When an object moves through a fluid, as when a ball falls through air or a glass sphere falls through water, the fluid exerts a viscous force F on the object. This force tends to slow the object. For a small sphere of radius r, moving slowly at a speed v, the magnitude of the viscous force is given by Stocke’s law, Fv = 6 πη rv. In this formula, η is the coefficient of viscosity of the fluid, which is the proportionality constant that determines how much tangential force is required to move a fluid layer at a constant speed v, when the layer has an area A and is located a perpendicular distance z from an immobile surface. The magnitude of the force is given by F = η Av/z. For a viscous fluid to move from location 2 to location 1 along a pipe of radius R and length L, the pressure at location 2 must exceed that at location 1. Poiseuille’s law gives the volume flow rate Q that results from such a pressure difference P 2 – P 1 . The flow rate is expressed by the formula:

Poiseuille’s law remains valid as long as the fluid flow is laminar. For a sufficiently high speed, however, the flow becomes turbulent. Flow is laminar as long as the Reynolds number is less than approximately 2000. This number is given by the formula:

Re =

in which is the average speed, ρ is the density. η is the coefficient of viscosity of the fluid, and R is the radius of the pipe. Take the density of water to be ρ = 1000 kg/m 3 .

(i)Which of the following may be concluded from the information in the passage?