---
title: "Meniscus"
date: 2021-03-11
categories:
- Physics
- Fluid mechanics
- Fluid statics
- Surface tension
layout: "concept"
---

When a fluid interface, e.g. the surface of a liquid,
touches a flat solid wall, it will curve to meet it.
This small rise or fall is called a **meniscus**,
and is caused by surface tension and gravity.

In 2D, let the vertical $y$-axis be a flat wall,
and the fluid tend to $y = 0$ when $x \to \infty$.
Close to the wall, i.e. for small $x$, the liquid curves up or down
to touch the wall at a height $y = d$.

Three forces are at work here:
the first two are the surface tension $\alpha$ of the fluid surface,
and the counter-pull $\alpha \sin\phi$ of the wall against the tension,
where $\phi$ is the contact angle.
The third is the [hydrostatic pressure](/know/concept/hydrostatic-pressure/) gradient
inside the small portion of the fluid above/below the ambient level,
which exerts a total force on the wall given by
(for $\phi < \pi/2$ so that $d > 0$):

$$\begin{aligned}
    \int_0^d \rho g y \dd{y}
    = \frac{1}{2} \rho g d^2
\end{aligned}$$

If you were wondering about the units,
keep in mind that there is an implicit $z$-direction here too.
This results in the following balance equation for the forces at the wall:

$$\begin{aligned}
    \alpha
    = \alpha \sin\phi + \frac{1}{2} \rho g d^2
\end{aligned}$$

We isolate this relation for $d$
and use some trigonometric magic to rewrite it:

$$\begin{aligned}
    d
    = \sqrt{\frac{\alpha}{\rho g}} \sqrt{2 (1 - \sin\phi)}
    = \sqrt{\frac{\alpha}{\rho g}} \sqrt{4 \sin^2\!\Big(\frac{\pi}{4} - \frac{\phi}{2}\Big)}
\end{aligned}$$

Here, we recognize the definition of the capillary length $L_c = \sqrt{\alpha / (\rho g)}$,
yielding an expression for $d$
that is valid both for $\phi < \pi/2$ (where $d > 0$)
and $\phi > \pi/2$ (where $d < 0$):

$$\begin{aligned}
    \boxed{
        d
        = 2 L_c \sin\!\Big(\frac{\pi/2 - \phi}{2}\Big)
    }
\end{aligned}$$

Next, we would like to know the exact shape of the meniscus.
To do this, we need to describe the liquid surface differently,
using the elevation angle $\theta$ relative to the $y = 0$ plane.
The curve $\theta(s)$ is a function of the arc length $s$,
where $\dd{s}^2 = \dd{x}^2 + \dd{y}^2$,
and is governed by:

$$\begin{aligned}
    \dv{x}{s}
    = \cos\theta
    \qquad
    \dv{y}{s}
    = \sin\theta
    \qquad
    \dv{\theta}{s}
    = \frac{1}{R}
\end{aligned}$$

The last equation describes the curvature radius $R$
of the surface along the $x$-axis.
Since we are considering a flat wall,
there is no curvature in the orthogonal principal direction.

Just below the liquid surface in the meniscus,
we expect the hydrostatic pressure
and the [Young-Laplace law](/know/concept/young-laplace-law/)
to agree about the pressure $p$,
where $p_0$ is the external air pressure:

$$\begin{aligned}
    p_0 - \rho g y
    = p_0 - \frac{\alpha}{R}
\end{aligned}$$

Rearranging this yields that $R = L_c^2 / y$.
Inserting this into the curvature equation gives us:

$$\begin{aligned}
    \dv{\theta}{s}
    = \frac{y}{L_c^2}
\end{aligned}$$

By differentiating this equation with respect to $s$
and using $\idv{y}{s} = \sin\theta$, we arrive at:

$$\begin{aligned}
    \boxed{
        L_c^2 \dvn{2}{\theta}{s} = \sin\theta
    }
\end{aligned}$$

To solve this equation, we multiply it by $\idv{\theta}{s}$,
which is nonzero close to the wall:

$$\begin{aligned}
    L_c^2 \dvn{2}{\theta}{s} \dv{\theta}{s}
    = \dv{\theta}{s} \sin\theta
\end{aligned}$$

We integrate both sides with respect to $s$
and set the integration constant to $1$,
such that we get zero when $\theta \to 0$ away from the wall:

$$\begin{aligned}
    \frac{L_c^2}{2} \Big( \dv{\theta}{s} \Big)^2
    = 1 - \cos\theta
\end{aligned}$$

Isolating this for $\idv{\theta}{s}$ and using a trigonometric identity then yields:

$$\begin{aligned}
    \dv{\theta}{s}
    = \frac{1}{L_c} \sqrt{2 (1 - \cos\theta)}
    = \frac{1}{L_c} \sqrt{4 \sin^2\!\Big( \frac{\theta}{2} \Big)}
    = - \frac{2}{L_c} \sin\!\Big( \frac{\theta}{2} \Big)
\end{aligned}$$

We use trigonometric relations on the equations
for $\idv{x}{s}$ and $\idv{y}{s}$ to get $\theta$-derivatives:

$$\begin{aligned}
    \dv{x}{\theta}
    &= \dv{x}{s} \dv{s}{\theta}
    = \bigg( 1 - 2 \sin^2\!\Big( \frac{\theta}{2} \Big) \bigg) \bigg( \dv{\theta}{s} \bigg)^{-1}
    = L_c \sin\!\Big( \frac{\theta}{2} \Big) - \frac{L_c}{2 \sin(\theta/2)}
    \\
    \dv{y}{\theta}
    &= \dv{y}{s} \dv{s}{\theta}
    = \bigg( 2 \sin\!\Big(\frac{\theta}{2}\Big) \cos\!\Big(\frac{\theta}{2}\Big) \bigg) \bigg( \dv{\theta}{s} \bigg)^{-1}
    = - L_c \cos\!\Big( \frac{\theta}{2} \Big)
\end{aligned}$$

Let $\theta_0 = \phi - \pi/2$ be the initial elevation angle $\theta(0)$ at the wall.
Then, by integrating the above equations, we get the following solutions:

$$\begin{gathered}
    \boxed{
        \frac{x}{L_c}
        = 2 \cos\!\Big(\frac{\theta_0}{2}\Big) + \log\!\bigg| \tan\!\Big(\frac{\theta_0}{4}\Big) \bigg|
        - 2 \cos\!\Big(\frac{\theta}{2}\Big) + \log\!\bigg| \tan\!\Big(\frac{\theta}{4}\Big) \bigg|
    }
    \\
    \boxed{
        \frac{y}{L_c}
        = - 2 \sin\!\Big(\frac{\theta}{2}\Big)
    }
\end{gathered}$$

Where the integration constant has been chosen such that $y \to 0$ for $\theta \to 0$ away from the wall,
and $x = 0$ for $\theta = \theta_0$.
This result is consistent with our earlier expression for $d$:

$$\begin{aligned}
    d
    = y(\theta_0)
    = - 2 L_c \sin\!\Big(\frac{\theta_0}{2}\Big)
    = 2 L_c \sin\!\Big( \frac{\pi/2 - \phi}{2} \Big)
\end{aligned}$$


## References
1.  B. Lautrup,
    *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
    CRC Press.