Fluid
• Concept of Fluid
• Density
• Pressure: Pressure in a Fluid. Pascal´s principle
• Buoyancy. Archimede´s Principle.
• Forces on submerged surfaces
Fluids in motion
• Continuity equation.
• Bernoulli´s Equation
• Viscous Flow. Viscosity. Poiseuille´s Law. Skin
drag
• turbulent flow.
References: Tipler; wikipedia,…
FLUID. Fluids in Motion
Fluids in Motion . Description of the Fluid Flow
A fluid flow may be described in two different ways: the Lagrangian approach
(named after the French mathematician Joseph Louis Lagrange), and the Eulerian
approach (named after Leonhard Euler, a famous Swiss mathematician). In the
Lagrangian approach, one particle is chosen and is followed as it moves through
space with time. The line traced out by that one particle is called a particle pathline.
A Eulerian approach is used to obtain a clearer idea of the flow at one particular
instant. The entire flow field is easily visualized. The lines comprising this flow field
are called streamlines. A streamline is a line drawn through a fluid in such a
manner that it has the direction of the fluid velocity at every point.
Thus, a pathline refers to the trace of a single particle in time and space whereas a
streamline presents the line of motion of many particles at a fixed time. Both
coincide in a steady flow.
Stream tube
m, V,ρ, v, ..
path line
streamline
FLUID. Fluids in Motion
Fluids in Motion . Description of the Fluid Flow
flowing from left to right, inside of a stream tube ….
Cross-sectional Area A1
Cross-sectional Area A2
Volume flow rate…
Mass flow rate…
through a cross-section
Terminology:
Compressible vs incompressible flow
Viscous vs No-Viscous flow
Steady vs unsteady flow
Laminar vs turbulent flow
Newtonian vs non-Newtonian fluids
What is the volume flow rate
going in [leaving from] the tube?
What is the volume flow rate
that travels through the tube?
FLUID. Basic Equations of Fluids in Motion. Continuity Equation
v1; v2 : Average velocity over the crosssections A1; A2. Then, no variation in velocity
in the cross-section will be considered
Continuity equation
A2
v2
v1
A1
The mass flow rate going in the tube through the cross section A1 :
ρ1 A1 v1
The mass flow rate leaving from the tube through the cross-section A2 ρ2 A2 v2
For a steady flow, from the general law of conservation of mass,
1 v1 A1  2 v2 A2
Mass flow rate has to be the
same through any cross-section
along the stream tube
For the incompressible flow case
1  2
Volume flow rate [discharge rate] passing any
cross-section along the stream tube will be the same
v1 A1  v2 A2
Blood flows in an aorta of radius 1 cm at 30 cm/s. What is the volume flow rate?. What is the pumping
rate of the heart?. Give this result in liters per minute. If the aorta reduce its radius to 0,5 cm because of
thickening of the arterial walls (arteriosclerosis), what is the speed of the blood in the narrower region?.
FLUID. Fluids in Motion. Steady and Incompressible flow
Steady vs unsteady flow
When all the magnitudes describing the flow in one point of space no
vary with time, the flow is considered to be steady. Otherwise, it is called
unsteady.
Compressible vs incompressible flow
All fluids are compressible to some extent, that is changes in pressure or
temperature will result in changes in density. However, in many situations
the changes in pressure and temperature are sufficiently small that the
changes in density are negligible. In this case the flow can be modeled as
an incompressible flow. In a incompressible flow the density is constant
throughout the fluid. Per example, to assume it for liquids in most
situations is an excellent approximation
FLUID. Fluids in Motion. Viscous and non viscous flow
Viscous vs inviscid flow
Viscous problems are those in which fluid friction has significant effects on the fluid
motion. Non viscous flow implies no dissipation of mechanical energy.
For fluids having relatively small viscosity- water, air,..-, the effect of internal friction is
appreciable only in a narrow region surrounding the fluid boundaries –boundary-layer
hypothesisThe no-slip condition for viscous fluid states that at a solid boundary, the fluid will
have zero velocity relative to the boundary. The fluid velocity at all liquid-solid
boundaries is equal to that of the solid boundary. Conceptually, one can think of the
outermost molecules of fluid stick to the surfaces past which it flows.
Drag: A object submerged in a flowing fluid is subjected to a fluid force component on
it in the direction of the approach velocity, called the drag. The drag force is caused by
skin friction drag, which is caused by shear stress component (viscous effect) in the
flow direction, and by form, or profile, drag, caused by the lower pressure on the
downstream side of the object.
FLUID. Fluids in Motion. Turbulent flow
Laminar vs
turbulent flow
Turbulence is
flow dominated
by recirculation,
eddies, and
apparent
randomness.
Flow in which
turbulence is not
exhibited is
called laminar.
