Viscous Fluid Slowing Down

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Homework Statement


A fluid of viscosity [itex]\nu[/itex] is rotating with uniform angular velocity [itex]\Omega[/itex] inside a cylinder of radius [itex]a[/itex] that is also rotating. At time [itex]t = 0[/itex], the cylinder is brought to a rest. The circular motion of the fluid is gradually slowed down due to the viscosity; show that [itex]u_\theta (r, t)[/itex] satisfies the following relation with a Fourier-series approach:

[itex]u_\theta (r,t) = - 2 \Omega a \sum_{n \geq 1}{\frac{J_1(\lambda_n r / a)}{\lambda_n J_0(\lambda_n)} \exp{(- \lambda_n^2 \frac{\nu t}{a^2}})}[/itex]
where [itex]\lambda_n[/itex] is the n-th root of the Bessel function [itex]J_1[/itex].

I'll write [itex]u[/itex] rather than [itex]u_\theta[/itex] from now on for simplicity.

Homework Equations


Boundary condition [itex]u(a, t) = 0[/itex]
Initial condition [itex]u(r, 0) = \Omega r[/itex]
The fluid equations eventually give [itex]\frac{\partial u}{\partial t} = \nu (\frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} - \frac{u}{r^2})[/itex]


The Attempt at a Solution


I won't go through all the details as they take a lot to write. After a standard separation of variables [itex]u(r, t) = A(r )*B(t)[/itex] an equation for [itex]A[/itex] is found that can be brought in the form that has [itex]J_1[/itex] as its solution. I arrived at:

[itex]u(r, t) = \sum_{n \geq 1}{A_n J_1(\lambda_n r / a) \exp{(- \lambda_n^2 \frac{\nu t}{a^2})}}[/itex]

So I need to find each [itex]A_n[/itex] by using the initial condition [itex]u(r, 0) = \Omega r[/itex]. I don't seem to be able to do it. I find myself doing integrals like:

[itex]\int_0^1{x^2 J_1(\lambda_n x) dx}[/itex]

which I can't do. Can you please show the right steps to take to find the [itex]A_n[/itex]? It might be something very easy but I haven't dabbled with Bessel's function as much as I should have in the past.

4. Where the problem comes from
This is a worked out example in Acheson's book Elementary Fluid Dynamics. He just says that the result comes from a standard Fourier-type analysis and I was trying to re-derive it.
 

Answers and Replies

  • #2
SteamKing
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Your Bessel function integral can be easily evaluated using a Table of Integrals. IDK if you have any other difficult integrals, but consulting references like:

Abramowitz & Stegun, "Handbook of Mathematical Functions"

or

Gradshteyn & Ryzhik, "Table of Integrals, Series, and Products"

are very handy to have. After all, real physicists don't spend their time trying to remember all the rules of integrating weird functions when someone has already done the work.

A & S can be found here: http://people.math.sfu.ca/~cbm/aands/

and G & R can be found on the web or in most college libraries.
 

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