From e2f6ff4487606f4052b9c912b9faa2c8d8f1ca10 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 18 Jun 2023 17:59:42 +0200
Subject: Improve knowledge base

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+---
+title: "Capillary length"
+sort_title: "Capillary length"
+date: 2021-03-29
+categories:
+- Physics
+- Fluid mechanics
+- Fluid statics
+- Surface tension
+layout: "concept"
+---
+
+**Capillary action** refers to the movement of liquid
+through narrow spaces due to surface tension, often against gravity.
+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 $$R$$.
+The hydrostatic pressure difference
+between the top and bottom of the drop
+is much smaller than the Laplace pressure:
+
+$$\begin{aligned}
+    2 R \rho g \ll 2 \frac{\alpha}{R}
+\end{aligned}$$
+
+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.
+Rearranging the inequality yields:
+
+$$\begin{aligned}
+    R^2 \ll \frac{\alpha}{\rho g}
+\end{aligned}$$
+
+From this, we define the **capillary length** $$L_c$$
+such that gravity is negligible if $$R \ll L_c$$:
+
+$$\begin{aligned}
+    \boxed{
+        L_c
+        \equiv \sqrt{\frac{\alpha}{\rho g}}
+    }
+\end{aligned}$$
+
+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** $$\mathrm{Eo}$$:
+
+$$\begin{aligned}
+    \boxed{
+        \mathrm{Bo}
+        \equiv \mathrm{Eo}
+        \equiv \frac{L^2}{L_c^2}
+    }
+\end{aligned}$$
+
+Capillary action is observed when $$\mathrm{Bo \ll 1}$$,
+while for $$\mathrm{Bo} \gg 1$$ surface tension is negligible.
+
+For an alternative interpretation of $$\mathrm{Bo}$$,
+let $$m \equiv \rho L^3$$ be the mass of a cube with side $$L$$
+such that its weight is $$m g$$.
+The tension force on its face is $$\alpha L$$,
+so $$\mathrm{Bo}$$ is simply the force ratio:
+
+$$\begin{aligned}
+    \mathrm{Bo}
+    = \frac{m g}{\alpha L}
+\end{aligned}$$
+
+
+
+## References
+1.  B. Lautrup,
+    *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
+    CRC Press.
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