Newtonian vs non-Newtonian fluids
Sir Isaac Newton showed how stress and the rate of strain are very close to linearly related for many familiar
fluids, such as water and air. These Newtonian fluids are modeled by a coefficient called viscosity, which
depends on the specific fluid.
However, some of the other materials, such as emulsions and slurries and some visco-elastic materials (eg.
blood, some polymers), have more complicated non-Newtonian stress-strain behaviours. These materials
include sticky liquids such as latex, honey, and lubricants which are studied in the sub-discipline of rheology.
FLUID. Basic Equations of Fluid in Motion: Bernoulli´s Equation
Bernoulli´s Equation
Consider a fluid flowing in a tube as shown in the
figure in a steady, incompressible, no viscous flow.
We apply the work-energy theorem to a sample of
fluid initially contained between point 1 and 2.
During time Δt this sample moves along the tube to
the region between points 1´and 2´. So,
Wall forces = ΔK [Change of kinetic Energy]
Wall forces include gravitational and pressure forces.
We neglecting internal frictional forces. Nonviscous flow. No mechanical dissipation energy is
considered.
Work of gravitational forces can be computed as
the variation of potential energy of sample.
Considering the continuity equation we can obtain
 U  (m) g (h1  h2 )   g V (h1  h2 )
Change of kinetic energy of sample will be
K  12 (m)(v22  v12 )  12  V (v22  v12 )
Fluid moving in a pipe that varies in both
height and cross-sectional area. The net
effect on the sample during a time Δt is that
a mass initially at height h1 and speed v1 is
transferred to a height h2 with speed v2
FLUID. Basic Equations of Fluid in Motion: Bernoulli´s Equation
The work done by the pressure forces exerted
by the fluid behind the sample and by the fluid
in front of the sample will be
WF1  P1 A1 v1t  P1V
WF2   P2 A2 v2 t   P2 V
Applying the work-energy theorem we can
obtain
( P1  P2 )V   V g (h2  h1 )  12  V (v22  v12 )
Units: Joule (energy)
If we divide both sides by ΔV
( P1  P2 )   g (h2  h1 )   (v  v )
1
2
2
2
2
1
Units: Joule (energy) per unit of
volume
We can rearrange the terms
P1   g h1  12  v12  P2   g h2  12  v22 or P   g h  12  v2  const
The Bernoulli Equation states that the energy is the same at any two
points along a streamline in steady, incompressible and frictionless flow.
FLUID. Basic Equations of Fluid in Motion: Bernoulli Equation
Remarks about Bernoulli Equation
P1   g h1  12  v12  P2   g h2  12  v22
P   g h  12  v 2  const
P
v2
h 
 const 2
g
2g
P
g
h
v2
2g
(2)
(1)
This combination of quantities has the
same value at any point along of the
stream tube.
Each term having the same units (1) [Energy per
unit of volume] (2) [Energy per unit of weight].
Expression (1) can also be considered as energy
per volume flow rate, and (2) as energy per
weight flow rate
Flow work or pressure energy; Is the portion of the potential energy
term that the fluid is capable of yielding of its sustained pressure. IS
Unit: Joule per Newton (meter). Dimension : Length
Potential energy, due to the gravitational field. IS Unit: meter
Kinetic energy. IS Unit: meter (Joule per Newton)
The Bernoulli equation aids in solving problem in which the losses due to internal friction
(viscous flow) can be neglected or corrected for application of an experimentally determined
coefficient.
Bernoulli´s equation can also be obtained by integrating the Euler´s equation, which is derived
applying directly the Newton´s laws to a particle of the fluid
FLUID. Basic Equations of Fluid in Motion:
Bernoulli Equation. Applications
P1   g h1   v  P2   g h2   v
1
2
2
1
1
2
2
2
The Bernoulli Equation states that the energy is the
same at any two points along a streamline in
steady, incompressible and frictionless flow.
Torricelli´s Law
In the case of a tank of liquid having a nozzle on the
side at a distance Δh below the surface of water, we
can consider that a streamline connects points a and
b, and applying Bernoulli equation to each point
Pa   g ha  12  va2  Pb   g hb  12  vb2
Considering same pressure in each point (free atmosphere)
and v1, velocity at the surface, almost nil (steady flow)
vb  2 g h
Torricelli´s law: The water emerges from
the nozzle with a speed equal it would
have if it dropped in free-fall distance Δh
Exercise. A large reservoir of water, open at the top has a hole of diameter 10 cm, placed a
distance of 10 m below of surface of the water. (a) Calculate the speed of the water as it
flows out of the nozzle. (b) What is the emerging volume flow rate? © What is the emerging
mass flow rate. If the emerging water moves a turbine (d) What is the maximum power can
we obtain if the efficiency of turbine were 100%.
