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
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