The Dynamics of Thin Sheets of Fluid. III. Disintegration of Fluid Sheets

Geoffrey Taylor


The free edge of a sheet of uniform thickness moves into it at the same speed, (2T/$\rho $t)$^{\frac{1}{2}}$, as antisymmetrical waves, sweeping the fluid into roughly cylindrical borders. Here T, $\rho $ and t are surface tension, density and thickness of the sheet. In a radially expanding sheet t decreases with increasing radius and beyond a radius R where (2T/$\rho $t)$^{\frac{1}{2}}$ is greater than u the radial velocity of the sheet, the edge moves inwards faster than it is convected outwards. Photographs show that the edge of an expanding sheet establishes itself near but inside the radius R. The sheet produced by a swirl atomizer expands as a cone but photographs show that its thickness fluctuates very greatly at the point where it emerges from the orifice. The edge of a conical sheet of varying thickness establishes itself at a point well inside the radius at which (2T/$\rho \overline{t}$)$^{\frac{1}{2}}$ = u, $\overline{t}$ being the mean thickness. A moving sheet of uniform thickness can be bounded by a stationary free edge at angle sin$^{-1}$ (W$^{\frac{1}{2}}$) to the direction of motion. Here W, the Weber number, is 2T/$\rho $tu$^{2}$. Photographs show free edges at this angle and therefore parallel to antisymmetrical waves. If this remained true in an expanding sheet the edges would coincide with the cardioids discussed in part II, but reasons are given to show that this is not the case. A small obstacle can divide an expanding sheet forming two edges which lie at the same angle to one another as the two cardioids, namely, 2 sin$^{-1}$ (W$^{\frac{1}{2}}$) but photographs show that these edges do not subsequently lie on cardioids.