Fluid Mechanics

Force of cohesion:

The force of attraction between molecules of same substances is called force of cohesion.

Force of adhesion:

The force of attraction between molecules of different substance is called force of adhesion.

Surface tension:

It is the property of liquid at rest by virtue of which its surface behaves as stretched membrane and tries to occupy minimum possible surface area.

Mathematically, surface tension is the force per unit length.

Surface tension$(T)=\frac{Force(F)}{Length(L)}$

Its SI unit is N/m.

Force= Tension * Length

Or, F= T*L

Dimension formula of surface tension is [MT^-2].

 

Surface Energy:

The potential energy per unit surface area is called surface energy. It is also defined as the amount of work done to increase the area of the film through unity.

Surface energy=$\frac{Work\,done\,in\,increasing\,surface\,area}{Increase\,in\,surface\,area}$

Relation between surface tension and surface energy:

Suppose a rectangular frame ABCD is dipped in a soap solution and then took out. A thin film is formed. If the movable BC is pulled through x against the surface tension force F, the work done is,

W= Fx……(i)

Again, surface tension force is,

F= T*2l……(ii)

As the length (BC)l is in contact with two force surfaces, 2l is taken for length.

From equations (i) and (ii), we get,

W=T2lx

We have,

Surface energy $(\sigma)=\frac{work done}{increase\,in\,area}$

Or, $\sigma=\frac{T2lx}{2lx}$

Or, $\sigma=T$

Therefore, surface energy is equal to the surface tension.

 

Molecular theory of surface tension:

A molecule lying well below the surface of the liquid is attached by molecules from all direction while the molecules lying on the surface of liquid experiences only downward force. As a result, the surface of the liquid is behaves as an stretched membrane.

 

Excess pressure inside curved surface:

 

As shown in the figure, the downward horizontal components of tension cancel to each other while the vertical component add up. As a result, pressure inside the convex surface increases.

 

Excess pressure inside liquid drop:

Suppose a spherical drop of radius R. Let the radius of the drop is increased through dR, due to the excess pressure inside the drop.

Work done to increase the drop= Force*displacement

                                                          = pressure* area* displacement

                                                          = $P4\pi{R^2dR}$

Increase in area= $4\pi{(R+dR)^2}-4\pi{R^2}$

=$4\pi{R^2}+8\pi{RdR}+dR^2-4\pi{R^2}$

=$8\pi{RdR}$[As dR is very small, dR² can be neglected]

We have,

Surface energy=$\frac{workdone}{increase\,in\,area}$

Or, $\sigma=\frac{P4\pi{r^2dR}}{8\pi{RdR}}$

Or, $T=\frac{PR}{2}$ $[\because\sigma=T]$

Or, $P=\frac{2T}{R}$

Or, $P_{in}-P_{out}=\frac{2T}{R}$

 

This gives excess pressure inside liquid drop.

Again, excess pressure inside the liquid bubble or soap bubble is,

$P_{in}-P_{out}=\frac{4T}{R}$

 

Angle of contact:

 It is the angle between the tangent on the liquid surface and solid surface inside the liquid at the point of contact.

 

In case of water and glass vessel the adhesive force is greater than the cohesive force so the concave meniscus is formed and the angle of contact is acute.

In case of mercury and glass vessel convex meniscus is formed because cohesive force of mercury is greater than the adhesive force of mercury and glass vessel. As a result the angle of contact is obtuse.

 

Capillarity or capillary action:

The rise or fall of liquid in a capillary tube is called capillarity. 

Capillary tube:

A tube of very fine bore is called capillary tube.

Applications:

• Oil rises in cotton wicks of lamp
• A nib of a pen is split so that the ink rises.
• Rising of sap in a tree due to capillary action.
• Using a towel to dry our body after bathing.