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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?


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.


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%.


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.

Coefficients of viscosity of various fluids
Fluid Temperature (ºC) Viscosity η (mPa·s)
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
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

how environment affect demand and supply of commodity ?
Amos Reply
Wht at the criteria for market ?
what is difference between monitory policy and fiscal policy?
Malik Reply
monetary policy is a policy thrust by National Govt(CBN) to influence government spending, purchase &taxes
necessity of economics
Pamela Reply
I will say want,choice,opportunity cost,scarcity,scale of preference
what is monopoly market.How price output are determined under monopoly market
b) Monopoly market is an impecfect market where s single firm having the innovation to produce a particular commodity.Prices are determined through output since there are no other competitive.
Monopoly market:firm has market power & does not respond to market price
Explain the process of price determination under perfect competition market with suitable diagram
bisham Reply
Price determination under perfect competition via this process :firms have no market power to influence price rather firms respond to market price.
price is different from demand- demand is amount of commodity
Effah Reply
demand is amount /quantity of commodity a potential buyer is willing to buy at a given price at market
demand is a desire of customer on commodity with the ability to pay it and willing to buy it at given price of commodity
demand is price of what
Faith Reply
show that shortrun average cost
Baby Reply
what is economics
Mbah Reply
what is money
what is money
Difine macro economics
money is a medium of exchange between goods and services,maybe inform of currency.
Economics is study of how human beings strive to satisfy numerous wants using limited available resources.
how do you find the maximum number of workers the firms should employ order to produce where there are increasing returns
what are implications of computing national income?.
what is the formulae for calculating national income
it calculated by value added method
classify the production units like agriculture, banking, transport etc
money is anything that is generally acceptetable for human
Estimate the net value added(NVA) at fixed cost by each industrial structure
definition of unemployment
Adam Reply
what are the causes of unemployment?
Mbubi Reply
The main causes of unemployment are listed below. 1. Frictional unemployment 2. Cyclical unemployment 3. Structural unemployment
We can also categorize the causes on a broader sense as: 1. Political and 2. Social cause As unemployeement root causes are embaded in this two.
would opportunity cost exist if there was no scarcity?
yes just because the opportunity cost arose when there is Alternative to choose among the alternatives.
I am thinking that, if our resources were unlimited, then there wouldn't be any need to forgo some wants. Hence the inexistence if opportunity cost
politics has done what?
consider time assani
I'm Emmanuel,...I taught the main cause is the change in gov't.
...Lack of capital to set up a firm respectively
I would like to bring in Educational levels can also be the cause the cause of the problem respectively
lack of skills among the new generation is the serious issue.
Where I come from , I don't see why education or personal aspects seem to do with unimployment, technically the motivation and eigerness in all works of live is there , dispite the cultural influence and physical bearriors;the thing we lacking is Government Support and open market ethics.
sorry about that-(repation). We have a over powering ethical political system that's displacing the marketing asspects of economy and causing large scale unemployment right across the board...
can someone Explain Expansionary Monetary Policy and Contractionary Monetary Policy Using one of the instrument of Monetary Policy? Please am kinda lost here?. ta
Emmanuel Reply
using a graph show the case of substitute and compliment goods
Ade Reply
can anyone give me a simple explanation to Five Sector Macroeconomics?
Can someone please define what economics is
jason Reply
economics simply is a social science subject that study human behavior.
economics is a social science which studies human behaviour as a relationship between ends and scarce means that has alternative uses
Can someone please tell me how to calculate GDP
emmanual kapal to calculate GDP (Gross Domestic Product) has three method in calculating it (1)income approach (2) expenditure approach (3) value added method
thanks Alae
u are welcome
in basic terms economics is revered to as battery system, it date back to when Men sees the need to exchange sapless goods and produce to gain , either wealth , basic necessities or to establish trading ties for personal benefit or social asspects in terms of coexistence and continuity, future .
what is the law of demand
Berlinda Reply
keep other thing constant, when the price increases demand decrease when the price decreases demand increases of the commodity.
all things being equal,quantity demanded decrease as price increase and increase as price decrease
there's practial joke to it ..." the higher the demand ; scarcity, increase in production and drop in quality"... quite the controversy - for example China vs Europe, United States and we are all boxed up in between somewhere...
Other thing remain constant the low price of commodity the high quantity of commodity and vice versa is true
Explain Effective demand
Anita Reply
What is effective demand
like Modi is in demand...best example of effective demand
Don't get you
Anita you mean you don't get me or who?
level of demand that represents a real intention to purchase by people with the means to pay
Difference between extinct and extici spicies
Amanpreet Reply
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Practice Key Terms 5

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Source:  OpenStax, Physics 105: adventures in physics. OpenStax CNX. Dec 02, 2015 Download for free at http://legacy.cnx.org/content/col11916/1.1
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