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)