Fluid mechanics.Blood circulation.

2018

Gábor Szabó

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a fluid is a substance that continually deforms (flows) under an applied shearstress, no matter how small. Fluids are a subset of the phases of matter andinclude liquids, gases, plasmas and, to some extent, plastic solids.

Fluids...

• have a definite volume but no definite shape (although fluids take the shape of the container)

• are incompressible (exception: gases - however in fluid dynamics gases can also be treated

as incompressible fluids, as long as the flow velocity is below a certain level)

• the molecules “wander” through the liquid in a random fashion (theintermolecular forces are not strong enough to keep the molecules in afixed position - hole theory of liquids)

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Physical quantities used in fluid mechanics

Quantity Symbol Type Unit

Pressure p Scalar Pa=N/m2

Velocity (speed) v Vector m/s

Density r Scalar kg/m3

Viscosity η Scalar kg/(m·s)

Body force b Vector N, N/kg, N/m3

Time t Scalar s

Flow rate IV Scalar m3/s

Constant Symbol Type Value

Gravitational

accelerationg Vector ≈9.81 m/s2

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Hydrostatic pressure

• the shape of the container does not affect the pressure!

~ a change in pressure applied to an enclosed fluid is transmitted undiminishedto every point of the fluid and to the walls of the container

Pascal’s principle:

the pressure exerted by a fluid at a givenpoint within the fluid, due to the force of gravity

0p p ghr

P0: atmospheric pressure

ρ: density of the fluid

h: depth

• all points at the same depth must be at the same pressure

• the pressure p at a depth h below the surfaceof a liquid open to the atmosphere is greaterthan atmospheric pressure by the amount ρgh

(if this were not the case, fluid would flow from the higher pressure region to the lowerpressure region)

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for the characterization of fluid flow streamlinesare used

• tangent: direction of fluid velocity • density: magnitude of the velocity

The flow of fluids

at low flow velocities liquids flow in parallel layers (without mixing)

laminar flow

exceeding a critical speed fluids undergo erraticflowing and mixing (in the middle of the tube)

turbulent flow5

Ideal fluids

• Deformable bodies, no internal shear stress. i.e. internally frictionless

Real fluids

• Internal friction exists

Mennyiség Szimbólum Típus Egység

sebesség v vektor m/s

áram(erősség) Q skalár m3/s

/v s t velocity: the velocity (speed) of a given particle in the streamline

volumetric flow rate (rate of fluid flow, IV): is the volume of fluid which passesthrough a given surface per unit time

Velocity and volumetric flow rate

cross section

velocity

(it is different among particles in the same flowing fluid!)

V

VI

t

Quantity Symbol Type Unit

Velocity (speed) v Vector m/s

Flow (volumetric flow rate) IV Scalar m3/s

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Parabolic velocity profilefor real liquids in a tube.

Equation of continuity

• flow velocity and the cross-section of the tube are inversely proportional (stationary or steady-state flow: fluid properties (velocity, pressure, cross-section) at a point in the

system do not change over time, rigid walls)

during the flow of an ideal fluid the velocity isuniform through the entire cross-section of thefluid, since...

• there is no friction between fluid layers

• velocity changes can not occur, layers move along together

at tighter sections flow is accelerated and vice versa

VI A v

• relationship between the volumetric flow rate and the flow velocity

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conservation of mass, incompressibility

real liquidsVI A v

constantVI A v

Textbook, pages 211-212

• in our body: AT↑, v↓

(although the walls of blood-vessels are not rigid and the blood flow is not stationary)

the flow velocity of blood decreases from the aorta towards the capillaries

• the continuity equation can accurately describe blood velocities

(due to the increase in the total cross-sectional area, AT)

Consequence of equation of continuity…

8

x = 100

the sum of the pressure, kinetic energy and potential energy (both per unitvolume) has the same value at all points along a streamline in ideal liquids

2

1~

rv

Bernoulli’s-law

• horizontal tube: static pressure is lower at the places of higher velocity

21.

