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If viscosity is zero, the fluid is frictionless and the resistance to flow is also zero. Comparing frictionless flow in a tube to viscous flow, as in [link] , we see that for a viscous fluid, speed is greatest at midstream because of drag at the boundaries. We can see the effect of viscosity in a Bunsen burner flame, even though the viscosity of natural gas is small.

The resistance R size 12{R} {} to laminar flow of an incompressible fluid having viscosity η size 12{η} {} through a horizontal tube of uniform radius r size 12{r} {} and length l size 12{l} {} , such as the one in [link] , is given by

R = 8 η l π r 4 . size 12{R= { {8η l} over {π r rSup { size 8{4} } } } "."} {}

This equation is called Poiseuille’s law for resistance    after the French scientist J. L. Poiseuille (1799–1869), who derived it in an attempt to understand the flow of blood, an often turbulent fluid.

Part a of the diagram shows a fluid flow across a rectangular non viscous medium. The speed of the fluid is shown to be same across the tube represented as same length of vertical rising arrows. Part b of the diagram shows a fluid flow across a rectangular viscous medium. The speed of the fluid speed at the walls is zero, increasing steadily to its maximum at the center of the tube represented as wave like variation for length of vertical rising arrows. Part c of the figure shows a burning Bunsen burner.
(a) If fluid flow in a tube has negligible resistance, the speed is the same all across the tube. (b) When a viscous fluid flows through a tube, its speed at the walls is zero, increasing steadily to its maximum at the center of the tube. (c) The shape of the Bunsen burner flame is due to the velocity profile across the tube. (credit: Jason Woodhead)

Let us examine Poiseuille’s expression for R size 12{R} {} to see if it makes good intuitive sense. We see that resistance is directly proportional to both fluid viscosity η size 12{η} {} and the length l size 12{l} {} of a tube. After all, both of these directly affect the amount of friction encountered—the greater either is, the greater the resistance and the smaller the flow. The radius r size 12{r} {} of a tube affects the resistance, which again makes sense, because the greater the radius, the greater the flow (all other factors remaining the same). But it is surprising that r size 12{r} {} is raised to the fourth power in Poiseuille’s law. This exponent means that any change in the radius of a tube has a very large effect on resistance. For example, doubling the radius of a tube decreases resistance by a factor of 2 4 = 16 size 12{2 rSup { size 8{4} } ="16"} {} .

Taken together, Q = P 2 P 1 R size 12{Q= { {P rSub { size 8{2} } - P rSub { size 8{1} } } over {R} } } {} and R = 8 η l π r 4 size 12{R= { {8ηl} over {π`r rSup { size 8{4} } } } } {} give the following expression for flow rate:

Q = ( P 2 P 1 ) πr 4 8 η l . size 12{Q= { { \( P rSub { size 8{2} } - P rSub { size 8{1} } \) πr rSup { size 8{4} } } over {8ηl} } } {}

This equation describes laminar flow through a tube. It is sometimes called Poiseuille’s law for laminar flow, or simply Poiseuille’s law    .

Using flow rate: plaque deposits reduce blood flow

Suppose the flow rate of blood in a coronary artery has been reduced to half its normal value by plaque deposits. By what factor has the radius of the artery been reduced, assuming no turbulence occurs?

Strategy

Assuming laminar flow, Poiseuille’s law states that

Q = ( P 2 P 1 ) πr 4 8 η l . size 12{Q= { { \( P rSub { size 8{2} } - P rSub { size 8{1} } \) πr rSup { size 8{4} } } over {8ηl} } } {}

We need to compare the artery radius before and after the flow rate reduction.

Solution

With a constant pressure difference assumed and the same length and viscosity, along the artery we have

Q 1 r 1 4 = Q 2 r 2 4 . size 12{ { {Q rSub { size 8{1} } } over {r rSub { size 8{1} } rSup { size 8{4} } } } = { {Q rSub { size 8{2} } } over {r rSub { size 8{2} } rSup { size 8{4} } } } } {}

So, given that Q 2 = 0 . 5 Q 1 size 12{Q rSub { size 8{2} } =0 "." 5Q rSub { size 8{1} } } {} , we find that r 2 4 = 0 . 5 r 1 4 size 12{r rSub { size 8{2} } rSup { size 8{4} } =0 "." 5r rSub { size 8{1} } rSup { size 8{4} } } {} .

Therefore, r 2 = 0 . 5 0 . 25 r 1 = 0 . 841 r 1 size 12{r rSub { size 8{2} } = left (0 "." 5 right ) rSup { size 8{0 "." "25"} } r rSub { size 8{1} } =0 "." "841"r rSub { size 8{1} } } {} , a decrease in the artery radius of 16%.

Discussion

This decrease in radius is surprisingly small for this situation. To restore the blood flow in spite of this buildup would require an increase in the pressure difference P 2 P 1 size 12{ left (P rSub { size 8{2} } - P rSub { size 8{1} } right )} {} of a factor of two, with subsequent strain on the heart.

