To double the flow rate, by how many times must the pressure be increased?

To understand why increasing the pressure by a factor of four is necessary to double the flow rate, it’s essential to reference the principles of fluid dynamics, specifically the relationship described by the Bernoulli's equation and the continuity equation.

In fluid mechanics, the flow rate through a pipe is proportional to the square root of the pressure difference. More explicitly, if we denote the flow rate as Q and the pressure difference as P, the relationship can typically be expressed in a manner that implies:

[ Q \propto \sqrt{P} ]

This means that if the pressure is increased, the flow rate increases with the square root of that change in pressure.

To determine the required increase in pressure to double the flow rate, we can set up the following relationship:

1. Let the initial flow rate be ( Q_1 ) at an initial pressure ( P_1 ).

2. To find the new pressure ( P_2 ) that will result in a flow rate of ( Q_2 = 2Q_1 ), we can write:

[ 2Q_1 \propto \sqrt{P_2} ]

3. From the proportional relationship, we can set up:

[ \frac{