# 12.4 Viscosity and laminar flow; poiseuille’s law  (Page 3/12)

 Page 3 / 12

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 $R$ to laminar flow of an incompressible fluid having viscosity $\eta$ through a horizontal tube of uniform radius $r$ and length $l$ , such as the one in [link] , is given by

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.

Let us examine Poiseuille’s expression for $R$ to see if it makes good intuitive sense. We see that resistance is directly proportional to both fluid viscosity $\eta$ and the length $l$ 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 $r$ 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 $r$ 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 ${2}^{4}=\text{16}$ .

Taken together, $Q=\frac{{P}_{2}-{P}_{1}}{R}$ and $R=\frac{8\eta l}{\pi {r}^{4}}$ give the following expression for flow rate:

$Q=\frac{\left({P}_{2}-{P}_{1}\right){\mathrm{\pi r}}^{4}}{8\eta l}\text{.}$

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

$Q=\frac{\left({P}_{2}-{P}_{1}\right){\mathrm{\pi r}}^{4}}{8\eta l}\text{.}$

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

$\frac{{Q}_{1}}{{r}_{1}^{4}}=\frac{{Q}_{2}}{{r}_{2}^{4}}\text{.}$

So, given that ${Q}_{2}=0\text{.}\text{5}{Q}_{1}$ , we find that ${r}_{2}^{4}=0\text{.}{5r}_{1}^{4}$ .

Therefore, ${r}_{2}={\left(0\text{.}5\right)}^{0\text{.}\text{25}}{r}_{1}=0\text{.}\text{841}{r}_{1}$ , 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 $\left({P}_{2}-{P}_{1}\right)$ of a factor of two, with subsequent strain on the heart.

Coefficients of viscosity of various fluids
Fluid Temperature (ºC) Viscosity $\eta \phantom{\rule{0.25em}{0ex}}\text{(mPa·s)}$
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

a rare fraction and adjacent contraction of a sound wave are travelling in air and are separated by distance of 15cm, taken that velocity sound is 330ms what is the frequency?
Jennifer Reply
u need answer
Abbas
yes pls
Jennifer
Is ans 64/3
yuvraaj
2200hertz
chiadikaobi
what is speed and velocity
Yaseer Reply
speed is scaler quantity and velocity is vector quantity speed has only magnitude and velocity has magnitude+direction both speed and velocity has same unit m/s
shubham
I am finding it hard to draw the velocity time graph, please can anyone assist me to know how to draw it, please?
bona
explain and draw how to measure length when using ruler, micrometer screw gauge and vernnier calliper
gift Reply
Calculate the average velocity in time interval 6sec to 12sec and determine the instantaneous velocity
Lekiisi Reply
v/6
chiadikaobi
the force is not constant in this case of tow car collide for short period of time ..why is the force is not constant?
Abel Reply
hmmm
Fortune
calculate the average velocity in time interval 6sec to 12sec ad determine the instantaneous velocity
Lekiisi
meaning of the term si units
Chali Reply
what what causes electric current
Chali
what what causes electric current
Chali
Charges(electron)
Caleb
Systeme international unitq@1qa@aaq
Caleb
Correct one international system of units.
Caleb
electric dynamo
albert
Electric dynamo is the sources of electric magnetic forces which utilize electromagnetic induction.
albert
A stone is dropped down a well, if it take 5 seconds to reach the water, how dip is the well
Mollamin Reply
125m dip
chiadikaobi
an aircraft at as steady velocity of 70m/so eastwards at a height of 800me drops a package of supplies .a, how long will it take for the package to rich the ground? b, how fast will it be going as it lands?
Ng Reply
what is hypothesis theory law
Tamba
144.57m/s
chiadikaobi
physics is the science of measurement
Jide Reply
What is physics
Victor Reply
what is physics
Obaro
Good question! Physics is the study of the nature world . Does this help?
Yonn
physics is the study of matter in relations to energy.
Enoch
physics is the science of measurements
Jide
physics is a science concern with nature and properties of matter and energy
Ugomma
what is a parallelogram law of motion?
Nancy
describe how you would find the area of an irregular shaped body?
Chali
physics is the study of natural phenomena at their most fundamental level
Yaseer
Definition for physics
Adesola Reply
It deal with matter and relation to energy
Soughie
physics is the Study of matter in relation to energy.
albert
physics is a natural science that study matter its behaviour and relation to energy.
mohammed
physic tells us more about quantities and measurement also
Kelly
life as we know it that can be measured and calculated
Jesus
what is a reference frame
Chukwu Reply
what is anatomy in relation to physics
Mubarak Reply
how does half life exist
Humble Reply
The amount of time it takes a radioactive isotope to decay into a stable isotope is different for each radioactive isotope, and is characterized by its “half-life”. An isotope's half-life is the amount of time it takes for half the number of atoms of that isotope to decay to another isotope.
Nardine
what is the difference between Mass and weight
Pjustin
mass is constant while weight varies. unit of mass is kg, unit of weight is newton
Faith
how can a coin float in water and what principle governs it
Mercy Reply
in my opinion that work of surface tension but restrictions on coin is that coin do not break surface energy of molecules but some days before scientists prove that's another types of force
Aman
which force hold floating coins together thats my confusion
Aman

### Read also:

#### Get the best College physics course in your pocket!

Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics' conversation and receive update notifications?

 By By By By By By Rhodes By