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TRANSCRIPT
20151112
1
Fluid flow
Taliaacuten Csaba Gaacutebor
University of Peacutecs
Department of Biophysics
20121209
Fluids
Fluids ndash continually deform (flow) under shear stress
Liquid
Gas
Plasma
Liquids
Short-range crystalline organisation that always reforms
Constant volume incompressible
Moderate resistance to deformationfit the shape of solid surrounding (container) or force field
No preferred direction
Fluid mechanics
Hydrostatics (resting fluids)
Hydrodynamics (moving fluids)
Ideal fluids (no internal friction)
Real (viscous) fluids
bull Newtonian fluids
bull Non-newtonian fluids
Flow
bull Laminary flow
bull Turbulent flow
bull Stationary flow (equal amounts of mass or volume through the cross section in a unit of time)
bull Changing in time
Pascalrsquos law
Fluids are incompressible
Blaise Pascal (1623-1662 FRA)
∙ ∆
∙ ∆ ∙
∙ ∙ ∙ ∙
13 rarr 13
Pressure exerted anywhere on a fluid in a confined space is transmitted to an equal extent in all directions
Hydrostatic pressure
In a gravitation field pressure is proportional to height (depth) because of the weight of the fluid column
∙
∙ ∙
∙ ℎ ∙ ∙
∙ ℎ ∙
∙
If also atmospheric pressure is considered
Independent of the fluidrsquos shape
For interconnected fuids in equilibrium
+ ∙ ℎ ∙
∙ ℎ ∙ ∙ ℎ ∙ ℎ
ℎ
1 mmHg (1 Torr) = 1333 Pa1 atm = 101 325 Pa = 101 kPa
20151112
2
Law of Archimedes
Objects immersed in fluid lose weight
Archimedes (~ aD 287-212 GRE)
weight of fluid = upthrust force
F1
F2
∙ ∙ ∙ ℎ ∙
∙ ∙ ∙ ℎ ∙
minus ∙ ∙ ℎ minus ℎ ∙ ∙ ∙ ∙
Flow
Movement of fluids in one direction
Driving force is the pressure difference
Foundational axioms conservation laws (mass energy momentum)
Continuum assumption
Fluids are continuous matter rather than made up of molecules
Physical properties are well-defined at infinitesimal points and vary continuously from one point to another
intensity of current or volumetric flow rate
~ 5Lmin in aorta
∆
∆
$
Law of continuity
Fluids are incompressible
In Δt time the flow volume throughany cross-section is the same
continuity equation
∆
∆ ∙ amp
∆ ∙ ∙ ∆
∆ ∙
∙ ∙
For rigid tubes ideal fluids and stationary flow(conservation of mass)
( ) ∙ +-0- 1 + 2 ∙3
3 +-0-
Bernoullirsquos law
mechanical work
static pressure dynamic pressure
conservation of energy
∙ 4 ∙ 4 ∆
∙ ∆4
∙∆
∙ ∆ ∙ ∆
minus 56 minus 56
∆ ∆567
+ 56 + 56
∙ ∆ + ∙
2 ∙ ∆ + ∙
2
+ ∙
2 + ∙
2∆
1+ 2 ∙3
3+ 2 ∙ ∙ +-0-
Bernoullirsquos law
potential energy
hydrostatic pressure
Daniel Bernoulli(1700-1782 NED-SUI)
∆ ∙ ∙ ℎ ∙ ∙ ℎ minus ∙ ∙ ℎ
∙ ∆ + ∙
2+ ∙ ∙ ℎ ∙ ∆ + ∙
2+ ∙ ∙ ℎ∆
+ ∙
2+ ∙ ∙ ℎ + ∙
2+ ∙ ∙ ℎ
Laminary flow in real fluids
constant cross-section
pressure decreasing in the direction of flow
proportional to distance
Fluid resists to the moving effect (flow) with a force
An internal friction between imaginary fluid layers sliding past each other decreasing displacement profile
VISCOSITY
20151112
3
Newtonrsquos law
ℎlt=gtgtlt
gt=gt=lt Δ
ΔB
∆4
∆ ∙ ∆B
CDEFGEDHI ℎlt=gtgtlt
gt=gt=lt
∙ΔB
Δ
Isaac Newton (1643-1727 ENG)
AΔx
Δy
F
Δv
J
∙
J ∙
K= ∙
L M ∙ ) ∙N
NO
Viscosity depends on matter temperature concentration pressure
Ideal fluid zero viscosity (~ certain liquid He species)
Newtonian-fluid (eg water)
Viscosity ~ shear stress
Non-newtonian fluid (eg blood)
Viscosity not proportional onlyto shear stress
It depends on flow velocity