2v gh p constr r

Kinetic energy per unit volume (kinetic pressure)

Potential energy per

unit volume (hydrostatic

pressure)

static pressure

9conservation of mechanical energy

Textbook, pages 213-214

Mennyiség Szimbólum Típus Egység

viszkozitás skalár kg/(m·s)

is a measure of the resistance of a real fluid to either shear (or tensile) stress. Ineveryday terms (and for fluids only) viscosity is "thickness" or "internal friction"

Viscosity

• let fluid flow between two parallel platesand examine what force (F) has to be exertedto slide the upper plate of surface area A withconstant velocity v over the fixed lower plate

• the frictional force between the layers offluid (or the force is needed for thetranslocation of the plate) is proportional tothe surface and the velocity gradient

vF A

h

Newton’s law of friction:

Quantity Symbol Type Unit

Viscosity η Scalar kg/(m·s)

• as the plates are very close together thevelocities change linearly from 0 to v → Δv/Δh is constant

v

vF A

h

10„Newtonian

fluids”

Textbook, pages 214-215

• viscosity of fluids decreases with rising temperature (and vice versa)

• IMPORTANT: viscosity and density are not the samequantities!

• η = coefficient of viscosity, SI unit: Pa·s (N/m2)

• The reciprocal of the dynamic viscosity is fluidity

(former unit: poise (P) named after Jean Louis Marie Poiseuille)

1 10Pa s poise

reason: the relative ‘concentration’ of irregularities (lattice-holes) which enable layersof fluid slide over one-another increases with rising temperature

~E

kTe

(Arrhenius–Andrade relationship)

circulation becomes poor in the limbs in cold temperature

Viscosity

(viscosity increases at lower T → flow rate is inversely proportional to the viscosity; see H-P equation) 11

This is a

factor e.g.

in Raynaud-

disease

Textbook, pages 215-216

consequence for blood flow: more cells travel within the axis of the tube

v: lowv: highv: low

(static pressure is the lowest there, the pressure difference drives them from thewalls towards the ‘mainstream’)

Bernoulli’s-law and Newton’s law of friction: consequences

21.

2v gh p constr r

12

v

13

Hagen–Poiseuille law• applies to newtonian fluids undergoing stationary and laminar flow only

4

8V

pI r

l

(negative sign denotes that fluids flow from the site ofhigher pressure to the lower pressure)

the volumetric flow rate is directly proportional to the pressure gradient Δp/Δl and the fourth power (!) of the radius (r) of the tube

• in the body a small change in the arterial radius can provide a effective control ofthe organs’ blood-supply

• arteriosclerosis → the radii of the vessels (arteries/arterioles) decrease → theheart can only ensure blood supply with much higher pressure → decreased bloodsupply → tissue damage

vessel Vp R I where2 2 2

8 8( )

vessel

l lR

r A

analogy with Ohm’s law

(although blood is a non-newtonian fluid, and in the arteries close to the heart the flow is not stationary, the H-P law is still a useful approximation)

14A: cross section area of the tube

Textbook, pages 217-222

http://www.promotemsc.org/results/CZ/Viscous_flow.pdf

Htk

Deviation from H-P in our blood stream

Velocity gradient

Vis

cosity

A B

PA PBif PA > PB

fluid flow is driven by the pressure difference, flow of electric charges is driven by the potential difference → the current corresponds to volumetric flow rate

• note! Relectric ~ 1/A ↔ Rtube ~ 1/A2

• Kirchhoff’s first law is also applicable: the flow-rate in the main tube equals to the sum of the flow rates in the branches (continuity equation!)

Flow in a pipe – analogy with conductors

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R1 R2 R3 R… Rn

1

n

e i

i

R R

the equivalent resistance of a series combination of resistors is the algebraic sum of the individual resistances and is always greater than any of the individual resistors

1

1

1e n

i i

R

R

the reciprocal of the equivalent resistance of two or more resistors connected in parallel is the algebraic sum of the reciprocals of the individual resistance and is always less than the

smallest resistor in the group

R1

R…

Rn

connection of tubes (‘resistors’)

• serial combination of tubes (different types of blood vessels form a series circuit)

• parallel combination of tubes (different blood vessels of the same type form aparallel circuit)

Flow in a pipe – analogy with conductors II.

