Final answer:
To determine the new flow rate in each scenario, one can apply the changes to the variables in Poiseuille's law, which relates flow rate to pressure difference, tube radius, fluid viscosity, and tube length.
Step-by-step explanation:
When considering the sensitivity of the flow rate to various factors in a tube, Poiseuille's law provides insight. According to the law, the flow rate Q is proportional to the pressure difference (ΔP) and the fourth power of the radius (r⁴), and inversely proportional to the viscosity (η) and the length (L) of the tube. The formula is Q = (πΔPr⁴) / (8ηL).
Given the scenarios:
- (a) Pressure difference increases by a factor of 1.50, the new flow rate will be 1.50 times the original, so Q_new = 1.50 * Q_original.
- (b) If a new fluid with 3.00 times greater viscosity is used, the new flow rate will be 1/3 that of the original, so Q_new = Q_original / 3.
- (c) If the tube length is increased by 4.00 times, the new flow rate will be 1/4 of the original, so Q_new = Q_original / 4.
- (d) If another tube is used with a radius 0.100 times the original, the new flow rate will be (0.100)⁴ = 0.0001 times the original, so Q_new = 0.0001 * Q_original.
- (e) With a radius of 0.100 times the original and half the length, and the pressure difference increased by a factor of 1.50, the new flow rate incorporates both changes. So Q_new = 1.50 * (0.100)⁴ / 0.50 * Q_original.
The new flow rates can be calculated by substituting the original flow rate (100 cm³/s) into each of these expressions.