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a. Poiseuille's equation is given as: Q = \({\Delta}\)P\({\pi}\)r4/8L\({\eta}\)?
b. Poiseuille's equation can be thought of the same way one can think about liquid moving through a drinking straw. The resistance of liquids within the straw is a function of the length of the tube (L), the radius of the tube (r), and the viscosity of the liquid (?). This resistance is given as: R = 8L\({\eta}\)/ \({\pi}\)r4.
c. By substituting the resistance relationship into the law of bulk flow, Q = \({\Delta}\)P/R, you can derive Poiseuille's equation. Based on the terms given in the equation, we know that flow can be affected by the pressure gradient, the radius of the vessel, the length of the vessel, and the viscosity of the fluid. The radius of the vessel (changed by vasoconstriction or vasodilation) can have a huge impact on the flow of the circulating fluid throughout the system, more than some of the other factors, because it is raised to the fourth power.
d. Poiseuille's equation makes a variety of assumptions regarding the physics of the circulatory system. For example, it assumes that the blood vessels are unbranched and rigid, and that the flow is steady. Violations to the equation include:
i. The circulatory system is branched, in both convergent and divergent ways, which makes the length of the system complicated. This violates the simple function of length (L).
ii. Vessel walls are flexible, or "compliant," in response to pressure, which makes the calculation of 'r' complicated. Moreover, there are some vessels that stretch easily when exposed to pressure (highly compliant), while others stretch less. Also, the compliance of a blood vessel is not constant, but becomes less compliant at higher pressures. Finally, blood vessels take time to stretch. Together, these factors add up to the violation of the assumption r.
iii. In larger vessels, pressure of blood flow is pulsatile, increasing when the heart contracts. Also, flow can sometimes be turbulent (instead of laminar), mainly in the heart and some vessel branching points. Additionally, the velocity profile of the blood is not identical across the diameter of the vessel. Instead, flow is slower near the walls, and faster near the center. Overall, flow is complex, and these issues add up to a violation of pressure (P).
iv. Finally, blood is a mixture of components with different viscosities. It acts as a non-Newtonian fluid, which means that its viscosity varies depending on the size of the tube that it flows through. The components of blood tend to separate in smaller blood vessels. That is, blood cells get swept into the high velocity flow in the center of the vessel, while it is mainly plasma at the walls, leading to "low viscosity" and "high viscosity" parts of the vessel. In even smaller vessels, blood cells take up almost the whole diameter of the vessel. Cells change shape to squeeze through, and blood vessels tend to stick to the vessel walls and to each other, leading to high viscosity in those areas. Therefore, fluid is a complex mixture, and viscosity is variable, which violates \({\eta}\).
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