PREMEDICAL COURSE – PHYSICS
10. FLUID MECHANICS
PREMEDICAL COURSE – PHYSICS
10. Fluid Mechanics
Veronika Kollár - University of Pécs, Medical School, Department of Biophysics, 2014 August
www.biofizika.aok.pte.hu
OVERVIEW
Introduction
Ideal and real fluids
Hydrostatics
Hydrostatic pressure
Pascal’s Principle
Archimedes' principle
Surface tension
Hydrodynamics
Viscosity
Continuity law
Bernoulli's Equation
INTRODUCTION
All liquids are fluids but not all fluids are liquids. The scientist will make that distinction but the nonscientist frequently doesn't. Fluids are a substance that can flow. They include liquids and gases.
Liquids are a type of fluid that flows and takes the shape of its container but does not expand to fill
its container. (Gases do that.) Liquid is the second state of matter, between solid and gas.
Liquids do not expand, gases do. The form of liquids depends on the container. They show resistance
against volume changing forces and are incompressible. Their volume is constant.
Fluid mechanics is the study of fluids either in motion (fluid dynamics, hydrodynamics) or at rest
(fluid statics, hydrodynamics).
Density
The density of a fluid is its mass per unit volume:
Pressure
Pressure is the (compression) force exerted by a fluid per unit area.
Differences or gradients in pressure drive a fluid flow, especially in ducts and pipes.
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PREMEDICAL COURSE – PHYSICS
10. FLUID MECHANICS
IDEAL AND REAL LIQUIDS
An ideal fluid is usually defined as a fluid in which there is no friction; its viscosity is zero. Although
such a fluid does not exist in reality, many fluids approximate frictionless flow at sufficient distances
from solid boundaries, and so we can often conveniently analyse their behaviors by assuming an
ideal fluid.
In a real fluid, either liquid or gas, tangential or shearing forces always develop whenever there is
motion relative to a body, thus creating fluid friction, because these forces oppose the motion of one
particle past another. These friction forces give rise to a fluid property called viscosity.
HYDROSTATICS
Hydrostatics is the branch of fluid mechanics that studies fluids at rest.
HYDROSTATICS PRESSURE
The pressure exerted by a static fluid depends only upon the depth of the fluid, the density of the
fluid, and the acceleration of gravity. Hydrostatic pressure increases in proportion to depth measured
from the surface because of the increasing weight of fluid exerting downward force from above. The
hydrostatic pressure is independent of the form of the container, also proportional to the height (h)
of the column and the density (ρf) of the fluid.
ph = ρ f ⋅ h ⋅ g
Example 1
A glass tube which holes are covered by only plastic membrane is in a water filled container.
h
The pressure at a given depth is independent of direction -- it is the same in all directions. This is
another statement of the fact that pressure is not a vector and thus has no direction associated with
it when it is not in contact with some surface.
PASCAL’S PRINCIPLE
Pressure applied to an enclosed fluid at rest is transmitted undiminished to every portion of the fluid
and to the walls of the containing vessel.
p=
F1 F2
=
A1 A2
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PREMEDICAL COURSE – PHYSICS
10. FLUID MECHANICS
Example 2
Hydraulic lift (hydraulic jack).
A multiplication of force can be achieved by the application of fluid pressure according to Pascal's
principle, which for the two pistons implies P1 = P2. This allows the lifting of a heavy load with a
small force.
BUOYANT FORCE AND ARCHIMEDES' PRINCIPLE
Buoyant force
When an object is placed in a fluid, the fluid exerts an upward force we call the buoyant force. The
buoyant force comes from the pressure exerted on the object by the fluid.
F1 = p1 ⋅ A = h1 ⋅ ρ ⋅ g ⋅ A
F2 = p2 ⋅ A = h2 ⋅ ρ ⋅ g ⋅ A
· · !
·
Fbuoyant = ρ fluid ⋅ g ⋅ Vsubmerged
Archimedes' principle
According to Vitruvius, a crown for a
temple had been made for King Hiero II,
who had supplied the pure gold to be
used, and Archimedes was asked to
determine whether some silver had
been substituted by the dishonest
goldsmith. Archimedes had to solve the
problem without damaging the crown.
While taking a bath, he noticed that the
level of the water in the tub rose as he
got in, and realized that this effect
could be used to determine the volume
of the crown. For practical purposes water is incompressible,[so the submerged crown would
displace an amount of water equal to its own volume. By dividing the mass of the crown by the
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PREMEDICAL COURSE – PHYSICS
10. FLUID MECHANICS
volume of water displaced, the density of the crown could be obtained. This density would be lower
than that of gold if cheaper and less dense metals had been added.
