Final answer:
The flow rate of a fluid in a tube changes in response to adjustments in pressure, viscosity, tube length, and radius due to their relationships outlined in Poiseuille's law. Each factor affects the rate either directly or inversely, in proportion to its relationship with the flow rate.
Step-by-step explanation:
This question relates to the study of fluid dynamics within the field of physics, specifically the behavior and flow rate changes in tubes and pipes due to various factors. Flow rate sensitivity can be calculated using principles from fluid mechanics, in which the variables such as pressure difference, fluid viscosity, tube length, and tube radius have distinct relationships with the flow rate.
The flow rate (Q) of a fluid through a tube can be described by Poiseuille’s law, which states that Q is directly proportional to the pressure difference (ΔP) and the fourth power of the tube's radius (r⁴), and inversely proportional to the viscosity (η) of the fluid and the length (L) of the tube. Mathematically, this can be expressed as:
Q = (π ΔP r^4) / (8 η L)
When analyzing the scenario in which the original conditions are modified, the new flow rate can be found by applying the changes to the corresponding variables within Poiseuille’s law. For instance:
If the pressure difference increases by a factor of 1.50, then the new flow rate would also increase by a factor of 1.50, since it's directly proportional.
If a new fluid with 3.00 times greater viscosity is used, the flow rate would be 1/3 of the original, reflecting the inverse relationship.
Increasing the tube length by 4 times would make the flow rate 1/4 of the original.
A change in tube radius to 0.100 times the original would have a dramatic effect due to the fourth power relationship, decreasing the flow rate to 0.0001 (or 1/10000) of the initial rate.
For the combined changes in scenario (e), the overall factor affecting the flow rate can be calculated by considering the simultaneous effects of pressure increase, reduction in radius, and the change in tube length. The combined result is a product of the individual factors.