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Nasal passages and scaling laws (again)



I have just asked a friend of mine who is in fluid dynamics about the
scaling laws for fluid flow through irregular tubes. (Of course, all
errors that may occur in the following are caused by me being to
stupid to understand what he told me.)

Mickey Rowe wrote, answering my question
>> Why is it clear that the width of nasal passages (I hope this is the
>> correct term) is strongly correlated with the metabolic rate?
>
>Because (as Guy Leahy seemed to fail to appreciate with respect to the
>implications of the following on concomitant rates of respiratory
>water loss) endotherms almost by definition have to pull in oxygen at
>about ten times the rate of similarly sized ectotherms.

This was indeed clear to me. I think, what I wanted to ask was: Is it
clear that there are no OTHER factors governing the WONP (which I
will use as a shorthand for "width of nosal passage)? Like, for
instance, something to do with sense of smelling. This seems not too
likely, but I just wanted to make sure I understand the arguments correctly

>> The fact that all these data points fall on the same line also tells us,
>> that there is a factor limiting the size of nasal passages FROM ABOVE.
>
>My impression (as above and as clearly shown in the data I have
>available to me) is that the biggest differences here are between the
>two groups (ectotherm and endotherm), and that differences in size
>within the groups are smaller.
(stuff snipped)
>However, with all due respect to complaints about
>the small sample sizes in the study, if the within group variances
>were large then you probably wouldn't have seen the plotted points
>fall so close to the regression lines.  That's not to say that data
>from all birds (or all endotherms) would fall close to the regression
>line created for the paper in question.

Actually, this seems not to answer my question: If the data of all
birds are relatively close on the same line, this means that no bird
exists with, e.g. a nasal passage being 3 or four times larger than it
should. And this means that something is limiting the WONP from above,
otherwise I would guess we would find such a critter somewhere.

Perhaps these two points are a little bit far-fetched - I'm just
trying to make the case waterproof.

>reference to physical laws.  However, as a physicist, can you tell us
>of any biologically relevant situation in which resistance to fluid
>flow increases with increasing tube size (all other things being
>equal)?

Not offhand, but I was interested in whether scaling law itself looks
plausible to me.

>The regression lines from the paper are:
>
>ectotherms:   A = 0.11 M^^0.76
>
>endotherms:   A = 0.57 M^^0.68
>
>where A is nasal cross-sectional area in square centimeters and M is
>mass in kilograms.

Now here comes the true point of my message:
Let us try to guesstimate what type of scaling we would expect.

For turbulent flows through a straight tube the mass flow F will go as
F = c* r**3.25 (where c is a proprtionality constant and r is the tube
radius) if the Reynolds number (a dimensionless number characterising
the behaviour of the flow) is not too high (about 100000).
If the Reynolds number is higher than this the dependency drops down.

For non-straight tubes with some obstacles inside things are depending
on the exact shape of the thing, espescially if the tube is strongly
curved, as I would expect for nasal passages. I was told that assuming
F = c* r**2 would be a reasonable guess for this kind of situation,
although the exact dependency may differ.

So let us assume this law.

Its oxygen need drops a bit with increasing mass, as far as I know
(larger beasts having smaller need of oxygen per mass). So lets take
oxygen need N proportional to M**q (abbreviated propto) , where q is a
number a little smaller than one.

If the way of intaking the air is similar in small and large animals,
we have  N propto F, so  F propto M**q  and on the other hand
F propto r**2 or r**3.25, so finally

r  propto M**(q/2)  or r propto M**(q/3.25) or some exponent in
between.
These exponents vary between about .5 and .3. The situation for the
large beasts would be perhaps even better if they breathe slower,
because the flow will get more steady this way.

So this is a (addmittedly very crude) estimate of the scaling law I
would expect.
The exponents found experimentally where something like .7, and this
is what I wanted to point out: Are my estimates so grossly
oversimplified that we should not be bothered by this difference, or
is there another factor coming in?

                   Dr. Martin Baeker
                   Institut fuer Werkstoffe
                   Langer Kamp 8
                   38106 Braunschweig
                   Germany
                   Tel.: 00-49-531-391-3066
                   Fax   00-49-531-391-3058
                   e-mail <martin.baeker@tu-bs.de>