# 12.1 Flow rate and its relation to velocity  (Page 3/6)

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${n}_{1}{A}_{1}{\overline{v}}_{1}={n}_{2}{A}_{2}{\overline{v}}_{2}\text{,}$

where ${n}_{1}$ and ${n}_{2}$ are the number of branches in each of the sections along the tube.

## Calculating flow speed and vessel diameter: branching in the cardiovascular system

The aorta is the principal blood vessel through which blood leaves the heart in order to circulate around the body. (a) Calculate the average speed of the blood in the aorta if the flow rate is 5.0 L/min. The aorta has a radius of 10 mm. (b) Blood also flows through smaller blood vessels known as capillaries. When the rate of blood flow in the aorta is 5.0 L/min, the speed of blood in the capillaries is about 0.33 mm/s. Given that the average diameter of a capillary is $8.0\phantom{\rule{0.25em}{0ex}}\mu \text{m}$ , calculate the number of capillaries in the blood circulatory system.

Strategy

We can use $Q=A\overline{v}$ to calculate the speed of flow in the aorta and then use the general form of the equation of continuity to calculate the number of capillaries as all of the other variables are known.

Solution for (a)

The flow rate is given by $Q=A\overline{v}$ or $\overline{v}=\frac{Q}{{\mathrm{\pi r}}^{2}}$ for a cylindrical vessel.

Substituting the known values (converted to units of meters and seconds) gives

$\overline{v}=\frac{\left(5.0\phantom{\rule{0.25em}{0ex}}\text{L/min}\right)\left({\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}\text{/L}\right)\left(1\phantom{\rule{0.25em}{0ex}}\text{min/}\text{60}\phantom{\rule{0.25em}{0ex}}\text{s}\right)}{\pi {\left(0\text{.}\text{010 m}\right)}^{2}}=0\text{.}\text{27}\phantom{\rule{0.25em}{0ex}}\text{m/s}.$

Solution for (b)

Using ${n}_{1}{A}_{1}{\overline{v}}_{1}={n}_{2}{A}_{2}{\overline{v}}_{1}$ , assigning the subscript 1 to the aorta and 2 to the capillaries, and solving for ${n}_{2}$ (the number of capillaries) gives ${n}_{2}=\frac{{n}_{1}{A}_{1}{\overline{v}}_{1}}{{A}_{2}{\overline{v}}_{2}}$ . Converting all quantities to units of meters and seconds and substituting into the equation above gives

${n}_{2}=\frac{\left(1\right)\left(\pi \right){\left(\text{10}×{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}\text{m}\right)}^{2}\left(0.27 m/s\right)}{\left(\pi \right){\left(4.0×{\text{10}}^{-6}\phantom{\rule{0.25em}{0ex}}\text{m}\right)}^{2}\left(0.33×{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}\text{m/s}\right)}=5.0×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\text{capillaries}.$

Discussion

Note that the speed of flow in the capillaries is considerably reduced relative to the speed in the aorta due to the significant increase in the total cross-sectional area at the capillaries. This low speed is to allow sufficient time for effective exchange to occur although it is equally important for the flow not to become stationary in order to avoid the possibility of clotting. Does this large number of capillaries in the body seem reasonable? In active muscle, one finds about 200 capillaries per ${\text{mm}}^{3}$ , or about $\text{200}×{\text{10}}^{6}$ per 1 kg of muscle. For 20 kg of muscle, this amounts to about $4×{\text{10}}^{9}$ capillaries.

## Section summary

• Flow rate $Q$ is defined to be the volume $V$ flowing past a point in time $t$ , or $Q=\frac{V}{t}$ where $V$ is volume and $t$ is time.
• The SI unit of volume is ${\text{m}}^{3}$ .
• Another common unit is the liter (L), which is ${\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}$ .
• Flow rate and velocity are related by $Q=A\overline{v}$ where $A$ is the cross-sectional area of the flow and $\overline{v}$ is its average velocity.
• For incompressible fluids, flow rate at various points is constant. That is,
$\left(\begin{array}{c}{Q}_{1}={Q}_{2}\\ {A}_{1}{\overline{v}}_{1}={A}_{2}{\overline{v}}_{2}\\ {n}_{1}{A}_{1}{\overline{v}}_{1}={n}_{2}{A}_{2}{\overline{v}}_{2}\end{array}}\text{.}$

## Conceptual questions

What is the difference between flow rate and fluid velocity? How are they related?

