Final answer:
The equation for the volume flow rate for laminar flow in a circular duct is dimensionally consistent since the dimensions derived from Poiseuille's law match those for flow rate, which are [L^3T^-1].
Step-by-step explanation:
To assess whether the equation for the volume flow rate for laminar flow of a Newtonian fluid in a circular duct is dimensionally consistent, we should consider the dimensions associated with flow rate (Q) and compare them with the dimensions of the other variables in the equation, such as the cross-sectional area (A), average velocity (v), and any other constants. Given that flow rate is defined as the volume of fluid passing through an area per unit time, its dimensions are [L3T-1], where L is length and T is time. Dimensional consistency in this context means that both sides of the equation must have the same dimensions.
According to Poiseuille’s law, for laminar flow through a tube, the flow rate is directly proportional to the fourth power of the radius of the tube, the pressure difference, and inversely proportional to the tube’s length and the viscosity of the fluid. Ensuring that the combination of these variables results in dimensions equivalent to flow rate would confirm that the equation is dimensionally consistent. Since these dimensions indeed coincide as expected in Poiseuille's law, the equation is dimensionally consistent. Therefore, the correct answer is (a) Yes, it is dimensionally consistent.