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Surface  tension  

Lecture 3 Fluid  surfaces  

At the boundary between two fluids (e.g. water and air) we get a free surface.

The free surface must be •  perpendicular to the force acting on the liquid

(sea level!)

•  at the same pressure as the fluid above (why?)

In our fluid model, we don’t predict any other effects.

Surface  tension  

In reality, the surface of any liquid behaves as though it is covered by a stretched membrane. Small insects can walk on water without getting wet. Why?

Recall that there is an attractive force between molecules.

net force into liquid

net force zero

This cohesive force makes the surface of the liquid behave like a elastic sheet under tension.

The strength of this surface membrane varies for different liquids.

e.g. the surface tension of soapy water is much less than for pure water.

Surface tension has dimensions [force]/[length]

The net inward force means that the surface is less favourable energetically than the bulk of the liquid, so there is an excess energy at the surface called the surface energy density. Liquids always assume a shape which minimises the total surface energy by minimising the surface area.

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This explains why liquid drops are spherical, because a sphere has the smallest possible surface area for a given volume.

For a liquid, the surface tension is equal to the surface energy density, e.g. for water

surface energy density = 0.072 J/m2

surface tension = 0.072 N/m Dimensions:

[force/length] = kg.m.s–2/m = kg.s–2

[energy/area] = kg.m2.s–2/m2 = kg.s–2

Example: Compare the surface energy of a drop of water with that of two drops, each with half the volume of the first.

Adherence  

Fluid in contact with a surface forms an angle of contact φ: this is called adherence or wetting. Water wets glass completely: φ ~ 0o

water mercury

glass

φ< 90o φ> 90o

water mercury

lead

Adherence depends on chemistry: if the fluid molecules are more strongly attracted to the solid than each other, the adherence or wetting is high. e.g. water molecules are strongly attracted to glass (hydrophilic) and repelled from materials like oil (hydrophobic).

Look at a close-up of the surface of water where it meets a glass wall. Water wets glass completely ⇒ φ small. Each surface water molecule exerts a force on its neighbour. Equilibrium ⇒ the glass exerts a force on the water.

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Note the direction that the surface tension pulls in: it pulls along the surface.

But when water is in contact with e.g. glass, the contact angle φ is so small ⇒ this force is vertical (parallel to the surface of the glass).

water

water + soap

water

glass slide

This is how surface tension is measured: using a glass slide dipped into the fluid.

Experimentally, it is found that the force exerted by a liquid on a body is proportional to the length of the line of contact:

F = γ l

The constant of proportionality γ is the surface tension of the liquid.

Surface tension has units of N/m. (values at 20o C)

Liquid Surface tension mN/m

ethanol 22.1

water 72.8

mercury 425.4

Pressure  inside  a  bubble  

Consider a bubble of radius r.

Imagine a plane separating the bubble into two hemispheres. The internal pressure P tends to push the hemispheres apart, but the surface tension acting around the circumference tends to pull them together. Fst

p radius r

In zero gravity, fluid behaviour is dominated by surface tension effects.

Capillarity  

Because of surface tension, water will creep up narrow tubes: capillary action.

h

r Upward force = γl = 2πrγ Downward force = mg = (πr2h) ρg

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Note that if the contact angle is greater than 90o (e.g. mercury in a glass container), the liquid will be depressed rather than lifted.

QuesAon  

Can capillary action explain the rise of sap in trees?

Assume sap is mainly water: ρ = 103 kg m–3, γ = 7.28×10–2 N m–1

Radius of capillaries r = 2.5×10–5 m

Next  lecture  

Fluid dynamics