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Viscous Fluid Flow (14 p.341) Viscous Fluid Flow A. Prof. Hamid NEBDI [email protected] Faculty of...
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Transcript of Viscous Fluid Flow (14 p.341) Viscous Fluid Flow A. Prof. Hamid NEBDI [email protected] Faculty of...
(14 p.341)
Viscous Fluid Flow
A. Prof. Hamid [email protected]
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