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Diffstat (limited to 'source/know/concept/capillary-action')
| -rw-r--r-- | source/know/concept/capillary-action/index.md | 42 |
1 files changed, 21 insertions, 21 deletions
diff --git a/source/know/concept/capillary-action/index.md b/source/know/concept/capillary-action/index.md index 1ee3cd4..4b9e76c 100644 --- a/source/know/concept/capillary-action/index.md +++ b/source/know/concept/capillary-action/index.md @@ -16,7 +16,7 @@ It occurs when the [Laplace pressure](/know/concept/young-laplace-law/) from surface tension is much larger in magnitude than the [hydrostatic pressure](/know/concept/hydrostatic-pressure/) from gravity. -Consider a spherical droplet of liquid with radius $R$. +Consider a spherical droplet of liquid with radius $$R$$. The hydrostatic pressure difference between the top and bottom of the drop is much smaller than the Laplace pressure: @@ -25,17 +25,17 @@ $$\begin{aligned} 2 R \rho g \ll 2 \frac{\alpha}{R} \end{aligned}$$ -Where $\rho$ is the density of the liquid, -$g$ is the acceleration due to gravity, -and $\alpha$ is the energy cost per unit surface area. +Where $$\rho$$ is the density of the liquid, +$$g$$ is the acceleration due to gravity, +and $$\alpha$$ is the energy cost per unit surface area. Rearranging the inequality yields: $$\begin{aligned} R^2 \ll \frac{\alpha}{\rho g} \end{aligned}$$ -From the right-hand side we define the **capillary length** $L_c$, -so gravity is negligible if $R \ll L_c$: +From the right-hand side we define the **capillary length** $$L_c$$, +so gravity is negligible if $$R \ll L_c$$: $$\begin{aligned} \boxed{ @@ -44,10 +44,10 @@ $$\begin{aligned} } \end{aligned}$$ -In general, for a system with characteristic length $L$, +In general, for a system with characteristic length $$L$$, the relative strength of gravity compared to surface tension -is described by the **Bond number** $\mathrm{Bo}$ -or **Eötvös number** $\mathrm{Eo}$: +is described by the **Bond number** $$\mathrm{Bo}$$ +or **Eötvös number** $$\mathrm{Eo}$$: $$\begin{aligned} \boxed{ @@ -58,16 +58,16 @@ $$\begin{aligned} } \end{aligned}$$ -The right-most side gives an alternative way of understanding $\mathrm{Bo}$: -$m$ is the mass of a cube with side $L$, such that the numerator is the weight force, +The right-most side gives an alternative way of understanding $$\mathrm{Bo}$$: +$$m$$ is the mass of a cube with side $$L$$, such that the numerator is the weight force, and the denominator is the tension force of the surface. -In any case, capillary action can be observed when $\mathrm{Bo \ll 1}$. +In any case, capillary action can be observed when $$\mathrm{Bo \ll 1}$$. The most famous example of capillary action is **capillary rise**, -where a liquid "climbs" upwards in a narrow vertical tube with radius $R$, +where a liquid "climbs" upwards in a narrow vertical tube with radius $$R$$, apparently defying gravity. Assuming the liquid-air interface is a spherical cap -with constant [curvature](/know/concept/curvature/) radius $R_c$, +with constant [curvature](/know/concept/curvature/) radius $$R_c$$, then we know that the liquid is at rest when the hydrostatic pressure equals the Laplace pressure: @@ -77,12 +77,12 @@ $$\begin{aligned} = 2 \alpha \frac{\cos\theta}{R} \end{aligned}$$ -Where $\theta$ is the liquid-tube contact angle, -and we are neglecting variations of the height $h$ due to the curvature +Where $$\theta$$ is the liquid-tube contact angle, +and we are neglecting variations of the height $$h$$ due to the curvature (i.e. the [meniscus](/know/concept/meniscus/)). -By isolating the above equation for $h$, +By isolating the above equation for $$h$$, we arrive at **Jurin's law**, -which predicts the height climbed by a liquid in a tube with radius $R$: +which predicts the height climbed by a liquid in a tube with radius $$R$$: $$\begin{aligned} \boxed{ @@ -91,7 +91,7 @@ $$\begin{aligned} } \end{aligned}$$ -Depending on $\theta$, $h$ can be negative, +Depending on $$\theta$$, $$h$$ can be negative, i.e. the liquid might descend below the ambient level. @@ -106,7 +106,7 @@ $$\begin{aligned} \approx 2 \pi R (\alpha_{sg} - \alpha_{sl}) \end{aligned}$$ -Where $\alpha_{sg}$ and $\alpha_{sl}$ are the energy costs +Where $$\alpha_{sg}$$ and $$\alpha_{sl}$$ are the energy costs of the solid-gas and solid-liquid interfaces. Thanks to the [Young-Dupré relation](/know/concept/young-dupre-relation/), we can rewrite this as follows: @@ -116,7 +116,7 @@ $$\begin{aligned} = 2 \alpha \cos\theta \end{aligned}$$ -Isolating this for $h$ simply yields Jurin's law again, as expected. +Isolating this for $$h$$ simply yields Jurin's law again, as expected. |