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R 4 ( P1 P2 ) 8 L
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(12-19)
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where R is the radius of the tube, (the greek letter, eta) is the viscosity of the fluid, L is the length of the tube from point 1 to point 2, and P1 2 P2 is the drop in pressure in going from point 1 to point 2 The interesting thing to note about Poiseuille s law is the dependence on the fourth power of the radius This means, for example, that doubling the radius will increase the volumetric flow rate 16 times It also means that for a given (constant) flow rate and viscosity (as we typically have in the case of blood in arteries), a small increase in radius will
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chapter 12 p h y s i o l o g i c a l a n d a n at o m i c a l B i o p h y s i c s
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make the pressure drop significantly less This is easy to see by rearranging Eq (12-19) to solve for the pressure drop P1 P2 = Q8 L R4 (12-20)
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Equation (12-20) tells us how much the pressure will drop, due to viscosity, as a fluid flows along a tube with a given Q, , L, and R (ie, with a given flow rate, viscosity, length of tubing, and radius) The pressure drop is inversely proportional to the fourth power of the radius This means that increasing the radius just a little bit will make the pressure drop much smaller (so the effect of viscosity is greatly reduced) It also means that for a liquid such as blood that has a relatively small viscosity to begin with it takes only a moderate radius for the viscous pressure drop to be small enough for Bernoulli s equation to become a very good approximation
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still struggling
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in the case of blood flowing in the body, the volumetric flow rate, Q, is constant yes it goes up and down slightly with the pumping of the heart but on average it is constant along the length of a given blood vessel; that is, the rate of blood flowing into one end of a vessel is the same as is flowing out of the other end in such a case where the flow rate is constant, viscous friction reduces the pressure the flow rate is the same at both ends of the vessel, but the pressure is different however, if this pressure change is very small, then the situation can be approximated by a frictionless fluid (Bernoulli s equation) since the pressure drop is inversely proportional to the radius to the fourth power, a small change in the radius can make a big change to the pressure drop A slight increase to the radius can make the pressure drop small and insignificant so that Bernoulli s equation can be applied conversely, a decrease in the radius can cause the pressure drop to be large enough to cause significant negative health effects
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Turbulent Flow
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As the velocity of fluid flow increases, a point is reached where the flow is no longer smooth and laminar Instead eddies are formed These are small currents of fluid that flow backward, opposite the direction of the overall flow See Fig 12-3
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B I O P H y S I C S D e mys tifie D
Figure 12-3 Turbulent fluid flow with eddies (circular currents)
The velocity at which flow changes from laminar to turbulent is called the critical flow velocity The critical flow velocity is proportional to the viscosity, and to a property of the fluid known as the Reynolds number It is inversely proportional to the fluid density and to the radius of the tube vc = 2 R (12-21)
From Eq (12-21) we see that increased viscosity and increased Reynolds number ( ) increase the critical velocity; this makes it less likely at a given velocity that the flow will be turbulent, whereas increasing the radius or the fluid density increases the likelihood of turbulent flow