Physics > Fluid Mechanics > 9.0 Angle of contact
Fluid Mechanics
1.0 Introduction
1.1 Ideal liquid
1.2 Density of a liquid $\left( \rho \right)$
1.3 Relative density of a liquid $(RD)$
1.4 Density of a mixture of two or more liquid
1.5 Density variation with temperature
1.6 Density variation with pressure
2.0 Fluid pressure
2.1 Atmospheric pressure
2.2 Pressure variation with depth
2.3 Measurement of pressure
2.4 Pressure difference in accelerating fluids
3.0 Pascal's law
4.0 Buoyant force
5.0 Flow of fluids
6.0 Viscosity
7.0 Stoke's law
8.0 Intermolecular forces
9.0 Angle of contact
9.1 Capillarity
1.2 Density of a liquid $\left( \rho \right)$
1.3 Relative density of a liquid $(RD)$
1.4 Density of a mixture of two or more liquid
1.5 Density variation with temperature
1.6 Density variation with pressure
2.2 Pressure variation with depth
2.3 Measurement of pressure
2.4 Pressure difference in accelerating fluids
The phenomenon of rise or fall of liquid in a capillary tube is known as capillarity.
Surface tension causes the elevation or depression of the liquid in a capillary tube.
Consider a glass capillary of radius $R$ dipped in water as shown in the figure.
The pressure below the meniscus of radius $r$ is given by,
From excess pressure we can write, $${P_O} - {P_1} = \frac{{2T}}{r}$$ So, $${P_1} = \left( {{P_O} - \frac{{2T}}{r}} \right)\quad ...(i)$$ We know, $${P_2} = {P_1} + \rho gh\quad ...(ii)$$ As the bigger concave surface is assumed to be flat. So, $${P_O} = {P_3}$$ Also, $${P_3} = {P_2}$$ So, $${P_2} = {P_O}\quad ...(iii)$$
From equation $(i)$, $(ii)$ and $(iii)$ we get, $$\begin{equation} \begin{aligned} {P_O} = {P_O} - \frac{{2T}}{r} + \rho gh \\ h = \frac{{2T}}{{r\rho g}} \\\end{aligned} \end{equation} $$
Relation between $R$ and $r$
$$\cos \theta = \frac{R}{r}$$ or $$r = \frac{R}{{\cos \theta }}$$
Therefore, $$h = \frac{{2T\cos \theta }}{{R\rho g}}$$ where,
$T$: Surface tension
$\theta $: Angle of contact
$\rho $: Density of the liquid
$R$: Radius of the capillary tube
$r$: Radius of the meniscus
$g$: Acceleration due to gravity
Note:
- If $\theta < 90^\circ $
The meniscus is concave. So, $h$ will be positive and the liquid will rise in the capillary tube.
- If $\theta > 90^\circ $
The meniscus is convex. So, $h$ will be negative and the liquid will rise in the capillary tube.
- If $\theta = 90^\circ $
The meniscus is concave, then $h=0$.
So, no phenomenon of capillarity.
