Mathematical Motivation: Consequences of Nonarusarity. The relationship between the flow F (say in cubic centimeters per second) of a liquid through a cylindrical tube of radius r and length L is given by the Hagen-Poiseuille equation
F = ΔP r4 / ηL (1)
where η is the viscecity of the fluid and ΔP is the pressure difference in the tube. What are the consequences of this nonlinear relationship between flow and radius?
Putting it Together: Consider the flaw of blood with *cocky n through an artery, which we model as a cylindrical tube of radius r, length L and a pressure difference ΔP produced by the beating heart. Writing k = ΔP / ηL, the Hagen-Poiseuille equation becomes
F(r) = kr4 (2)
1. The radius of a typical open artery is 1.5 mm. In the "mild" case of CAD, plaque may Fla the walls so that it rcducc, the radius of the open lumen of this occluded artery to 0.75 min. Assuming that the other variables affecting blood flow remain the same (that is, it in equation (2) remains the same), use equation (2) to determine the ratio of blood flow in the 5M% laded artery vs. the oPen artery. Show your calculations.
2. Death by Nonlinearity. Om body attempts to compensate with reduced blood flow (that is, volume/time) in part by increasing the blood pressure. Normal blood pressure varies between 120 mm Hg maximum (systolic) just after pumping of the leh ventricle and 80 mm Hg just before the left ventricle pumps again. If your systolic pressure gets above 140 mm Hg, you will be dimmosed with incipient high blood prams. Higher pressures can mean 'gnificant damage to the body's or
How much would the systolic blood pressure have to increase in order to restore the flow in the 50% occluded artery so that it matches the flow in an open artery with a systolic prams of 120 mm Hg, Show your calculations using equation (1). (Note: the flow in an open artery at the systolic pressure of 120 mm Hg is fowid using ΔP = 120 and r = 1.5 in equation (1); you need to find the value of ΔP to equal the flow in an open artery if r = 0.75. That's, you need to solve for ΔP.
(120/ηL) (1.5)4 = (ΔP/ηL) (0.75)4