Question

Water flowing through a cylindrical pipe suddenly comes to a section of the pipe where the diameter decreases to 86% of its previous value. If the speed of the water in the larger section of the pipe was 32 m/s what is its speed in this smaller section if the water behaves like an ideal incompressible fluid?

Answer 1

The speed in the smaller section is
`43.2\,(m)/(s)`

Step-by-step explanation:

Assuming all the parts of the pipe are at the same height, we can use continuity equation for incompressible fluids:

`\Delta Q=0` (1)

With Q the flux of water that is
`Av` with A the cross section area and v the velocity, so by (1):

`A_(2)v_(2)-A_(1)v_(1)=0`

subscript 2 is for the smaller section and 1 for the larger section, solving for
`v_(2)`:

`v_(2)=(A_(1)v_(1))/(A_(2))` (2)

The cross section areas of the pipe are:

`A_(1)=(\pi)/(4)d_(1)^(2)`

`A_(2)=(\pi)/(4)d_(2)^(2)`

but the problem states that the diameter decreases 86% so
`d_(2)=0.86d_(1)`, using this on (2):

`v_(2)=((\pi)/(4)d_(1)^(2)v_(1))/((\pi)/(4)d_(2)^(2))=\frac{\cancel{(\pi)/(4)d_(1)^(2)}v_(1)}{\cancel{(\pi)/(4)}(0.86\cancel{d_(1)})^(2)}\approx1.35v_(1)`

`v_(2)\approx(1.35)(32)\approx43.2\,(m)/(s)`