FLUID. Basic Equations of Fluid in Motion:
Bernoulli Equation. Applications
P1   g h1   v  P2   g h2   v
1
2
2
1
The venturi meter
1
2
2
2
The Bernoulli Equation states that the energy is the
same at any two points along a streamline in
steady, incompressible and frictionless flow.
1
2
The venturi meter shown in the figure is used to
measure the rate of flow of a fluid in a close conduit
(pipe). It consists of a converging portion from 1, to a
throat section at 2. The Bernoulli equation for point 1
and 2 can be written as above, and considering that h1
is equal to h2, we can obtain
When the speed of a fluid increase, the pressure
1
1
2
2
drops (in a horizontal close conduit). This effect is
P1  2  v1  P2  2  v2
called “Venturi effect”
By use of continuity equation
The speed in the throat section increase,
A1v1  A2v2
then the pressure at section 2 drops.
The discharge rate, Q, (volume flow
rate), will be
Q  A1v1  A2v2  A1 A2
2( P1  P2 )
 ( A12  A22 )
To obtain the discharge rate we need to measure the pressure difference
between the section 1 and the section 2
FLUID. Basic Equations of Fluid in Motion:
The Venturi meter
Q  A1v1  A2v2  A1 A2
Bernoulli Equation. Applications
2( P1  P2 )
 ( A12  A22 )
Measuring the pressure difference P1-P2
h1
1
2
h2
1.-Inserting a vertical pipe open at the top in each
section we can measure the pressure at section 1
and at section 2.
Applying hydrostatic equation, since the column is
at rest
P P  gh
1
atm
1
P2  Patm   g h2
P1  P2   g (h1  h2 )   g h
2.- We can insert a U-tube manometer partially
filled with a liquid of density ρL, connecting the
section 1 and section 2, as shown in figure.
In this case the difference of pressure will be
determined from the difference of height at the
manometer
P1  P2  L g h
FLUID. Basic Equations of Fluid in Motion: Bernoulli Equation
Energy equation
The Bernoulli equation aids in solving problem in which the losses due to internal
friction (viscous flow) can be accounted experimentally by a determined coefficient.
So, we can write
P1   g h1  12  v12  P2   g h2  12  v22  losses 1  2
Energy per
unit of volume
The available energy at 1 is equal to the available energy at 2, plus all the losses
between the two sections.
Pumps, Turbines
If a pump were in action between the sections 1 and 2, its total dynamic
head would appear in the loss term as a negative quantity. Similarly a
turbine would be treated as a loss.
So, having in account that
Power/Volume flow rate = Energy per unit of volume
P1   g h1  12  v12  P2   g h2  12  v22 
Power
Q
FLUID. Basic Equations of Fluid in Motion:
Bernoulli Equation. Applications
siphon
A siphon is a device for transferring a liquid from
one container to another . The tube shown in
figure must be filled to start the siphon, but once
this has been done, fluid will flow through the
tube. (a) Using Bernoulli´s equation, show that
the speed of water in the tube is v = √(2gd). (b)
What is the pressure at the highest part of the
tube?. (c) When the flow in the siphon will stop?
Pitot-static tube
Figure shows a Pitot-static tube, a device used
for measuring the speed o a gas. The inner
pipe faces the incoming fluid, while the ring of
holes in the outer tube is parallel to the gas
flow. Show that the speed of the gas is given
by v2= 2gh(ρL-ρg)/ ρg
ρL density of the liquid used in the manometer
ρg density of the gas
FLUID. Basic Equations of Fluid in Motion:
Bernoulli Equation. Applications
Water flows through the pipe in figure and
exits to the atmosphere at the right end of
section C. The diameter of the pipe is 2.0
cm at A; 1.00 cm at B and 0.800 cm at C.
The gauge pressure in the pipe at the
center of section A is 1.22 atm and the flow
rate is 0.8 liter/s. The vertical pipes are
open to the air at the top. Find the level of
the liquid-air interfaces in the two vertical
pipes. Assume laminar non-viscous flow.
FLUID. Basic Equations of Fluid in Motion:
Bernoulli Equation. Applications.
Airplane in flight
(a) Field velocity of the flowing
air in a wing (b) Field pressure
http://www.inta.es/descubreAprende/H
echos/Hechos05.htm
With the airplane in flight, an air flow passes over
the wing, and the shape of the wing's cross
section--curved above, flat or nearly flat below-reduces the pressure on top, causing the extra
pressure from below to exert a lifting force ("lift").
The lift increases if the front of the wing is raised
slightly, biting into the moving air at a small angle
("angle of attack"), and for a given lifting force,
this kind of wing produces much less air resistance
("drag") than a flat kite.
