Lecture
Fluids
Lecture questions
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Ideal fluid . Viscous fluid. Stationary flow
The continuity equation
Bernoulli’s law
Poiseuille's Law
Surface tension
Capillary Action
Fluids
Fluids include liquids, gases, and plasmas.
•
All fluids have the property of fluidity, the ability to flow
(also described as the ability to take on the shape of the
container).
• An ideal fluid (perfect fluid) has no viscosity. It is a
frictionless fluid. The flow of a fluid that is assumed to have
no viscosity is called inviscid flow. The ideal fluids can
only be subjected to normal, compressive stress which is
called pressure. Real fluids display viscosity and so are
capable of being subjected to low levels of shear stress.
• Steady-state (stationary) flow refers to the condition
where the fluid properties at any single point in the system
do not change over time. These fluid properties include
temperature, pressure, and velocity.
The continuity equation
• In the steady-state case for incompressible and inviscid
flow product of a cross-sectional area (S) of a tube and
fluid velocity (V) is constant. Hence, a reduced cross
section gives a greater velocity.
SV=const
• The product SV is the volume flowrate.
Bernoulli’s law
• In a stationary flow of ideal liquid
p
v 2
2
 gh  const
• where p is the static pressure (in Newtons per square meter), ρ
is the fluid density (in kg per cubic meter), v is the velocity of
fluid flow (in meters per second) and h is the height above a
reference surface. The second term in this equation is known
as the dynamic pressure.
Viscosity
• The viscosity of a fluid can be defined as the
measure of how resistive the fluid is to flow. It
is analogous to the friction of solid bodies in
that it also serves as a mechanics for
transforming kinetic energy into thermal
energy. Given two plane parallel plates
separated by a distance and with a fluid
between them. The resistance to flow in a
liquid can be characterized in terms of the
viscosity of the fluid if the flow is smooth. In
the case of a moving plate in a liquid, it is
found that there is a layer or lamina which
moves with the plate, and a layer which is
essentially stationary if it is next to a stationary
plate.
Viscosity
Fluid Velocity Profile
Under conditions of laminar flow in a viscous fluid, the velocity
increases toward the center of a tube.
The velocity profile as a function of radius is
Poiseuille's Law
• In the case of smooth flow (laminar flow), the volume
flowrate is given by the pressure difference divided
by the viscous resistance. This resistance depends
linearly upon the viscosity and the length, but the
fourth power dependence upon the radius is
dramatically different. Poiseuille's law is found to be
in reasonable agreement with experiment for uniform
liquids (called Newtonian fluids) in cases where there
Surface tension
The cohesive forces among the liquid molecules
are responsible for phenomenon of surface
tension. In the bulk of the liquid, each molecule is
pulled equally in every direction by neighboring
liquid molecules, resulting in a net force of zero.
The molecules at the surface do not have other
molecules on all sides of them and therefore are
pulled inwards. This creates some internal pressure
and forces liquid surfaces to contract to the
minimal area.
Surface tension is a property of the surface of a
liquid that allows it to resist an external force. It is
revealed in floating of some objects on the surface
of water, even though they are denser than water,
in the ability of some insects (e.g. water striders)
to run on the water surface.
Surface tension (in terms of energy)
Another way to view it is in terms of
energy. A molecule in contact with a
neighbor is in a lower state of energy than
if it were alone (not in contact with a
neighbor). The interior molecules have as
many neighbors as they can possibly
have, but the boundary molecules are
missing neighbors (compared to interior
molecules) and therefore have a higher
energy. For the liquid to minimize its
energy state, the number of higher energy
boundary molecules must be minimized. Water beading on a leaf
The minimized quantity of boundary
molecules results in a minimized surface
area.
Two definitions
There is a flat soap film bounded on one
side by a taut thread of length, L. Surface
tension, represented by the symbol σ is
defined as the force along a line of unit
length, where the force is tangent to the
surface but perpendicular to the line. The
thread will be pulled toward the interior of
the film by a force equal to 2σL (the factor
of 2 is because the soap film has two sides,
hence two surfaces).Surface tension is
therefore measured in forces per unit length.
An equivalent definition is work done per
unit area. As such, in order to increase the
surface area a quantity of work A is needed.
This work is stored as potential energy.
Since mechanical systems try to find a state
of minimum potential energy, a free droplet
of liquid naturally assumes a spherical
shape, which has the minimum surface area
for a given volume.
Diagram shows, in cross-section, a
needle floating on the surface of
water. Its weight, Fw, depresses the
surface, and is balanced by the
surface tension forces on either side,
Fs, which are each parallel to the
water's surface at the points where it
contacts the needle. Notice that the
horizontal components of the two Fs
arrows point in opposite directions, so
they cancel each other, but the
vertical components point in the same
direction and therefore add up to
balance Fw.
Surface curvature and pressure
LaPlace's Law
 1

1

pL    
R R 
y 
 x
p is the pressure
σ is surface tension.
Rx and Ry are radii of
curvature in each of the axes
that are parallel to the surface
Cohesion and Adhesion
Molecules
liquid
state
experience
strong
intermolecular
attractive
forces. When those forces are
between like molecules, they
are referred to as cohesive
forces. When the attractive
forces are between unlike
molecules, they are said to be
adhesive forces. The adhesive
forces
between
water
molecules and the walls of a
glass tube are stronger than
the cohesive forces lead to an
Forces at contact point shown for contact
angle greater than 90° (left) and less
than 90° (right)
Capillary Action
Capillary action is the result of
adhesion and surface tension.
Adhesion of water to the walls of a
vessel will cause an upward force
on the liquid at the edges and result
in a meniscus which turns upward.
The surface tension acts to hold the
surface intact, so instead of just the
edges moving upward, the whole
liquid surface is dragged upward.
Capillary Action
Capillary action is the result of
adhesion and surface tension.
Adhesion of water to the walls of a
vessel will cause an upward force
on the liquid at the edges and result
in a meniscus which turns upward.
The surface tension acts to hold the
surface intact, so instead of just the
edges moving upward, the whole
liquid surface is dragged upward.
Thank you for your attention !