Viscosity of gases increases at higher temperatures that of fluids decreases
matter T (degC) η (Pas)
air 0 1710-6
ethanol 20 12510-3
water 20 10-3
mercury 20 0017
blood 37 4-2510-3
honey 20 10
asphalt 20 108
glass 20 1040
Hagen-Poiseuille law
Force because of pressure difference = frictional force(laminary stationary flow rigid tube)
~
∆ ∙ minus ∙ minusQ ∙ R ∙Δ
Δℎ
~gtS~
1
amp~
1
Q
(V )X ∙ gtS
8 ∙ Q∙∆
amp
pressure gradient
Z 8 ∙ Q ∙ amp
X ∙ gtSresistance Gotthilf Hagen (1797-1884 GER)
Jean Poiseuille (1797-1869 FRA)
∆ ∙ Z
Turbulent flow
Laminar flow
Low speed
No swirling
On smooth surface
Turbulent flow
High speed for viscosity
Swirling no bdquolayersrdquo
On rough surface (blood vessels)
[ ∙ 2 ∙
M
tube radius
Osborne Reynolds(1842-1912 IRE)
Reynolds number
] Z] ∙ Q
^ ∙ gtcritical velocity for smooth tubes Rcrit asymp 1160
Hydrostatic resistance
A medium (gas or liquid) exerts an opposite force to the direction of the movement on the objects moving in it
_ ∙ ∙ ∙
shape factor largest cross-section
Streamline bodies (small k) flow layers unite behind the object low resistance
Non-streamline bodies (great k) the medium flows rapidly behind the object rarr low pressure rarr suction effect rarr great counter-force
Stokesrsquo law
Movement of bodies in real fluids
Frictional force on moving sphaerical objects
r radius
George Stokes (1819-1903 IRE)
2
9∙Uabc W
Q∙ ∙ gt
L d ∙ e ∙ M ∙ ∙
] fg3 ∙ Q
^ ∙ gt
Terminal (settling) velocity a = 0 v = const
h Ri
20151112
4
Summary
Pascalrsquos law transmittion of pressure
Continuity equation relationship of surface and flow velocity
Bernoullirsquos law relationship of pressure and velocity
Newtonrsquos law internal friction
Hagen-Poiseuille law flow of real fluids
Reynolds-number critical velocity of the turbulent flow
Stokesrsquo law objects moving in a medium
( ) ∙ +-0-
1 2 ∙3
3 2 ∙ ∙ +-0-
L M ∙ ) ∙N
NO
UV WX ∙ gtS
8 ∙ Q∙∆
amp
[ ∙ 2 ∙
M
L d ∙ e ∙ M ∙ ∙
THANK YOU FOR ATTENTION
Torricellirsquos law
Specific case of Bernoullirsquos law
at the top and at the opening p = patm
at the top v = 0 at the opening h = 0
∙ ∙ ∙
2
3 ∙ ∙
Evangelista Torricelli (1608-1647 ITA)
Venturi effect
Flow through a constriction
Gain in kinetic energy
Loss in pressure
parfume spray chimney Giovanni Venturi (1746-1822 ITA)
20151112
2
Law of Archimedes
Objects immersed in fluid lose weight
Archimedes (~ aD 287-212 GRE)
weight of fluid = upthrust force
F1
F2
∙ ∙ ∙ ℎ ∙
∙ ∙ ∙ ℎ ∙
minus ∙ ∙ ℎ minus ℎ ∙ ∙ ∙ ∙
Flow
Movement of fluids in one direction
Driving force is the pressure difference
Foundational axioms conservation laws (mass energy momentum)
Continuum assumption
Fluids are continuous matter rather than made up of molecules
Physical properties are well-defined at infinitesimal points and vary continuously from one point to another
intensity of current or volumetric flow rate
~ 5Lmin in aorta
∆
∆
$
Law of continuity
Fluids are incompressible
In Δt time the flow volume throughany cross-section is the same
continuity equation
∆
∆ ∙ amp
∆ ∙ ∙ ∆
∆ ∙
∙ ∙
For rigid tubes ideal fluids and stationary flow(conservation of mass)
( ) ∙ +-0- 1 + 2 ∙3
3 +-0-
Bernoullirsquos law
mechanical work
static pressure dynamic pressure
conservation of energy
∙ 4 ∙ 4 ∆
∙ ∆4
∙∆
∙ ∆ ∙ ∆
minus 56 minus 56
∆ ∆567
+ 56 + 56
∙ ∆ + ∙
2 ∙ ∆ + ∙
2
+ ∙
2 + ∙
2∆
1+ 2 ∙3
3+ 2 ∙ ∙ +-0-
Bernoullirsquos law
