# Why doesn't Bernoulli's Equation apply in this problem?

• azure kitsune
In summary, we are trying to find the pressure at a distance r from the rotation axis of a cylindrical bucket of liquid rotating at an angular velocity of ω. By considering the net force on a mass element undergoing centripetal acceleration, we can derive the equation P = P0 + (1/2)ρω^2r^2, where P0 is the pressure at r = 0. This is because the equivalent "g" in a rotating reference frame is ω^2r and this potential difference can be found using integration. Bernoulli's equation is not applicable in this scenario, as it works along a streamline, but a proper application would be considering a small portion of the water traveling along its path of
azure kitsune

## Homework Statement

A cylindrical bucket of liquid (density ρ) is rotated about its symmetry axis, which is vertical. If the angular velocity is ω, show that the pressure at a distance r from the rotation axis is

$$P = P_0 + \frac{1}{2} \rho \omega^2 r^2$$

where P0 is the pressure at r = 0.

P = F/A

## The Attempt at a Solution

I was able to get the correct answer by considering the net force on a mass element dm since it is undergoing centripetal acceleration.

$$P = P_0 - \frac{1}{2} \rho \omega^2 r^2$$

Bernoulli's equation works along a streamline, which means the path of fluid flow. In this case, you took the radial pressure difference but used the tangential velocity v instead of the radial velocity of 0. If you use 0 and put back the gravitational potential term, because centripetal acceleration is equivalent to gravity, you'll get the right answer.

I'm sorry. I've been thinking about what you said but I do not understand what to do. I tried to rewrite ρgy as ρ(v2/r) but this isn't working either.

Also, to make sure I understand what streamline means, would this be a proper application of Bernoulli's equation?

If you consider a small portion of the water dm traveling along its path of flow (in a circle), then v1 = ωr = v2 and y1 = y2. Thus P1 = P2, so it's pressure remains constant.

azure kitsune said:
I'm sorry. I've been thinking about what you said but I do not understand what to do. I tried to rewrite ρgy as ρ(v2/r) but this isn't working either.

ρgy is meant to represent the difference in gravitational potential. In the rotating reference frame of the liquid, the equivalent "g" is w^2r, and you'll have to find the potential difference between the center of the liquid and the sides of the cylinder using integration. If you haven't learned integration yet, the potential difference you'd get is (1/2)ρw^2*r^2.

Also, to make sure I understand what streamline means, would this be a proper application of Bernoulli's equation?

If you consider a small portion of the water dm traveling along its path of flow (in a circle), then v1 = ωr = v2 and y1 = y2. Thus P1 = P2, so it's pressure remains constant.

Yup, that would be a proper application of the equation. It doesn't have any practical use, but you've gotten the concept of the streamline.

## 1. Why is Bernoulli's Equation not applicable in this problem?

Bernoulli's Equation is based on the assumptions of inviscid and incompressible fluid flow, which are not always true in real-world situations. Therefore, it may not accurately describe the behavior of fluids in certain scenarios.

## 2. Can Bernoulli's Equation be used for all types of fluid flow?

No, Bernoulli's Equation is only valid for steady, incompressible, and inviscid fluid flow. It cannot be applied to situations where these assumptions do not hold, such as compressible or turbulent flow.

## 3. What are some common limitations of Bernoulli's Equation?

Besides the assumptions of incompressibility and inviscid flow, Bernoulli's Equation also does not consider frictional losses, energy dissipation, and changes in fluid density. These factors can significantly affect the accuracy of the equation in certain cases.

## 4. Are there any alternative equations that can be used instead of Bernoulli's Equation?

Yes, there are other equations, such as the Navier-Stokes equations, that can be used to describe fluid flow in a more comprehensive and accurate manner. However, these equations are more complex and require more extensive calculations.

## 5. Can Bernoulli's Equation be modified to account for the limitations mentioned above?

Yes, modifications have been made to the original Bernoulli's Equation to account for certain limitations, such as the inclusion of a friction factor to consider frictional losses. However, these modifications may only be applicable in specific cases and may not provide a complete solution to the problem.

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