Many figures in the text show streamlines. Explain why fluid velocity is greatest where streamlines are closest together. (Hint: Consider the relationship between fluid velocity and the cross-sectional area through which it flows.)

Identify some substances that are incompressible and some that are not.

## Problems&Exercises

What is the average flow rate in ${\text{cm}}^{3}\text{/s}$ of gasoline to the engine of a car traveling at 100 km/h if it averages 10.0 km/L?

$\text{2.78}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}\text{/s}$

and yea about that hypothesis thing....For electrons and photons and other microscopic particles ...The dual nature applicable but Macroscopic object follows De Broglie's Hypothesis or Theory
Duh !! And guys I can't see any conversation! All I can guess what is going on ,on board is through notifications ! otherwise this screen is blank ! Does anybody know what is happening ?
Bibekbir
I am facing problem ....I am not able to view the latest conversation in group discussion. ..its just showing a blank page even after reinstalling the app...although the Internet connection is also working
Bibekbir
its ok
Salima
Therefore it was summarised that electrons or any particle for instance (*De Broglie's hypothesis ) shows Dual nature
ok i get it but it is still a hypothesis
Salima
Salima
yup it was as if electrons knew they were being watched ...
Which I personally find very creepy !
Bibekbir
Which I personally find very creepy !
Bibekbir
it is creepy
Salima
Which confused scientists so much ...so tried the same experiment using a microscope this time to watch the motion of the electrons...and surprisingly this time two slits were formed. !
really
Salima
Well what common sense says is ...There will be two slits on photographic plate...
But what was observed was totally unexpected. ..There were wave like imprint on the photographic plate....like water waves...
Bibekbir
ok then...
Salima
Well to explain the dual nature of light ...We need to study Young's double slit experiment .In which,.. okay I want you to imagine I have gun loaded with electrons. ..and before me I have this sheet with two slit in it and behind this sheet a photographic plate. .....what do you think my observation will be ?
can't say anything
Salima
Well light can exist as wave as well as a particle....There is something called Dual nature of light
yes you got it
i know about it but how is it possible
Salima
it depends upon given condition to photon
Bashir
Bashir
Mr. Bashir there is no question that have no answer
Salima
ok
Bashir
Why the Big Bang create more matter then antimatter?
there are a few theories but it is still mostly a mystery
James
Is it possible that the whole concept of antimatter is wrong after all its just theories
Salima
no its not theory
ghulam
but there is no strong evidence about it they are trying to make antimatter what if they are searching for, turns out something different other then antimatter
Salima
half of physics does not consists of the strong evidence
ghulam
salima are you pakistani ?
ghulam
yeah
Salima
Salima
And i am not talking about evidence
Salima
Actually there is an evidence .....its called cosmic inflation. ..
Bibekbir
can you explain
Salima
sure
Bibekbir
It's like this....okay imagine for a second that all the galaxies and stars ...and planet with all their energies vanishes
Bibekbir
Salima
even the black holes dark matter and dark energies also vanishes...
Bibekbir
Can anyone tell me what should be left ?
Bibekbir
ok
Salima
nothing but space
Salima
You will say an empty void ...but that's not the case after this called universe mapping we got this an active energetic background ! which was expanding at a very fast rate
Bibekbir
Now if something is expanding at a speed of nearly speed of light !! then it must have stated from a very small sized object and there you have big bang theory
Bibekbir
yeah i get it
Salima
You will say a void right ? But that was not what mathematical calculations said ....even there is something called universe mapping which mapped the energies in this background. ... and we found that this universe as whole itself was increasing with the speed faster than light !!!
Bibekbir
faster then light!!!
Salima
interesting question. would love a proper explanation.
Beatrax
questions are always intersting only if you know the answer
Salima
or understand the question
Salima
yup faster than light ...now here some people will ask that Einstein had claimed nothing can be faster than light right ? ....but the Universe doesn't count in 'nothing ' and everything is in universe including light ...
Bibekbir
Beatrax do you have a question?
Bibekbir
universe is expending right so there must be a limit for its expension and after that would it be possible that it start contracting again?
Salima
Well we simply don't know to what universe is expanding to .....so here comes string theory....which is story of another time though. ... yes there is a theory though which claims that the universe will again collapse in a small mass just as it started in big bang