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Coefficients of viscosity of various fluids
Fluid Temperature (ºC) Viscosity η (mPa·s)
Gases
Air 0 0.0171
20 0.0181
40 0.0190
100 0.0218
Ammonia 20 0.00974
Carbon dioxide 20 0.0147
Helium 20 0.0196
Hydrogen 0 0.0090
Mercury 20 0.0450
Oxygen 20 0.0203
Steam 100 0.0130
Liquids
Water 0 1.792
20 1.002
37 0.6947
40 0.653
100 0.282
Whole blood The ratios of the viscosities of blood to water are nearly constant between 0°C and 37°C. 20 3.015
37 2.084
Blood plasma See note on Whole Blood. 20 1.810
37 1.257
Ethyl alcohol 20 1.20
Methanol 20 0.584
Oil (heavy machine) 20 660
Oil (motor, SAE 10) 30 200
Oil (olive) 20 138
Glycerin 20 1500
Honey 20 2000–10000
Maple Syrup 20 2000–3000
Milk 20 3.0
Oil (Corn) 20 65

Questions & Answers

Give an example (but not one from the text) of a device used to measure time and identify what change in that device indicates a change in time.
David Reply
hour glass, pendulum clock, atomic clock?
S.M
tnks
David
how did they solve for "t" after getting 67.6=.5(Voy + 0)t
Martin Reply
Find the following for path D in [link] : (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish.
David Reply
the topic is kinematics
David
can i get notes of solid state physics
Lohitha
just check the chpt. 13 kinetic theory of matter it's there
David
is acceleration a fundamental unit.
David Reply
no it is derived
Abdul
no
Nisha
K thanks
David
hi guys can you teach me how to solve a logarithm?
Villaflor Reply
how about a conceptual framework can you simplify for me? needed please
Villaflor
Hello what happens when electrone stops its rotation around its nucleus if it possible how
Afzal
I think they are constantly moving
Villaflor
yep what is problem you are stuck into context?
S.M
not possible to fix electron position in space,
S.M
Physics
Beatriz
yes of course Villa flor
David
equations of kinematics for constant acceleration
Sagcurse Reply
A bottle full of water weighs 45g when full of mercury,it weighs 360g.if the empty bottle weighs 20g.calculate the relative density of mercury and the density of mercury....pls I need help
Lila Reply
well You know the density of water is 1000kg/m^3.And formula for density is density=mass/volume Then we must calculate volume of bottle and mass of mercury: Volume of bottle is (45-20)/1000000=1/40000 mass of mercury is:(360-20)/1000 kg density of mercury:(340/1000):1/50000=(340•40000):1000=13600
Sobirjon
the latter is true
Sobirjon
100g of water is mixed with 60g of a liquid of relative density 1.2.assuming no changes in volume occurred,find the average relative density of the mixture...take density of water as 1g/cm3 and density of liquid 1.2g/cm3
Lila
plz hu can explain Heisenberg's uncertainty principle
Emmanuel Reply
who can help me with my problem about acceleration?
Vann Reply
ok
Nicholas
how to solve this... a car is heading north then smoothly made a westward turn during the travel the speed of the car remains constant at 1.5km/h what is the acceleration of the car? the total travel time of the car as it smoothly changed its direction is 15 minutes
Vann
i think the acceleration is 0 since the car does not change its speed unless there are other conditions
Ben
yes I have to agree, the key phrase is, "the speed of the car remains constant...," all other information is not needed to conclude that acceleration remains at 0 during the entire time
Luis
who can help me with a relative density question
Lila
1cm3 sample of tin lead alloy has mass 8.5g.the relative density of tin is 7.3 and that of lead is 11.3.calculate the percentage by weight of tin in the alloy. assuming that there is no change of volume when the metals formed the alloy
Lila
morning, what will happen to the volume of an ice block when heat is added from -200°c to 0°c... Will it volume increase or decrease?
adefenwa Reply
no
Emmanuel
hi what is physical education?
Kate
BPED..is my course.
Kate
No
Emmanuel
I think it is neither decreases nor increases ,it remains in the same volume because of its crystal structure
Sobirjon
100g of water is mixed with 60g of a liquid of relative density 1.2.assuming no changes in volume occurred,find the average relative density of the mixture. take density of water as 1g/cm3 and density of liquid as 1.2g/cm3
Lila
Sorry what does it means"no changes in volume occured"?
Sobirjon
volume can be the amount of space occupied by an object. But when an object does not change in shape it will still occupy the same space. Thats why the volume will still remain the same
Ben
Most soilds expand when heated but if it changes state at 0C it will have less volume. Ice floats because it is less dense ie a larger mass per unit volume.
Richard
how to calculate velocity
Okwethu Reply
v=d/t
Emeka
his about the speed?
Villaflor
how about speed
Villaflor
v=d/t
Nisha
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Mine is good. How about you?
Chase
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lawan Reply
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Okwethu
hello
akinmeji
Hello
Mishael
hello
Jerry
hi
Sakhi
hi
H.C
so, what is going on here
akinmeji
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Ajayi
good morning ppl
ABDUL
If someone has not studied Mathematics enough yet, should theu study it first then study Phusics or Study Basics of Physics whilst srudying Math as well?
Riaz Reply
whether u studied maths or not, it is advisable to start from d basics cuz it is essential to know dem
Nuru
yea you are right
Badmus
wow, you got this w/o knowing math
Thomas
I guess that's it
Thomas
later people
Thomas
mathematics is everywhere
Anand
thanks but dat doesn't mean it is good without maths @Riaz....... Maths is essential in sciences particularly wen it comes to PHYSICS but PHYSICS must be started from the basic which may also help in ur mathematical ability
Nuru
A hydrometer of mass 0.15kg and uniform cross sectional area of 0.0025m2 displaced in water of density 1000kg/m3.what depth will the hydrometer sink
Lila
16.66 meters?
Darshik
16.71m2
aways
,i have a question of let me give answer
aways
the mass is stretched a distance of 8cm and held what is the potential energy? quick answer
aways
oscillation is a to and fro movement, it can also be referred to as vibration. e.g loaded string, loaded test tube or an hinged door
Olatunji Reply
Practice Key Terms 5

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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