---
title: "Capillary rise exceeding tube height"
subject: "Fluid Mechanics"
chapter: 1
key: "semester_3/fluid_mechanics:capillary-rise-exceeds-tube-height"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/capillary-rise-exceeds-tube-height
site: BCT Engineering
---

# Capillary rise exceeding tube height

Fluid Mechanics · Chapter 1: Fluid and its physical properties

- Asked in: 2079 Baisakh, 2 marks
- Video: [CAPILLARITY (PROBLEM SOLVING)](https://www.youtube.com/watch?v=fZP4HOrNX5U) — E PADASALA
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/capillary-rise-exceeds-tube-height
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-1

## Question

Suppose the water rise predicted by the capillarity formula exceeds the height of the capillary tube. Does the water overflow? Explain with mathematical expression.

## Short answer

No, the water does not overflow. If the tube is shorter than the calculated capillary rise, the meniscus flattens, increasing the contact angle so that the reduced vertical pull of surface tension balances the shorter column’s weight. The water rises only to the top.

## Answer

**No, the water does not overflow.** If the tube is shorter than the calculated rise, the meniscus simply becomes less curved, so that the pull of surface tension is still balanced by the weight of the (shorter) column.

For a tube of radius $r$ with contact angle $\theta$, equating the vertical surface-tension pull to the weight of the column gives

$$
2\pi r\sigma\cos\theta=\rho g\,\pi r^{2}h\quad\Rightarrow\quad h=\frac{2\sigma\cos\theta}{\rho g r}
$$

If the tube height is $h_t<h$, the water rises to the top and the meniscus at the rim adjusts to a new apparent angle $\theta'$ (larger than $\theta$) such that

$$
\cos\theta'=\frac{\rho g r\,h_t}{2\sigma}<\cos\theta
$$

Equivalently, the radius of curvature of the meniscus becomes $R'=r/\cos\theta'$, which is larger than before (a flatter meniscus), so the vertical pull $2\pi r\sigma\cos\theta'$ just equals the weight of the shorter column. The column height stays $h_t$ and the water stays in the tube. However short the tube, the meniscus only flattens further ($\theta'\to90^\circ$ as $h_t\to0$); surface tension at the rim always holds the water, so it never flows over.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. If the calculated capillary rise is more than the tube height, what happens?
   A. Water overflows from the tube top
   B. Water does not overflow; the meniscus flattens to a larger contact angle
   C. Water rises to the calculated height anyway
   D. Water does not rise at all
   Answer: B. The meniscus becomes less curved (larger theta') so the pull balances the weight of the shorter column. (Key point: Overflow or not)

2. Equating surface tension pull to the column weight gives which height?
   A. h = 2 sigma cos(theta) / (rho g r)
   B. h = 2 sigma / (rho g r cos(theta))
   C. h = sigma cos(theta) / (rho g r^2)
   D. h = rho g r / (2 sigma cos(theta))
   Answer: A. 2 pi r sigma cos(theta) = rho g pi r^2 h gives h = 2 sigma cos(theta)/(rho g r). (Key point: Capillary rise formula)

3. For tube height h_t less than h, how does the new contact angle theta' compare with theta?
   A. theta' is smaller, cos(theta') is larger
   B. theta' equals theta
   C. theta' is larger, cos(theta') = rho g r h_t / (2 sigma) is smaller
   D. theta' becomes zero
   Answer: C. cos(theta') = rho g r h_t/(2 sigma) < cos(theta), so theta' > theta and the meniscus radius R' = r/cos(theta') increases. (Key point: New contact angle)

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