Final answer:
The pressure difference must increase by a factor of about 1.50 to maintain the same flow rate when the radius is reduced to 90% of its original value. Turbulence would further impair the flow, increasing the required pressure to sustain flow rate.
Step-by-step explanation:
The subject in question involves the physics of fluid dynamics, specifically within the context of the human circulatory system. The relationship between flow rate, pressure, and vessel radius is described by Poiseuille's law, which shows the flow rate (Q) is directly proportional to the fourth power of the radius (r) of the blood vessel and the pressure difference (ΔP) and inversely proportional to the viscosity (η) and the length (L) of the vessel:
Q = (πΔPr⁴) / (8ηL)
If a blood vessel's radius decreases to 90% of its original size, we can denote the new radius as 0.9r. Because the flow rate is proportional to the radius to the fourth power, the new flow rate would be (0.9r)⁴, which is 0.6561 times the original flow rate. To keep the flow rate constant, we need to compensate for this reduction by increasing the pressure difference by the inverse proportion. Thus, the pressure difference must be increased by 1/0.6561, which equals approximately 1.524. Therefore, the pressure difference must increase by a factor of roughly 1.50 to maintain the same flow rate when the radius is reduced by 10%.
In part (b), the creation of turbulence from an obstruction would further reduce the flow rate, as turbulent flow is less efficient than laminar flow, dissipating more energy and requiring additional pressure to maintain the same flow rate.