If the radius of a tube is doubled, how will the laminar flow through the tube change?

When the radius of a tube is doubled, the laminar flow through the tube changes significantly due to the principles of fluid dynamics, particularly the Poiseuille's Law, which describes how the flow rate through a tube relates to various factors, including the radius.

Laminar flow is characterized by smooth and orderly fluid motion, and for a cylindrical tube, the flow rate (Q) can be represented by the equation:

[ Q = \frac{\pi r^4 (P_1 - P_2)}{8 \eta L} ]

In this equation, ( r ) is the radius of the tube, ( P_1 ) and ( P_2 ) are the pressures at either end of the tube, ( \eta ) is the dynamic viscosity of the fluid, and ( L ) is the length of the tube. From this relationship, it can be seen that the flow rate is proportional to the fourth power of the radius.

When the radius is doubled (i.e., ( r ) becomes ( 2r )), the new flow rate can be calculated as follows:

1. Substitute ( 2r ) into the radius portion of Poiseuille's Law:

[ Q'