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heart

vascular system

the whole circulatory system can be seen as an analogue to an electrical circuitwhere the heart is the battery and the blood vessels are resistors

• pulmonary circulation (through the lungs whereblood is oxygenated) and systemic circulation (thecirculation of the oxygenated blood to all parts of thebody and back to the heart) form a series circuit,therefore the (volumetric) blood flow in the twocirculation systems is the same, but differentpressure levels (e.g.) can prevail in the two systems

The human circulatory system

PC: right ventricule → pulmonary artery → lungs → pulmonary vein →left atrium

SC: left ventricule → aorta → arteriolae, capillaries → tissues → veins(vena cava) → right atrium 20

PA - PB = IV * R

MAP MAP: Mean Arterial Pressure

CO: Cardiac Output

TPR: Total Peripheral Resistance

Paorta - Pvena cava = CO * TPR

MAP = CO * TPR

Pressure levels in the systemic circulation

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cső Vp R I

22

23

http://cnx.org/contents/03841c4c-9e9a-4822-95b2-12273c843a4e@3/Blood-Flow-Blood-Pressure-and-

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How can we explain that the total resistance increases while the total cross-section also increases??

the pressure drop is largest in the arterioles, sincethese have the largest overall resistance

28vessel

lR

A

resistance of one blood-vessel:

resistance of n identical blood-vessels connected in parallel:

1

total vessel

n

R R

2 2 2 28 8 8total

total

l n l n lR

nA n A A

• in the huma circulation n is rising more quickly than Atotal

2, thus Rtotal rises (narteriola >> naorta)

• blood vessels contain large amount of smoothmuscle → their diameter is variable

Pressure levels in the systemic circulation

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(small musc. artery)

Textbook, pages 219-220

Resistance and pressure conditionsin the systemic circulation

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Turbulent flowFt: force of kinetic pressure

FS: frictional force (viscosity)

Ft

Fs

vcrit

Textbook, pages 222-224

Turning on turbulence.

B Eckhardt Science 2011;333:165-166

Published by AAAS

Turbulent flow

if the flow exceeds a certain velocity (vcrit) it becomes turbulent

• in the case of turbulent flow the same perfusion pressure results in

LOWER flow, i.e. higher pressure is needed to maintain the same level of

flow in the pipe

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• Reynolds’ number (Re) is the measure of the tendency for turbulence to

occur in a tube

•Re < 1000 means a low probability of turbulent flow, Re>2000- turbulence

•Re depends on the flow velocity (v), the diameter of the vessel (d) and the

density (r) and the viscosity (h) of the fluid in the vessel

vcrit

• the elasticity of the aorta and the arterial branches plays an important role since:

- it blunts the pressure-fluctuation caused by the heart

- the flow is more uniform, it is almost stationary in the arterioles and capillaries

- the reduced vmax reduces the risk of exceeding vcrit, therefore turbulent flow isavoided

the heart pumps blood in pulses into the aorta

blood circulation is not uniformly stationary

Flow in solid and elastic pipes

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In a distensible tube, an increase in pressure stretches walls lowering resistance

→ tendency for resistance to fall with increasing pressure

→ if pressure falls towards 0, vessel collapses and flow ceases (transmural pressure must be

>0 to permit vessel to be open, i.e. intraluminal pressure must be larger than extraluminal

pressure)

The distensibility of blood vessels gives them capacitance - as vessels widen with increasing pressure,

transiently more blood will flow in than out

The vessel will store blood - the more distensible the more blood will be stored

Veins are particularly distensible (high compliance) and hold ~67% of the circulating blood volume

• in rigid pipes the flow intensity/volumetric flow rate is linearly proportional tothe pressure gradient

Flow in rigid and elastic pipes

• critical closing pressure is the internal pressure at which a blood vessel collapsesand closes completely

• at the same Δp → higher volumetric flow rate in the case of an elastic pipe

(blood vessels)

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Rigid pipe (made of steel)

Elastic pipe (blood vessel)

critical closing pressureP

Measuring blood pressure using Riva–Rocci’s method

blood pressure is measured with a sphygmomanometer

• a rubber bulb forces air into a cuff wrapped tightlyaround the upper arm and simultaneously into amanometer

• when the pressure in the cuff and brachial artery is justbelow the maximum value produced by the heart (at thesystolic pressure), the artery opens

• the pressure in the cuff is increased until the flow ofblood through the brachial artery in the arm is stopped

• at this point, the blood flow is turbulent which makesnoise (Korotkoff sounds)

• if the handcuff pressure is below the ‘diastolic bloodpressure’, the flow of blood in the artery becomes laminarand does not make noise

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Oscillometric measurement of blood pressure

Fluids:

•ideal or real (viscosity)

Newtonian non-Newtonian (v!)

Flow:

•laminar or turbulent•stacionary or non-stacionary (t!)•in elastic or in non-elastic tubes