Prove: Rotation of Flows Satisfies Equation of Continuity

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In summary, the conversation discusses how to prove that the equation of continuity and boundary conditions are satisfied by V = (omega) x r when liquid rotates within a closed circular cylinder. It is mentioned that 'omega' is the angular velocity of the fluid, which is dependent on time only, and r is the position vector measured from a point on the axis of rotation. The continuity equation in cylindrical coordinates is also attached. The person asking for help requests a hint on how to start the solution.
  • #1
banerjeerupak
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Homework Statement



If liquid contained within a closed circular cylinder rotates about the axis of the cylinder, prove that the equation of continuity and boundary conditions are satisfied by V = (omega) x r, where 'omega' is the angular velocity of fluid supposed to be dependent on time only, and r is the position vector measured from a point on the axis of rotation.

Homework Equations



V = 'omega' x r

The continuity equation in the cylindrical co-ordinates is attached as a word document

The Attempt at a Solution



I need a hint as to how to start the solution.
 

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  • #2
Sorry but did you write the question exactly as it was posed and with everything included? I'm a bit startled.
 
  • #3
Yes i wrote the question out exactly as it was given to me.
 

1. What is the equation of continuity?

The equation of continuity is a fundamental principle in fluid mechanics that states that the mass of a fluid flowing through a given area must remain constant. In other words, the rate at which fluid enters a certain region must be equal to the rate at which it leaves that region.

2. How does the equation of continuity relate to fluid rotation?

The equation of continuity is a fundamental principle that applies to all types of fluid flow, including rotational flow. It states that the total mass of fluid entering a given region must be equal to the total mass of fluid leaving that same region. This means that for rotational flow, the amount of fluid entering and leaving a region must take into account the effects of rotation.

3. What is the implication of proving that rotation of flows satisfies the equation of continuity?

The implication of proving that rotation of flows satisfies the equation of continuity is that it helps to validate the mathematical models and equations used to describe fluid motion. It also helps to better understand the behavior of rotational flows and their impact on the overall fluid dynamics of a system.

4. How is the equation of continuity used in practical applications?

The equation of continuity is used in a variety of practical applications, such as designing pumps, turbines, and other fluid machinery. It is also used in the analysis of pipe networks, open channel flows, and other fluid systems. In addition, it is an important tool in meteorology and oceanography for understanding large-scale fluid motion.

5. Are there any limitations to the equation of continuity?

While the equation of continuity is a fundamental principle in fluid mechanics, it does have some limitations. It assumes that the fluid is incompressible and has a consistent density throughout the system. In reality, there may be variations in density due to changes in temperature or composition, which can affect the accuracy of the equation of continuity. Additionally, it does not take into account external forces such as friction or turbulence, which can also impact the accuracy of the equation.

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