How many times does the flow increase if the radius of a tube is quadrupled, assuming laminar flow?

When considering laminar flow through a cylindrical tube, the flow rate is significantly influenced by the radius of the tube, as described by the Hagen-Poiseuille equation. This equation states that the volumetric flow rate (Q) is directly proportional to the fourth power of the radius (r) of the tube. Specifically, the equation can be expressed as:

Q ∝ r^4

This means that if the radius of the tube is increased, the flow rate will change according to the fourth power of the new radius.

In this scenario, if the radius is quadrupled (multiplied by 4), we can represent the new radius as 4r. Plugging this new radius into the equation gives us:

New Flow Rate ∝ (4r)^4 = 256r^4

To find out how many times the flow has increased, we compare the new flow rate to the original flow rate:

Original Flow Rate ∝ r^4

New Flow Rate ∝ 256r^4

Now we can calculate the increase in flow rate:

Increase in Flow Rate = New Flow Rate / Original Flow Rate = 256r^4 / r^4 = 256

This indicates that the flow rate increases