Taylor column

Taylor columns were first observed by William Thomson, Lord Kelvin, in 1868.[1][2] Taylor columns were featured in lecture demonstrations by Kelvin in 1881[3] and by John Perry in 1890.[4] The phenomenon is explained via the Taylor–Proudman theorem, and it has been investigated by Taylor,[5] Grace,[6] Stewartson,[7] and Maxworthy[8]—among others.

Taylor column in fluid

Taylor columns have been rigorously studied. For Re<<1, Ek<<1, Ro<<1, the drag equation for a cylinder of radius, a, the following relation has been found.[7][9]

${\displaystyle F={\frac {16}{3}}\rho a^{3}\Omega U}$

To derive this, Moore and Saffman solved the linearised Navier–Stokes equation along in cylindrical coordinates,[9] where some of the vertical and radial components of the viscous term are taken to be small relative to the Coriolis term:

${\displaystyle -2\Omega v=-{\frac {1}{\rho }}{\frac {\partial p}{\partial r}}}$

${\displaystyle 2\Omega u=\nu \left({\frac {\partial ^{2}v}{\partial r^{2}}}+{\frac {1}{r}}{\frac {\partial v}{\partial r}}-{\frac {v^{2}}{r}}\right)}$

${\displaystyle 0=-{\frac {1}{\rho }}{\frac {\partial p}{\partial z}}+\nu \left({\frac {\partial ^{2}w}{\partial r^{2}}}+{\frac {1}{r}}{\frac {\partial w}{\partial r}}\right)}$

To solve these equations, we incorporate the volume conservation condition as well:

${\displaystyle {\frac {1}{r}}{\frac {\partial (ur)}{\partial r}}+{\frac {\partial w}{\partial z}}=0}$

We use the Ekman compatibility relation for this geometry to restrict the form of the velocity at the disk surface:

${\displaystyle w-U=\pm {\frac {1}{2}}{\sqrt {\frac {\nu }{\Omega }}}{\frac {1}{r}}{\frac {\partial (rv)}{\partial r}}}$

The resultant velocity fields can be solved in terms of Bessel functions.

${\displaystyle u=-{\frac {\nu }{2\Omega }}\int \limits _{0}^{\infty }k^{2}A(k)J_{1}(kr)e^{-\nu k^{3}z/2\Omega }dk}$

${\displaystyle v=\int \limits _{0}^{\infty }A(k)J_{1}(kr)e^{-\nu k^{3}z/2\Omega }dk}$

${\displaystyle w=-\int \limits _{0}^{\infty }A(k)J_{0}(kr)e^{-\nu k^{3}z/2\Omega }dk}$

whereby for Ek<<1 the function A(k) is given by,

${\displaystyle A(k)={\frac {2Ua}{\pi }}\left(\cos ka-{\frac {\sin ka}{ka}}\right)}$

Integrating the equation for the v, we can find the pressure and thus the drag force given by the first equation.

• Brenner, Michael P.; Stone, Howard A. (May 2000). "Modern Classical Physics Through the Work of G. I. Taylor". Physics Today. 53 (5): 30–35. Bibcode:2000PhT....53e..30B. doi:10.1063/1.883100.

• Taylor columns (Martha Buckley, MIT)
• Record Player Fluid Dynamics: A Taylor Column Experiment (UCLA Spin Lab)