Oblique liquid curtains with a large Froude number

Abstract

This paper examines two-dimensional liquid curtains ejected at an angle to the horizontal and affected by gravity and surface tension. The flow in the curtain is, generally, sheared. The Froude number based on the injection velocity and the outlet’s width is assumed large; as a result, the streamwise scale of the curtain exceeds its thickness. A set of asymptotic equations for such (slender) curtains is derived and its steady solutions are examined. It is shown that, if the surface tension exceeds a certain threshold, the curtain – quite paradoxically – bends upwards, i.e. against gravity. Once the flow reaches the height where its initial supply of kinetic energy can take it, the curtain presumably breaks up and splashes down.

On oblique liquid curtains

“…In what follows, we first demonstrate in § 2 that the high-viscosity system proposed by Benilov (2019) is identical to that of the inviscid equation of Finnicum et al. (1993) for cases where the ambient gas dynamics are neglected and there is no pressure drop across the curtain.…”

“…The starting point for our analysis is equations (4.2)–(4.6) of Benilov (2019), which govern the shape of the curtain, and are written here in dimensional form (see figure 1 for definition sketch): Here is the local speed in the curtain, is the curtain thickness, is the arclength coordinate, is the angle that the curtain centreline makes with the horizontal, and are the horizontal and vertical coordinates, is the gravitational acceleration, is the constant surface tension of the air–liquid interfaces, and is the density of the liquid. In the derivation of (2.1), the slender curtain approximation is applied, in that the characteristic length of the curtain (in the direction) is assumed to be much larger than .…”

“…We focus here on the configuration of the die examined by Benilov (2019), where the die centreline is angled as shown in figure 1( a ), for which (note that is negative in the orientation shown in figure 1 b ). In this configuration, the distance downwards from the slot is given as , and in accordance with (2.4), the curtain speed increases in the direction monotonically.…”

“…Because curtains are thin, and more specifically because the characteristic lengths are much greater than the curtain thickness, the equations governing the liquid flow may be simplified significantly. Recently, Benilov (2019) used asymptotic methods to derive equations that describe the flow of a widthwise-invariant curtain and analysed configurations in which the die slot is not vertical. Figure 1( a ) is a sketch of Benilov’s configuration using his coordinate system notation.…”

“…The angle of the curtain centreline at the slot is denoted as , where as drawn, , and . The coordinate system is chosen to be consistent with that of Benilov (2019). …”