potential energy
hydrostatic pressure
Daniel Bernoulli(1700-1782 NED-SUI)
∆ ∙ ∙ ℎ ∙ ∙ ℎ minus ∙ ∙ ℎ
∙ ∆ + ∙
2+ ∙ ∙ ℎ ∙ ∆ + ∙
2+ ∙ ∙ ℎ∆
+ ∙
2+ ∙ ∙ ℎ + ∙
2+ ∙ ∙ ℎ
Laminary flow in real fluids
constant cross-section
pressure decreasing in the direction of flow
proportional to distance
Fluid resists to the moving effect (flow) with a force
An internal friction between imaginary fluid layers sliding past each other decreasing displacement profile
VISCOSITY
20151112
3
Newtonrsquos law
ℎlt=gtgtlt
gt=gt=lt Δ
ΔB
∆4
∆ ∙ ∆B
CDEFGEDHI ℎlt=gtgtlt
gt=gt=lt
∙ΔB
Δ
Isaac Newton (1643-1727 ENG)
AΔx
Δy
F
Δv
J
∙
J ∙
K= ∙
L M ∙ ) ∙N
NO
Viscosity depends on matter temperature concentration pressure
Ideal fluid zero viscosity (~ certain liquid He species)
Newtonian-fluid (eg water)
Viscosity ~ shear stress
Non-newtonian fluid (eg blood)
Viscosity not proportional onlyto shear stress
It depends on flow velocity
Viscosity of gases increases at higher temperatures that of fluids decreases
matter T (degC) η (Pas)
air 0 1710-6
ethanol 20 12510-3
water 20 10-3
mercury 20 0017
blood 37 4-2510-3
honey 20 10
asphalt 20 108
glass 20 1040
Hagen-Poiseuille law
Force because of pressure difference = frictional force(laminary stationary flow rigid tube)
~
∆ ∙ minus ∙ minusQ ∙ R ∙Δ
Δℎ
~gtS~
1
amp~
1
Q
(V )X ∙ gtS
8 ∙ Q∙∆
amp
pressure gradient
Z 8 ∙ Q ∙ amp
X ∙ gtSresistance Gotthilf Hagen (1797-1884 GER)
Jean Poiseuille (1797-1869 FRA)
∆ ∙ Z
Turbulent flow
Laminar flow
Low speed
No swirling
On smooth surface
Turbulent flow
High speed for viscosity
Swirling no bdquolayersrdquo
On rough surface (blood vessels)
[ ∙ 2 ∙
M
tube radius
Osborne Reynolds(1842-1912 IRE)
Reynolds number
] Z] ∙ Q
^ ∙ gtcritical velocity for smooth tubes Rcrit asymp 1160
Hydrostatic resistance
A medium (gas or liquid) exerts an opposite force to the direction of the movement on the objects moving in it
_ ∙ ∙ ∙
shape factor largest cross-section
Streamline bodies (small k) flow layers unite behind the object low resistance
Non-streamline bodies (great k) the medium flows rapidly behind the object rarr low pressure rarr suction effect rarr great counter-force
Stokesrsquo law
Movement of bodies in real fluids
Frictional force on moving sphaerical objects
r radius
George Stokes (1819-1903 IRE)
2
9∙Uabc W
Q∙ ∙ gt
L d ∙ e ∙ M ∙ ∙
] fg3 ∙ Q
^ ∙ gt
Terminal (settling) velocity a = 0 v = const
h Ri
20151112
4
Summary
Pascalrsquos law transmittion of pressure
Continuity equation relationship of surface and flow velocity
Bernoullirsquos law relationship of pressure and velocity
Newtonrsquos law internal friction
Hagen-Poiseuille law flow of real fluids
Reynolds-number critical velocity of the turbulent flow
Stokesrsquo law objects moving in a medium
( ) ∙ +-0-
1 2 ∙3
3 2 ∙ ∙ +-0-
L M ∙ ) ∙N
NO
UV WX ∙ gtS
8 ∙ Q∙∆
amp
[ ∙ 2 ∙
M
L d ∙ e ∙ M ∙ ∙
THANK YOU FOR ATTENTION
Torricellirsquos law
Specific case of Bernoullirsquos law
at the top and at the opening p = patm
at the top v = 0 at the opening h = 0
∙ ∙ ∙
2
3 ∙ ∙
Evangelista Torricelli (1608-1647 ITA)
Venturi effect
Flow through a constriction
Gain in kinetic energy
Loss in pressure
parfume spray chimney Giovanni Venturi (1746-1822 ITA)
20151112
3
Newtonrsquos law
ℎlt=gtgtlt
gt=gt=lt Δ
ΔB
∆4
∆ ∙ ∆B
CDEFGEDHI ℎlt=gtgtlt
gt=gt=lt
∙ΔB
Δ
Isaac Newton (1643-1727 ENG)
AΔx
Δy
F
Δv
J
∙
J ∙
K= ∙
L M ∙ ) ∙N
NO
