Velocity & Volume Change in Ideal Fluids

[Gear2d]

Homework Statement

In an ideal fluid, the pipe length is doubled, while radius is decreased by factor 2 what will have to the velocity and volume that is flowing?

Homework Equations

Q=Av = pi*r^2*v

The Attempt at a Solution

From this the volume that will pass, in ideal fluid cases, is constant (same throughout), but the velocity will increase by a factor of 4 since: Q/v = pi*r^2. Is this correct?

Also if I were considering the radius decreasing by a factor of 2, in real fluids, the velocity would increase by it would not increase as much as seen in ideal fluids?

 

[Topher925]

Makes sense to me. Assuming the fluid is incompressible and the volumetric flow rate remains constant. The problem statement doesn't make those assumptions so I will. In the real world, and making those assumptions, there would be a pressure difference across the pipe due to friction affects and an increase of kinetic energy, however the velocity would remain the same as an ideal fluid. This is of course assuming constant volumetric flow rate.

 

[baba89]

Your attempt at a solution is partially correct. In an ideal fluid, the volume that is flowing will remain constant regardless of changes in pipe length or radius. This is a result of the equation for volume flow rate, Q=Av, where A is the cross-sectional area of the pipe and v is the velocity of the fluid. Since the volume flow rate is constant, any changes in pipe length or radius will result in a corresponding change in velocity.

In the case of doubling the pipe length and decreasing the radius by a factor of 2, the cross-sectional area of the pipe will also decrease by a factor of 2. According to the equation Q=Av, this means that the velocity will increase by a factor of 2 to maintain a constant volume flow rate. So your statement that the velocity will increase by a factor of 4 is not entirely correct.

In real fluids, the velocity will also increase when the radius is decreased, but it will not increase as much as in ideal fluids due to the effects of viscosity and friction. These factors will cause some energy loss and decrease the velocity compared to what would be expected in an ideal fluid. However, the overall trend of velocity increasing with decreasing radius still holds true in both ideal and real fluids.