Archimedes' principle states that any object immersed in a fluid is acted upon by an upward, or
buoyant, force equal to the weight of the fluid displaced by the object.
SURFACE TENSION
A liquid, being unable to expand freely, will form an interface
with a second liquid or gas. The cohesive forces between liquid
molecules are responsible for the phenomenon known as
surface tension. The molecules at the surface do not have other
like molecules on all sides of them and consequently they
cohere more strongly to those directly associated with them on
Liquid-gas-solid interface
the surface.
Surface tension: the work done per unit increase in the surface area of the liquid.
"
∆$ %
∆ HYDRODYNAMICS
Laminar flow: laminar flow (or streamline flow) occurs when a
fluid flows in parallel layers, with no disruption between the
layers.
Turbulent flow: turbulent flow is a flow regime characterized by
chaotic property changes. This includes low momentum
diffusion, high momentum convection, and rapid variation of
pressure and velocity in space and time.
Flow rate (IV)
The intensity of current is the quotient of the across flowing volume and the required time.
IV =
∆V
∆t
VISCOSITY
Viscosity is a measure of a fluid’s resistance to flow. Viscosity is due to the internal frictional force
that develops between different layers of fluids as they are forced to move relative to each other.
The behavior of a fluid in laminar flow between two parallel plates when the upper plate moves with
a constant velocity:
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PREMEDICAL COURSE – PHYSICS
10. FLUID MECHANICS
Shear force:
F =η * A*
∆v
∆h
EQUATION OF CONTINUITY
When fluids move through a full pipe, the volume of fluid that enters the pipe must equal the volume
of fluid that leaves the pipe, even if the diameter of the pipe changes. This is a restatement of the
law of conservation of mass for fluids. The volume of fluid moving through
gh the pipe at any point can
be quantified in terms of the volume flow rate, which is equal to the area of the pipe at that point
multiplied by the velocity of the fluid. This volume flow rate must be constant throughout the pipe,
therefore you can write the
he equation of continuity for fluids (also known as the fluid continuity
equation) as:
IV =
∆V A ⋅ v ⋅ ∆t
=
= A ⋅ v = const .
∆t
∆t
This equation says that as the cross-section
cross section of the pipe gets smaller, the velocity of the fluid
increases, and as the cross-section
section gets larger, the fluid velocity decreases.
Bernoulli's Equation
onservation of energy, when applied to fluids in motion, leads to Bernoulli’s Principle.
P
Bernoulli’s
Conservation
Principle states that fluids moving at higher velocities lead to lower pressures, and fluids moving at
lower velocities result in higher pressures. This principle is also used in sailboats, carburetors, gas
delivery systems, and
d even water-powered
water
sump pumps.
p1 +
ρ ⋅ v12
2
+ ρ ⋅ g ⋅ h1 = p2 +
ρ ⋅ v22
2
http://physics
http://physics.tutorcircle.com/fluid-dynamics/
+ ρ ⋅ g ⋅ h2 = const.
static
dynamics
hydrostatic energgy
Here P1 and P2 are the given pressures before and after the
flow, v1 and v2 are velocities of the two fluids, h1 and h2 are the height at which fluid flows, ρ is the
density of the fluid which is constant.
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PREMEDICAL COURSE – PHYSICS
10. FLUID MECHANICS
Symbols and units:
Symbol
ρ
p
ph
Unit
kg/m3
N/m2 or Pa (Pascal)
Pa
Formula
m/V
F/A
Fbuoyant
N (Newton)
Fbuoyant = ρ fluid ⋅ g ⋅Vsubmerged
Surface tension
α
N/m or J/m2
Viscosity
Flow rate
η (éta)
IV
Pa·s
m3/s or l/min
Density
Pressure
Hydrostatic
pressure
Buoyant force
ph = ρ f ⋅ h ⋅ g
α=
∆W
∆A
IV =
∆V
∆t
QUESTIONS
1. The current flow of the water is 3 cm3 per 1 s. Determine the velocity of the water where the
diameter of the tube is 0.5 cm or 0.8 cm.
2. 10 % of ball is dipped in the water. How big is the radius of the ball, if it weights 55 g?
3. With the help of an injection needle - which has 1 mm inner diameter and 10 cm length - we
want to give the patient 20 cm3 medicine which viscosity is 10-3 Pas. We have to do this in 4
minutes against 1600 Pa venous pressure. How much pressure do we have to use?
http://physicstutorsinterest.blogspot.hu/2010/06/hydraulic-machines.html
http://hyperphysics.phy-astr.gsu.edu/hbase/fluid.html#flucon
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