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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.