Surface Tension

Some consequences of

surface tension.

At the interface between a liquid and a gas, or

between immiscible liquids, cohesive forces acting

on the liquid molecules are unbalanced.

The apparent physical consequence of this

unbalanced force is the creation of a surface

tension/hypothetical skin.

A tensile force due to molecular attraction may be

considered to be acting in the plane of the surface

along any line.

For a given liquid, the surface tension depends on

the temperature as well as the other fluid in

contact.

Due to surface tension, a steel needle may float on

a liquid surface.

Similarly, a small droplet of mercury will form into a

sphere due to large cohesive forces holding the

molecules together.

The origin of surface tension

Figure shows a liquid in contact with its vapor.

There is a gradual change of properties in the interface region

between the phases, which is about 1nm thick. Molecules are

attracted to each other: those at the surface of a liquid have no

molecules above them, resulting in a net attractive force

inwards.

We just noted that:

At the interface between a liquid and a gas,

or between immiscible liquids, cohesive

forces acting on the liquid molecules are

unbalanced.

Capillary action of water

compared to mercury

When the lower end of a vertical glass

tube is placed in a liquid such as

water, a concave meniscus forms.

Surface tension pulls the liquid column

up until there is a sufficient mass of

liquid for gravitational forces to

overcome the intermolecular forces.

With some pairs of materials, such as

mercury and glass, the interatomic

forces within the liquid exceed those

between the solid and the liquid, so a

convex meniscus forms and capillary

action works in reverse

The edge of the water that sticks up above

the water's surface is called a "meniscus."

2r

2rs

The weight of the liquid column is approximately:

W=mg=Vg= g(r2h)

Equating the vertical component of the surface tension

force to the weight gives:

W=Fsurface

g(r2h)= 2rs cos

The height h of a liquid column is given by:

where:

s is the liquid-air surface tension (energy/area)

is the contact angle

is the density of liquid (mass/volume)

g is acceleration due to gravity (length/time2)

r is radius of tube (length).

2r

2rs

Example:

For a water-filled glass tube in air at sea level, using SI units:

s is 0.0728 N/m at 20C

is 20 (0.35 rad)

is 1000 kg/m3

g is 9.81 m/s

therefore, the height of the water column is given by:

Thus for a 2 m wide (1 m radius) tube, the water would rise an unnoticeable 0.014 mm.

For a 2 cm wide (0.01 m radius) tube, the water would rise 1.4 mm, For a 0.2 mm wide (0.0001 m radius) tube, the water would rise 140 mm.

h

r1 r2

Water Mercury

h

h

Because of the surface tension, some liquids can

adhere to solid surfaces. In such fluid-surface

systems, the fluid is said to wet the surface.

Other fluids can be nonwetting because of strong

cohesion between liquid molecules, liquid in a tube is

depressed. A ball-like mercury droplets form solid

surfaces, mercury exhibits nonwetting behaviour.

Wetting forces equilibrium diagram

SV, the surface tension between the solid (for example, a laminate surface) and the surrounding vapour (for example, gas or flux)

SL, the surface tension between the solid and the liquid (for example, solder or adhesive)

LV, the surface tension between the liquid and the surrounding vapour

SVLV

SL

The capillary rise of water and the capillary

fall of mercury in a small-diameter glass tube.

The forces acting on a liquid column that

has risen in a tube due to the capillary

effect.

For non-wetting fluids like Mercury

h

Viscosity

Viscosity

Consider a fluid placed between plates separated by a

small distance h as shown:

Assume a force P is applied on the upper plate, and the

plate moves with a constant velocity U.

P

U

h

Viscosity

A closer inspection reveals that the fluid in contact with the

upper plate moves with the plate velocity, U, and the fluid

in contact with the bottom surface has a zero velocity.

The experimental observation that the fluid sticks to the

solid boundaries is usually referred as the no-slipcondition. All fluids satisfy no-slip condition.

P

U

h

The fluid between the plates moves with velocity

u=u(y)

that would be found to vary linearly

The velocity gradient developed in the fluid between plates

would be a constant

P

U

h

P: is the force applied to the upper plate

The experimental data show that for common fluids such as

water, oil and air, the shearing stress, , is linearly

proportional to the U/h ratio, that is the velocity gradient:

or

Where the constant of proportionality is designed by

(mu) and is called Absolute (dynamic) viscosity of the fluid

Viscosity is a property of a fluid that affects the shear

stress developed within the fluid as a result of its

motion.

In fluids, shear resistance is independent of the normal

force (pressure) acting within the fluid. In contrast, in

solids, shear resistance is totally dependent on the

normal force.

Consider a fluid flow, where all the fluid is moving in the

same direction but with a speed that varies in a

perpendicular direction;

That is the only non-zero component of the velocity is the x-

component, u, and it is a function of y coordinate u(y).

x

y

u (y)

A B

Shear stress

Across any plane perpendicular to y within the fluid a stress

will act (see line AB)

The faster fluid above the plane will drag the fluid below

forward, and the slower fluid below will drag the fluid

above back.

x

y

u (y)

A B

Shear stress

xy

u (y)

A B

Shear stress

Equal and opposite forces will thus act on

the fluid above and below.

The generation of this internal stress is

known as viscous stresses

If viscosity is zero, the thermal conductivity of the fluid is also zero

The dimension of viscosity () is FTL-2.

In SI units, it is given as N.s/m2

In flow problems, the viscosity often appears combined with density in the form:

This ratio is called the kinematic viscosity (nu)

Its dimension is L2/T

In SI units: m2/s

Newtonian Fluids

Fluids for which the shearing stress is

linearly related to the rate of shearing

strain (also known as the rate of angular

deformation) is designated as Newtonian

Fluids.

Non-Newtonian behavior can arise in

liquids with long molecules, suspensions,

emulsions such as blood, paints. Non-

Newtonian fluids are classified as: shear

thickening, shear thinning, ideal Bingham

plastic. Most non-Newtonian fluids are of

shear thinning.

Bingham plastic, such as

toothpaste, can withstand a

finite shear stress without any

motion, however it moves like

a fluid once this yield stress is

exceeded

(Shear thinning)

(Shear

thickening)

The actual value of viscosity depends on the

particular fluid, and for a particular fluid the

viscosity is also highly dependent on temperature.

The effect of pressure is usually neglected.

The viscosity of liquids decreases with an increase in temperature

The viscosity of gases increase with an increase in temperature.

The viscosity of gases increase with an increase in temperature.

For gases the empirical Sutherland equation

gives the effect of temperature on viscosity.

Where C and S are empirical constants.

T is temperature in Kelvin

The rate of deformation (velocity gradient)

of a Newtonian fluid is proportional to

shear stress, and the constant of

proportionality is the viscosity.

Variation of shear stress with the

rate of deformation for Newtonian

and non-Newtonian fluids (the slope

of a curve at a point is the apparent

viscosity of the fluid at that point).

Shear thinning

Shear thickening

33

The variation of

dynamic

(absolute)

viscosity of

common fluids

with temperature

at 1 atm

Standard Atmosphere

Although pressure and density in the earths atmosphere varies with altitude, a

standard atmosphere is used in the design of aircraft, missiles, and spacecraft, and in comparing their performance under standard conditions.

Properties of Standard Atmosphere at sea level:

Temperature, T 288.15 K (15C)Pressure, p 101325 Pa (absolute pressure)

Density, 1.225 kg/m3

Specific Weight, s 12.014 N/m3

Viscosity, 1.789x10-5 Ns/m2

gravity, g 9.81 m/s2