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Consider a tube with two different diameters (one section wider than the other). Because what comes in must go out (conservation of mass), the flow rate in the wider section must be the same as the flow rate in the smaller section (you can't loses some or what your put in or gain some of what you've not put in). Also, the flow of a liquid follows the pressure gradient (liquid flows from high pressure to low pressure in the tube). The left end of the tube is at a high pressure and the right end is at a lower pressure (consider the fluid moving from left to right). According to continuity,

$\Phi_1$ = flow rate 1 $= \Phi_2$ = flow rate 2

or

$A_1v_1 = A_2v_2$

where $A$ and $v$ are the cross sectional area and velocity respectively. Now consider the a disease/condition where the blood vessels are contracted (think of a normal blood vessel as the bigger diameter tube and the contracted blood vessel as the smaller diameter tube above). I forgot what this disease is called but that's beside the point. The main thing is that blood flow is reduced in the bad section of the blood vessel (small diameter section), requiring surgery to change out the bad section for a normal one. Does this not contradict the continuity equation, which states that the flow rate should be the same in each section?

My response: After talking to a professor, he gave the response (I didn't follow but goes something like) the continuity equation does not take into account the amount of pressure in the bigger diameter section and the amount of pressure in the smaller diameter section (it just takes into account the difference in pressure between the two ends). Can anyone help shed light on this? Please discuss what the continuity equation does and does not do as well as what's going on in the blood vessel please. Final note:

$\phi = \frac{\pi (P_A -P_B)}{8\eta L}R^4$

This equation describes flow rate taking into account the viscous drag force. It was derived considering a cylindrical tube. The derivation starts (I'm giving you this because even if you don't know what the equation is, I'm giving you context. I don't understand this equation either. It considers a tube all with a single diameter. $A = \pi r^2$, $SA$ stands for surface area. Note, velocity is greatest at the center and smallest along the tube walls due to friction)

(driving force) - ($F_{viscous}$) = 0 (at equilibrium)

$\Delta PA$ - $\eta (SA)\frac{dv}{dr}$ = 0

Anyways, the final equation for flow rate shows that $\phi \propto R^4$. I thought flow rate was constant? In summary, why is blood flow reduced in the contracted blood vessel and said to be the same in a pipe with different cross sectional areas?

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The continuity equation only means that the mass flow rate in equals the mass flow rate out. It does not mean that that flow rate never changes; flow in and flow out can both change simultaneously.

For a given pressure drop (pressure upstream minus pressure downstream), the flow rate is proportional to the 4th power of radius, according to the Hagen–Poiseuille equation. If you decrease radius, flow in AND flow out decrease. The continuity equation is still obeyed.

So if there is a decreased radius of a blood vessel, either flow rate decreases, or pressure increases to maintain the orginal flow rate, or there is a combination of pressure increase and flow decrease.

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