Viscosity depends on matter temperature concentration pressure
Ideal fluid zero viscosity (~ certain liquid He species)
Newtonian-fluid (eg water)
Viscosity ~ shear stress
Non-newtonian fluid (eg blood)
Viscosity not proportional onlyto shear stress
It depends on flow velocity
Viscosity of gases increases at higher temperatures that of fluids decreases
matter T (degC) η (Pas)
air 0 1710-6
ethanol 20 12510-3
water 20 10-3
mercury 20 0017
blood 37 4-2510-3
honey 20 10
asphalt 20 108
glass 20 1040
Hagen-Poiseuille law
Force because of pressure difference = frictional force(laminary stationary flow rigid tube)
~
∆ ∙ minus ∙ minusQ ∙ R ∙Δ
Δℎ
~gtS~
1
amp~
1
Q
(V )X ∙ gtS
8 ∙ Q∙∆
amp
pressure gradient
Z 8 ∙ Q ∙ amp
X ∙ gtSresistance Gotthilf Hagen (1797-1884 GER)
Jean Poiseuille (1797-1869 FRA)
∆ ∙ Z
Turbulent flow
Laminar flow
Low speed
No swirling
On smooth surface
Turbulent flow
High speed for viscosity
Swirling no bdquolayersrdquo
On rough surface (blood vessels)
[ ∙ 2 ∙
M
tube radius
Osborne Reynolds(1842-1912 IRE)
Reynolds number
] Z] ∙ Q
^ ∙ gtcritical velocity for smooth tubes Rcrit asymp 1160
Hydrostatic resistance
A medium (gas or liquid) exerts an opposite force to the direction of the movement on the objects moving in it
_ ∙ ∙ ∙
shape factor largest cross-section
Streamline bodies (small k) flow layers unite behind the object low resistance
Non-streamline bodies (great k) the medium flows rapidly behind the object rarr low pressure rarr suction effect rarr great counter-force
Stokesrsquo law
Movement of bodies in real fluids
Frictional force on moving sphaerical objects
r radius
George Stokes (1819-1903 IRE)
2
9∙Uabc W
Q∙ ∙ gt
L d ∙ e ∙ M ∙ ∙
] fg3 ∙ Q
^ ∙ gt
Terminal (settling) velocity a = 0 v = const
h Ri
20151112
4
Summary
Pascalrsquos law transmittion of pressure
Continuity equation relationship of surface and flow velocity
Bernoullirsquos law relationship of pressure and velocity
Newtonrsquos law internal friction
Hagen-Poiseuille law flow of real fluids
Reynolds-number critical velocity of the turbulent flow
Stokesrsquo law objects moving in a medium
( ) ∙ +-0-
1 2 ∙3
3 2 ∙ ∙ +-0-
L M ∙ ) ∙N
NO
UV WX ∙ gtS
8 ∙ Q∙∆
amp
[ ∙ 2 ∙
M
L d ∙ e ∙ M ∙ ∙
THANK YOU FOR ATTENTION
Torricellirsquos law
Specific case of Bernoullirsquos law
at the top and at the opening p = patm
at the top v = 0 at the opening h = 0
∙ ∙ ∙
2
3 ∙ ∙
Evangelista Torricelli (1608-1647 ITA)
Venturi effect
Flow through a constriction
Gain in kinetic energy
Loss in pressure
parfume spray chimney Giovanni Venturi (1746-1822 ITA)
20151112
4
Summary
Pascalrsquos law transmittion of pressure
Continuity equation relationship of surface and flow velocity
Bernoullirsquos law relationship of pressure and velocity
Newtonrsquos law internal friction
Hagen-Poiseuille law flow of real fluids
Reynolds-number critical velocity of the turbulent flow
Stokesrsquo law objects moving in a medium
( ) ∙ +-0-
1 2 ∙3
3 2 ∙ ∙ +-0-
L M ∙ ) ∙N
NO
UV WX ∙ gtS
8 ∙ Q∙∆
amp
[ ∙ 2 ∙
M
L d ∙ e ∙ M ∙ ∙
THANK YOU FOR ATTENTION
Torricellirsquos law
Specific case of Bernoullirsquos law
at the top and at the opening p = patm
at the top v = 0 at the opening h = 0
∙ ∙ ∙
2
3 ∙ ∙
Evangelista Torricelli (1608-1647 ITA)
Venturi effect
Flow through a constriction
Gain in kinetic energy
Loss in pressure
parfume spray chimney Giovanni Venturi (1746-1822 ITA)