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The continuity equation (2012)

Page history last edited by wlosert 7 years, 7 months ago

Working content >MacroModels > Fluids

 

Prerequisites

 

 

Continuity

In most of the cases we will consider, the matter of the fluid is neither created or destroyed.  We call this continuity, which means the amount of matter stays the same.  This has some interesting consequences for the flow of fluid through a pipe.  The rate at which fluid goes into the pipe must be equal to the rate at which fluid comes out.  There isn’t anywhere inside the pipe where fluid accumulates or where new fluid is generated. 

 

We have to treat gases (and compressible fluids) differently from incompressible fluids (such as water). First let's consider the harder situation -- a compressible gas.  For a pipe with area, A, the rate at which matter flows into the pipe is I = ρAv, where I is the matter flow rate (kg-m3/s), ρ is the density of the gas, A is the cross sectional area in m2, and v is the fluid velocity in m/s. Let's consider a specific example where the pipe changes its cross section. 
 

 

If the system is a continual flow with nothing changing inside, the mass of gas flowing into and out of the pipe must be the same in each instant of time.  The volume going in during the time Δt is A1Δx1 = A1v1Δt and the volume going out is A2Δx2 = A2v2Δt.  So setting the amount of matter going in equal to the amount going out gives the continuity equation

This is actually pretty complicated since both the density and the speed can change to compensate for the change in area. We would have to bring in more physics to figure out which changed and how.

 

The situation simplifies considerably when we switch to considering a fluid that can be treated as an incompressible fluid. In that case, the density of the fluid doesn't change and the volumes going in and out have to be the same.  Therefore, the volumetric currents must be equal

This tells us that the ratio v2 / v1 is equal to A1 / A2.  Since in our example A2 < A1, this means the flow velocity out of the pipe's narrow end will be faster than the flow velocity into the pipe's broader end.  This makes sense.  In order to keep a constant volume of fluid moving through the pipe, when the pipe’s area gets smaller, the fluid has to move faster. This relation is called the continuity equation for incompressible fluids.

 

Internal flow can be used by organisms to draw fluids into their body, such as by breathing (compressible fluid), or to transport fluids around the body, as in blood circulation (incompressible fluid).  In moving these flows around the body, should they be fast of slow?  In some situations, it is important to get the flow quickly to where it needs to go, say from the heart to the brain, in which case fast flow is useful.  However, once it gets there, it might help to slow the flow down, in order to transfer oxygen or nutrients from the flow to the surrounding tissues -- especially since that transfer takes place by diffusion, and diffusion is slow.

 

Biological implications

Animals have complex blood circulatory systems in which an initial flow into a vessel of large width (the aorta) is branched into smaller and smaller vessels with larger and larger total areas (the capillaries). In order to slow down the flow velocity in the small blood capillaries so diffusion of nutrients and waste products can take place, there must be enough capillaries to increase the overall cross sectional area.  The total capillary area, across all of the small vessels, must be greater than that of the aorta which delivers the heart’s flow. Otherwise, the flow will move just as fast as it does in the aorta.

 

Qaorta = Aaorta* vaorta = Qcap= Acap * vcap 

 

For vcap << vaorta   requires Acap >> Aaorta


 

Some data for the dog circulatory system suggests that the total cross sectional area of the smallest capillaries is 600 cm2 compared to the 2 cm2 of the large aorta, an increase of 300 fold.*  Thus the flow velocity in the capillaries will be 300 times slower.  Flow velocities in human capillaries have been measured to be 0.7 mm/s.  The typical time to travel one cell length (~21 microns) is then x/v= 21 x 10-6 m / (7 x 10-4 m/s) = 3 x 10-2s.  It turns out that this is comparable to the time it takes oxygen, which has a diffusion coefficient of D=1.8 x 10-9 m2 /s  - to diffuse on average across one cell:  t=x2/(6D) =4.1x 10-2s.   Therefore, at the slowest part of the bloodstream, the oxygen can move one cell length just as fast via diffusion as via fluid flow.  One can speculate that this convergence of fluid flow time and diffusion time is no accident:   the slowdown of fluid flow with decreasing capillary size may be the reason that no finer capillaries with even smaller diameters and slower fluid flow speeds exist.  It becomes faster for oxygen to diffuse than to flow on such small lengthscales.  If the capillaries were finer and more closely spaced, it would be faster for oxygen to simply diffuse; if the capillaries were larger the delivery of chemicals to the cells would get less effective! 

 

*Vogel table 10.1 CB pg 211

 

Joe Redish and Karen Carleton 10/26/11

Wolfgang Losert 11/3/2012

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