Final answer:
The new volume flow rate will be 0.6561 times the original flow rate if a blood vessel constricts to 90% of its original diameter, with no change in pressure, according to Poiseuille's law.
Step-by-step explanation:
The ratio of the new volume flow rate to the original flow rate, when a blood vessel narrows to 90% its original diameter with constant pressure across the vessel, can be found using Poiseuille's law. Poiseuille's law states that the flow rate (Q) is proportional to the fourth power of the radius (r) of the vessel. Therefore, if the radius decreases to 90% of its original value, the new flow rate will decrease to (0.90)4 = 0.6561 times the original flow rate.
This is because the resistance to flow changes drastically with small changes in diameter. The flow rate in a blood vessel is inversely proportional to the fourth power of its radius. In this case, the vessel narrows to 90% of its original diameter, which means the radius decreases to 0.9 times its original radius.