Calculate curl of rotating bucket of water

In summary, a bucket of water rotating at a steady state with angular velocity w exhibits a velocity field v(r) similar to a rigid body. The divergence of v is 0, indicating the absence of sources or sinks. However, the curl of v is 2w, suggesting the presence of a curl. This vector field cannot be represented as the gradient of a scalar function.
  • #1
joe:)
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Homework Statement



A bucket of water is rotated slowly with angular velocity w about its vertical axis..When a steady state has been reached the water rotates with a velocity field v(r) as if it were a rigid body. Calculate div(v) and interpret the result. Calculate curl (v). Can the flow be represented in terms of a velocity potential such that v = grad phi? If so, what is phi?

Homework Equations





The Attempt at a Solution



Not sure how to do this...
I think somewhere in my notes it says that v here should = (-wy,wx,0)

Is this right?

In which case, div(v) = 0 - what is the interpretation?? No sinks or sources??
curl v = (0,0,2w)...

Can I write it as the gradient of a scalar function? If so, what is the function?
 
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  • #2


I just don't see how to write it as the gradient of a scalar function basically..
 
  • #3


anyoneee?
 
  • #4


joe:) said:
I just don't see how to write it as the gradient of a scalar function basically..

The most simple approach here would be to draw a diagram showing the velocity vector, and how it's always perpendicular to the radius vector. (Every droplet goes around in a circle with its energy constant)

This yields:
[tex]\vec v= -\omega y \hat x +\omega x \hat y[/tex]

Taking the divergence of this function, does indeed get you 0. That means no sources or sinks are present.

Taking the curl, however, shows you that this vector field does have a curl!

[tex]\nabla \times \vec v= 2\omega \hat z[/tex]

A vector field that can be represented as the gradient of some scalar function must always have 0 curl. (Some googling brought up this: http://en.wikipedia.org/wiki/Conservative_vector_field)

Note that it is a mathematical identity that [tex]\nabla\times\nabla\phi=0[/tex] making it impossible for a gradient to have non-zero curl.
If a vector field can be represented as the gradient of a scalar function, then it MUST have 0 curl.
However, if a vector field has 0 curl, the converse is not true in general, you may be able to represent it as the gradient of a scalar function, and you might not be able to do so.
 
Last edited:

What is the curl of a rotating bucket of water?

The curl of a rotating bucket of water is a measure of the rotation or circulation of the water within the bucket. It is a vector quantity that describes how the water is rotating around a central axis.

How do you calculate the curl of a rotating bucket of water?

The curl of a rotating bucket of water can be calculated using the fluid dynamics equations, specifically the Navier-Stokes equations. These equations take into account factors such as the velocity, pressure, and density of the water to determine the curl.

What factors affect the curl of a rotating bucket of water?

Several factors can affect the curl of a rotating bucket of water, including the speed and direction of rotation, the shape and size of the bucket, and the properties of the water such as viscosity and density. Other external factors, such as friction and turbulence, can also play a role.

What is the significance of calculating the curl of a rotating bucket of water?

Understanding the curl of a rotating bucket of water is important in various fields, including fluid dynamics, meteorology, and oceanography. It can help predict and analyze the movement of water in natural and man-made systems, and can also aid in designing efficient water systems.

Are there any practical applications of knowing the curl of a rotating bucket of water?

Yes, there are many practical applications of knowing the curl of a rotating bucket of water. For example, it can be used to study and predict weather patterns, design efficient water turbines, and understand the dynamics of ocean currents. It also has applications in industries such as marine transportation and hydraulic engineering.

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