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Rotation in a cylindrical container with a free surface
Theoretically a mug of tea can be described as a cylindrical container of radius R, partially filled with a liquid to a height h0 at rest. It begins rotating (being stirred) about its axis at constant angular velocity ω. After some time, the tea moves as a rigid body together with the mug; every fluid particle in the mug moves with the same angular velocity and therefore there is no relative motion. If the angular velocity is great enough the height of the free surface will reach the edge of the mug and the tea will spill; this is the point where the distance from the centre of the base, r, is equal to R and the height of the free surface, zs, is equal to the height of the mug. The maximum angular velocity the tea can be stirred at without spilling can then be determined (in radians per second) using the following equation:
Example:
An 8cm diameter, 12cm high vertical mug (assumed to be a cylindrical container) shown in picture is a partially filled with 11cm high tea whose density is 963kg/m3. The cylinder is rotated (as if being stirred) at constant speed. Determine the rotational speed at which the tea will start spilling from the edges of the mug.
Your turn:
When milk is added, the tea is 11.5cm high in the same mug and the density becomes 966kg/m3. At what rotational speed will the milky tea now start spilling from the edges of the mug?
Read more:
Joseph, D.D. & Fosdick, R.L., (1973). The free surface on a liquid between cylinders rotating at different speeds part I. Archive for Rational Mechanics and Analysis [online]. 49(5), 321-380. [Viewed 4 March 2019]. Available from: dio: 10.1007/BF00253044
Piva, M. & Meiburg, E., (2005). Steady axisymmetric flow in an open cylindrical container with a partially rotating bottom wall. Physics of Fluids [online]. 17(6), 063603. [Viewed 4 March 2019]. Available from: dio: 10.1063/1.1932664
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