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+---
+title: "Jurin's law"
+sort_title: "Jurin's law"
+date: 2023-06-15
+categories:
+- Physics
+- Fluid mechanics
+- Fluid statics
+- Surface tension
+layout: "concept"
+---
+
+A well-known example of *capillary action* is
+when a liquid climbs up a narrow vertical tube with radius $$R$$,
+apparently defying gravity.
+Indeed, this occurs when the liquid's surface tension can overpower gravity;
+specifically, when the [capillary length](/know/concept/capillary-length/) $$L_c > R$$.
+
+Let us assume that the liquid-air interface has a spherical shape,
+which may point up or down depending on the liquid.
+This interface then has a constant [curvature radius](/know/concept/curvature/) $$r$$
+determined by the contact angle $$\theta$$ of the liquid to the tube:
+$$r = R / \cos{\theta}$$. We know that the liquid is at rest
+when the [hydrostatic pressure](/know/concept/hydrostatic-pressure/)
+equals the resulting [Laplace pressure](/know/concept/young-laplace-law/):
+
+$$\begin{aligned}
+    \rho g h
+    = \alpha \frac{2}{r}
+    = 2 \alpha \frac{\cos{\theta}}{R}
+\end{aligned}$$
+
+Note that $$h$$ is the height of interface's highest/lowest point;
+we neglect the [meniscus](/know/concept/meniscus/).
+By isolating the above equation for $$h$$, we arrive at **Jurin's law**:
+
+$$\begin{aligned}
+    \boxed{
+        h
+        = \frac{2 \alpha \cos{\theta}}{\rho g R}
+        = 2 \frac{L_c^2}{R} \cos{\theta}
+    }
+\end{aligned}$$
+
+Where $$L_c \equiv \sqrt{\alpha / \rho g}$$.
+This predicts the height climbed by a liquid in a narrow tube.
+If $$\theta > 90\degree$$, then $$h$$ is negative,
+i.e. the liquid descends below the ambient level.
+
+An alternative derivation of Jurin's law balances the forces instead of the pressures.
+On the right, we have the gravitational force
+(i.e. the energy-per-distance to lift the liquid),
+and on the left, the surface tension force
+(i.e. the energy-per-distance of the liquid-tube interface):
+
+$$\begin{aligned}
+     \pi R^2 \rho g h
+     \approx 2 \pi R (\alpha_{sg} - \alpha_{sl})
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+     R \rho g h
+     = 2 \alpha \cos\theta
+\end{aligned}$$
+
+Isolating this for $$h$$ simply yields Jurin's law again, as expected.
+
+
+
+## References
+1.  B. Lautrup,
+    *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
+    CRC Press.