Final answer:
The rate of increase in fluid velocity in a tapered pipe can be determined by the continuity equation for incompressible fluids, which states that the product of cross-sectional area and velocity must remain constant. As the diameter decreases by a certain factor, the area decreases by the square of that factor, causing the velocity to increase inversely by the square of that factor to maintain constant flow rate.
Step-by-step explanation:
The problem described involves the principles of fluid dynamics specifically relating to the continuity equation for incompressible fluids. Given a pipe with different diameters at each end and a known initial velocity of fluid at the larger end, we must determine the rate of increase in velocity as the fluid flows towards the smaller end. This situation is similar to the behavior of an ideal incompressible fluid in a Venturi tube, where the flow rate (Q), which is the product of cross-sectional area (A) and velocity (u), must remain constant throughout the pipe. The continuity equation for an incompressible fluid is Q = A1u1 = A2u2. Here, A1 and u1 are the cross-sectional area and velocity at the larger end of the pipe, while A2 and u2 are the respective values at the smaller end.
To calculate the rate of increase in velocity, we need to establish the relationship between the velocities and the diameters at the two ends. The cross-sectional areas are related to the diameters by the formula A = (πd2)/4. Since the diameter decreases by a factor (let's say α), the area decreases by a factor of α2, due to the square in the area formula. By substituting area in terms of diameter into the continuity equation, we find that the velocity must increase by a factor inversely proportional to α2 to maintain constant flow.