FLUID. Basic Equations of Fluid in Motion:
Atomizer
Bernoulli Equation. Applications
FLUID. Basic Equations of Fluid in Motion:
Viscous flow. Viscosity
Shear Stress Fs/A
Fs
v

A
z
in a rigourous way
Shear Strain
∆X/L
Shear stress on a solid is
proportional to shear strain
Fs
dv

A
dz
Internal friction between layer is
explained as a consequence of
molecular agitation (case of laminar
flow). There is a constant
interchange of molecules between
adjacent horizontal layers. F
In a laminar flow, the coefficient of (dynamic)
viscosity is the proportionally constant between
the shear stress and the relative velocity of
deformation.
IS Units: Pascal . Second; 1 Pa. s = 10 poise
Velocity profile. Turbulent flow
Adjacent layer
interchange
“parcels” of
fluid (eddies)
Boundary
layer
For fluids having
relatively small
viscosity- water,
air,..-, the effect of
internal friction is
appreciable only in a
narrow region
surrounding the fluid
boundaries
FLUID. Basic Equations of Fluid in Motion:
Fs
v

A
z



Viscous flow. Viscosity
Typically the coefficient of
viscosity of a liquid increase as
the temperature decrease.
In the case of gases the
variation is in opposite way.
Kinematic viscosity
Exercises
Calculate the kinematic viscosity of the
water at 20 ºC.
The viscosity of olive oil is 80 cP -10-2
Poise- at 20ºC and its specific gravity
at the same temperature is 0.915.
Express the viscosity of olive oil in Pa.s
and calculate its kinematic viscosity.
Calculate the shear stress required to
maintain a relative speed of deformation
of 1 m/s through a distance of z = 1 cm
(a) in water; (b) in engine oil
FLUID. Basic Equations of Fluid in Motion:
Viscous Flow. Poiseuille´s Law
Horizontal pipe, steady flow,
constant cross section, laminar flow
P1
P2
According Bernoulli´s equation the
pressure along the pipe would have to be
constant.
In practice, we observe a pressure drop
ΔP = P1-P2.
ΔP = P1-P2 = Q R
R: resistance to flow; it depends on the
length L, the radius r, and the viscosity of
the fluid.
Q: volume flow rate
Poiseuille´s Law
As a result of viscous forces, the velocity of
the fluid is not constant across the diameter
of the pipe The velocity is a maximum at the
centre line and varies (parabolically) to zero
at the wall (non slip condition).
P 
8L
Q
4
r
8L
R
 r4
Laminar,
steady flow,
horizontal
pipe
FLUID. Basic Equations of Fluid in Motion:
Turbulence. Reynolds number
Turbulence: Reynolds number
When the flow speed becomes sufficiently great, laminar flow breaks down and
turbulence sets in. Turbulence is flow dominated by recirculation, eddies, and apparent
randomness.
Turbulence implies a increasing of internal friction because adjacent layer interchange
“parcel” of fluid by eddies and so, a greater pressure drop per unit of length.
The flow (laminar or tubulent) can be characterized by a dimensionless number called
the Reynolds number , Re, which is defined by
Re 
2r  v

r radius of the pipe; ρ Density;
v: speed (average across the
section); η viscosity
In the case of flow through a pipe:
Re < 2000 Laminar
Re > 3000 Turbulent
Exercise: Crude oil has a viscosity of about 0.200 Pa.s at normal temperature. You are the
engineering in charge of constructing a 50.0 km horizontal pipeline to deliver oil at a rate of
500 liter/s. The flow through the pipeline must be laminar. Assuming that the density of
crude oil is 700 kg/m3 (a) estimate the diameter of pipeline that should be used. (b) What
the power of the pumps should be to maintain constant the flow ?
Water flows through a pipe of diameter D =1 inch. It is required maintain laminar
flow. What the maximum volume flow rate will be?. Calculate the average velocity in
a cross section. 1 inch = 2.54 cm
FLUID. Basic Equations of Fluid in Motion:
L= 10m
Olive oil flows through a horizontal pipe of radius r = 10
cm in a laminar, steady flow. The volume flow rate is 20
liter per second. (a) What is the average velocity in a
cross-section?. (b) What is the pressure drop over a
length L = 10 m?. Estimate the maximum volume flow
rate that is possible carry out in the pipe maintaining the
flow as laminar. In this last case, estimate the pressure
drop over a length L= 10 m. Physical properties of the
olive oil at 20º: Viscosity: 80 cP; density: 915 kg/m3.
Linear momentum equation. Not considered in this text
A firefighter holds a hose with a bend in
it as in Figure. Water is expelled from
the hose in a stream of radius 1.5 cm
at a speed of 30 m/s. (a) What mass of
water emerges from the hose in 1 s?.
(b) What is the horizontal momentum
of this water; (c) Find the force exerted
on the water by the hose.
http://www.it.iitb.ac.in/vweb/engr/civil/fluid_mech/course.html
FLUIDS:
Summary
http://hyperphysics.phy-astr.gsu.edu/hbase/fluid.html