T09:58:18+02:00 +draft: false +layout: "category" +--- + +This page will fill itself. 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 -- cgit v1.2.3