(14 p.341)

Viscous Fluid Flow

A. Prof. Hamid NEBDIhbnebdi@uqu.edu.sa

Faculty of Applied Science. Department of Physics. Room: 315 second floor. Phone: 3192

Umm Al-Qura University القرى أم معة جاFaculty of Applied Scienceكـلية العلوم التطبيقية Department of Physics قسم الفيزياء General Physics (For medical students): 403104-3 403104-3 : المدخل للفيزياء الطبية

Course’s Topics

14.1- Viscosity

14.4- Flow in the Circulatory System

14.1 Viscosity- The viscosity of a fluid is a measure of its

resistance to flow under an applied force.

- The greater the viscosity, the larger the force required to maintain the flow, and the more energy that is dissipated.

- Molasses has a high viscosity, water a smaller viscosity, and air a still smaller viscosity.

Viscosity is readily defined by considering a simple experiment.

Figure 1 shows two flat plates separated by a thin fluid layer.

Figure 1

• The lower plate is held fixed, a force is required to move the upper plate at a constant speed.

• This force is needed to overcome the viscous forces due to the liquid and is greater for a highly viscous fluid

Moving plate

Fixed plate

- The force F is observed to be proportional to the area of the plates A and to the velocity of the upper plate ∆v (the moving one) and inversely proportional to the plate separation ∆y.

(1)

- The proportionality constant (Greek letter “eta”) is called the viscosity.

- The larger the viscosity, the larger force needed to move the plate at a constant speed.

y

vAF

Dimensions of Viscosity

sPasmkgTML

LLT

LMLT

yvA

F

. 1111

1

2

2

M, L, and T stand for mass, length, and time respectively.

The S.I. Unit of viscosity is: 1 kg m-1 s-1 = 1 Pa s.

The Relation between Viscosity and Temperature

• As the temperature increases viscosity decreases, for liquids.

• As the temperature increases viscosity increases, for gases.

• Because viscous forces are usually small, fluids are often used as lubricants to reduce friction.

Example 14.1

An air track used in physics lecture demonstrations, supports a cart that rides on a thin cushion of air 1mm thick and 0.04 m2 in area. If the viscosity of the air is 1.8 x 10-5 Pa.s, find the force required to move the cart at a constant speed of 0.2 m/s.Solution 14.1:

The force required is:

NF

m

msmsPaF

y

vAF

4

3

125

1044.1

10

2.0)04.0)(.108.1(

This is a very small force and is consistent with the observation that an air track is nearly frictionless

The blood

- The circulatory system transports the substances required by the body and take off the waste products of metabolism.

- In order to perform a large number of functions, the blood contains many different constituents, including red blood cells, white blood cells, platelets, and proteins.

- However, for our purposes, it is sufficient to treat the blood as a uniform fluid with viscosity and density.

sPa.10084.2 3 33 100595.1 mkg

14.4 Flow in The Circulatory System

The Cardiovascular System

- This system includes the heart, and an extensive system

of arteries, vascular beds containing capillaries, and

veins.

- A particularly interesting compound of the cardiovascular

system is the arteriovenous anastomosis (AVA), or shunt.

- These shunts are particularly important, since the

surrounding muscle tissue can adjust the diameter of the

blood flow to various organs as conditions change.

- Smaller shunts in the skin are open if the body needs to

release heat or to increase skin temperature.

Flow Resistance

- The flow resistance Rf is defined in general, as the ratio of the pressure

drop to the flow rate:

- Rf defines the flow resistance whether the flow is laminar or not.

- The basic S.I. Unit of Rf is the : Pa.s.m-3

But we use : kPa s m-3

In text and literature on physiology pressures are usually measured in torr and lengths in cm:

1 torr s cm-3 = 1.333 x 105 kPa s m-3

- Usually the flow resistance in a large artery is small. Consequently, the

pressure drop in such arteries is small.

- The following relation is applicable only if the flow is laminar:

Where l is the length of the tube, and R is the Radius of the tube.

Q

PR f

4

8

R

lR f

Example 14.5:

• The aorta of an average adult human

has a radius 1.3 x 10-2 m. What are

the resistance and the pressure drop

over a 0.2 m distance, assuming a

flow rate of 10-4 m3s-1?

3

34

42

3

4

..2.37

..1072.3

)103.1(

)2.0)(.10084.2(8

8

mskPaR

msPaR

m

msPaR

R

lR

f

f

f

f

Solution 14.5:From Table 14.3, =2.084 x 10-3 Pa s, so the flow resistance of the aorta is:

The pressure drop over the 0.2 m distance is then:

kPaP

smmskPaP

QRP f

00372.0

)10)(..2.37( 1343

- This is very small value of the pressure drop, compared to the

total pressure drop in the system, which is about 13.3 kPa.

- Most of the flow resistance and pressure drops occur in the

smaller arteries and vascular beds of the body (Table 14.4).

- The flow resistance of a collection of arteries can be calculated. The

calculation can be done by considering each category of artery

separately.

- We assume that all of the arteries of a given size are in parallel; each

artery carries its equal share of the Rf1 and Q1:

P is the pressure across all of the arteries.

- If there are N identical arteries, the total flow is

Where Rp is the equivalent flow resistance of this arrangement.

11

fR

PQ

N

RR

R

PQ

R

PNQ

QQQQQ

fp

p

f

n

1

1

321 ...

Example 14.6

From Table 14.2, the radius of a single

capillary is 4 x 10-6 m and its length is

10-3 m. What is the resistance of 4.73

x 107 capillaries in the mesenteric

vascular bed of a dog if they are

assumed to be parallel?

3131

46

33

1

41

..10073.2

)104(

)10)(.10084.2(8

8

mskPaR

m

msPaR

R

lR

f

f

f

With =2.084x10-3 Pa s, the resistance of one capillary is:

Solution 14.6

There are N= 4.73x107 capillaries in parallel, so there effective resistance is :

35

7

313

1

..1038.4

1073.4

..10073.2

mskPaR

mskPaR

N

RR

f

f

ff

Flow Ratem3/s

Flow ResistancekPa.s/m3

Brain 12.5 x 10-6 9.3 x 105

Heart 4.2 x 10-6 2.8 x 106

Legs 13.7 x 10-6 8.5 x 105

Some approximate flow rates and resistances for the resting, reclining adult.

- Suppose we know the resistances of N sections, each of which leads into the next.

- The total pressure drop is as follows

- Each pressure drop is the total flow rate Q times the resistance of that section.

- The effective flow resistance Rs of these sections which are said to be in series, is the sum of the resistances.

fNff PPPP ...21

)...( 21

111

fNff

f

RRRQP

RQP

fNffs RRRR ...21