Bibekbir
so we can say that big bang will happen again
Salima
Let's come back to your question on antimatter right ? ....well we simply don't have a lot of information about anti matter...though there is a theory which claims that at the time of big bang antimatter was made with matter ....but not in the same quantity. ..
Bibekbir
Because if it were to happen .....The matter won't simply exist ...but we know it does exist....
Bibekbir
there must be some reason that why it creat more matter
Salima
and no the concept of antimatter is NOT wrong!!! Don't ever say that !!! Because some scientists spent their whole life studying on this .....and you will be surprised to know that there are antimatter found on the exosphere of Earth.
Bibekbir
and Ghulam you wrong there Bro. ..70% of physics is mathematics.....which language of universe....which is kinda right every time. ..Einstein gave all his theory solely based on thoughts and mathematics...
Bibekbir
really how is it possible was it natural i know that they are making antimatter but i don't believe it
Salima
And before something become a solid point ... ..It has to go through a long process of scientists in panels and all those stuff. ....and surprisingly it's all observable phenomenons....so yeah physics is actually defination of logic....mind that ..
Bibekbir
Do anyone have any question? feel free to ask ...
Bibekbir
yeah i also think that physics is the basic of all the universe and every thing we see in it
Salima
oh yeah i will
Salima
Yeah we kinda don't know....a lot on this ....moreover....I have not been keeping tap on this topic....so maybe they have got some stuff up their sleeves but I just don't know...
Bibekbir
what is actully light photons or electromagnetic waves
Salima
Well main topic which I focus on is dark matter dark energy and exoplanets. ..
Bibekbir
and maybe little bit of quantum mechanics
Bibekbir
ah there good question !! very good question! !
Bibekbir
are you a student of physics and where are you from
Salima
thanks
Salima
okay now ....We don't only have proton electron and neutron duh ! that's what students are taught!
Bibekbir
Actually from CERN till now 35 subatomic particles have been found !!!
Bibekbir
and now let's come back to your question about what is light photons....I guess you all from 12th grade am I right ?
Bibekbir
oh whats the 35 subatomic particals
Salima
Salima
What? I am actually umm. ...In my first year of college ....Sem1 so yea I am 18 years old....and I am doing undergraduate course in Maths Chemistry and Physics ... I am from India ! Duh ! I thought my name would give away my nationality !
Bibekbir
There are a lot of those subatomic particles like higgs boson particle is one of them
Bibekbir
oh
Salima
Not a lot 35
Bibekbir
i am physics professor any problm dear friend
ghulam
Salima
Salima you know a lot according to your age. ...don't worry you have a while chapter in 11th and 12th physics in electromagnetic waves and light photon ....
Bibekbir
*whole
Bibekbir
i am 16 years old
Salima
Mr. Ghulam No hard feelings I just said what I felt right . And I just love physics a lot can't just read any bad comment about it.Again I said what I felt right. If you are offended by anything I said Sir,then I am sorry . I am not here offend anyone,.just give my honest answers andwhatever I study
Bibekbir
Mr.Ghulam are you a Pakistani
Salima
why physics is more difficult for commerce student?
Bashir
i am a fsc pre medical student
Sameer
hello
James
am james
James
physics is not difficult but when you don't have any interest it seems difficult
Salima
James can you tell us about big bang theory?
Bashir
Salima
Yes but is very necessary for us but how to understand that cause our direction is another side
Bashir
hi Bibekbir are you there
Salima
stay on your direction Mr. Bashir
Salima
Yes
Bashir
Salima
Yes can but ..
Bashir
but..
Salima
i just want answer not "but"
Salima
Bashir
?
Bashir
what light actually photon or electromagnetic waves
Salima
Both
how? can a partical be wave at the same time
Salima
yes photon depend upon condition
photon is the energy packet ..light consist of photons in the form of EM waves
ghulam
What is the difference between potential difference and electromotive force
basically potentiometer is series circuit or parallel circuit?
What is half-life
the life in which half of the radioactive element decay
Arif
what is fluid
anything that flows is Liquid.
prakash
a substance that has no specific shape
Saleemulhaq
How submarines floats one water the same time sink in water
A submarine has the ability to float and sink. The ability to control buoyancy comes from the submarine'strim or ballast tanks which can be filled with either water or air, depending on whether the submarine needs to floator sink. When the submarine floats it means its trim tanks are filled with air
Arif
Anthony
what is work
Force times distance
Karanja
product of force and distance...
Arif
Is physics a natural science?
Nature is all physics
Salima