## Spin down of coffee in a bottomless cup.

Posted by peeterjoot on April 25, 2012

# Motivation.

Here’s a variation of a problem outlined in section 2 of [1], which looked at the time evolution of fluid with initial rotational motion, after the (cylindrical) rotation driver stops, later describing this as the spin down of a cup of tea. I’ll work the problem in more detail than in the text, and also make two refinements.

- I drink coffee and not tea.
- I stir my coffee in the interior of the cup and not on the outer edge.

Because of the second point I’ll model my stir stick as a rotating cylinder in the cup and not by somebody spinning the cup itself to stir the tea. This only changes the solution for the steady state part of the problem.

# Guts

We’ll work in cylindrical coordinates following the conventions of figure (1).

We’ll assume a solution that with velocity azimuthal in direction, and both pressure and velocity that are only radially dependent.

Let’s first verify that this meets the non-compressible condition that eliminates the term from Navier-Stokes

Good. Now let’s express each of the terms of Navier-Stokes in cylindrical form. Our time dependence is

Our inertial term is

Our pressure term is

and our Laplacian term is

Putting things together, we find that Navier-Stokes takes the form

which nicely splits into an separate equations for the and directions respectively

## Steady state solution

Before we seek the steady state, the solution of

We’ve seen that

is the general solution, and can now fit this to the boundary value constraints. For the interior portion of the cup we have

so is required. For the interface of the “stir-stick” (moving fast enough that we can consider it having a cylindrical effect) at we have

so the interior portion of our steady state coffee velocity is just

Between the cup edge and the stir-stick we have to solve

or

Subtracting we find

so our steady state coffee flow is

latex r \in [0, R_1]$} \\ \frac{\Omega R_1^2}{R_2^2 – R_1^2} \left( \frac{R_2^2}{r} -r \right)\hat{\boldsymbol{\phi}}& \quad \mbox{} \\ \end{array}\right.\end{aligned} \hspace{\stretch{1}}(2.19)$

## Time evolution.

We can use a separation of variables technique with to find

which gives us

and specified by

Checking [2] (9.1.1) we see that this can be put into the standard form of the Bessel equation if we eliminate the term. We can do that writing , and noting that and , which gives us

The solutions are

From (9.1.5) of the handbook we see that the plus and minus variations are linearly dependent since and , and from (9.1.8) that is infinite at the origin, so our general solution has to be of the form

In the text, I see that the transformation (where was the radius of the cup) is made so that the Bessel function parameter was dimensionless. We can do that too but write

Our boundary value constraint is that we require this to match 2.19 at . Let’s write , , , so that we are working in the unit circle with . Our boundary problem can now be expressed as

latex z \in [0, a]$} \\ \frac{1}{\frac{R^2}{a^2} – 1} \left( \frac{1}{{z}} – z\right)& \quad \mbox{} \\ \end{array}\right.\end{aligned} \hspace{\stretch{1}}(2.27)$

Let’s pull the factor into and state the problem to be solved as

latex z \in [0, a]$} \\ \frac{a^2}{1 – a^2} \left( \frac{1}{{z}} – z\right)& \quad \mbox{} \\ \end{array}\right..\end{aligned} \hspace{\stretch{1}}(2.28c)$

Looking at section 2.7 of [3] it appears the solutions for can be obtained from

where are the zeros of .

To get a feel for these, a plot of the first few of these fitting functions is shown in figure (2).

Using Mathematica in bottomlessCoffee.cdf, these coefficients were calculated for . The approximations to the fitting function are plotted with a comparison to the steady state velocity profile in figure (3).

As indicated in the text, the spin down is way too slow to match reality (this can be seen visually in the worksheet by animating it).

# References

[1] D.J. Acheson. *Elementary fluid dynamics*. Oxford University Press, USA, 1990.

[2] M. Abramowitz and I.A. Stegun. {\em Handbook of mathematical functions with formulas, graphs, and mathematical tables}, volume 55. Dover publications, 1964.

[3] H. Sagan. *Boundary and eigenvalue problems in mathematical physics*. Dover Pubns, 1989.

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