Final answer:
For Poiseuille flow through an elliptic pipe, the flow rate is proportional to (a²+b²), which is the product of the squares of the semiaxes of the ellipse since the cross-sectional area of the pipe is proportional to π times the product of the semiaxes.
Step-by-step explanation:
For Poiseuille flow through an elliptic pipe with semiaxes a and b, the flow rate depends on the cross-sectional area of the pipe, which is the area of the ellipse. The formula for the area of an ellipse is πab, where π is Pi, and thus we can infer that the flow rate through an elliptic pipe is proportional to both semiaxes. More specifically, since the cross-sectional area is proportional to Pi times the product of the semiaxes a and b, the flow rate should be proportional to the product of the squares of the semiaxes, given by (a²+b²).
According to Poiseuille's law, for laminar flow of an incompressible fluid of viscosity through a tube, the flow rate Q is directly proportional to the pressure difference and inversely proportional to the length of the tube and the viscosity of the fluid. The flow rate increases with the fourth power of the radius for circular pipes. However, when interpreting for an elliptic pipe, since the area is a more complex shape, we consider it proportional to (a²+b²) where a and b are the semiaxes of the elliptic cross-sectional area.