Understanding Proportional Changes in Tube Radius and Flow Rate
Understanding Laminar Flow and Fluid Dynamics in Tubes
When analyzing the behavior of fluids in tubes, it is essential to consider factors such as flow rates, pressure, viscosity, and tube dimensions. The student's question revolves around specific proportional changes in the radius of a tube and how that impacts the laminar flow.
The Hagen-Poiseuille Equation:
According to the Hagen-Poiseuille equation which governs laminar flow through a cylindrical pipe, the flow rate Q is directly proportional to the fourth power of the radius r of the tube, given by Q = (ÏÎP r^4) / (8ηL), where ÎP is the pressure difference, η is the viscosity, and L is the length of the pipe.
Calculation Examples:
For part (a), a 5.00% decrease in radius will result in a decrease in flow rate. The new flow rate is proportional to (0.95r)^4, which equals approximately 81.45% of the original flow rate, resulting in an 18.55% decrease.
For part (b), a 5.00% increase in radius will result in an increase in flow rate. The new flow rate is proportional to (1.05r)^4, equivalent to an increase to approximately 121.55% of the original flow rate, indicating a 21.55% increase in flow.
Conclusion:
Moreover, different scenarios in the provided examples demonstrate how flow rate is influenced by changing just one parameter while keeping others constant. The sensitivity of flow rate to changes in pressure, viscosity, tube length, and radius emphasize the complexity in real-world applications like engineering and medicine.