The equilibrium and stability of two dimensional pendent drops

Abstract

The paper gives a mathematical account of the equilibrium and the stability of equilibrium for two dimensional pendent drops hanging under the action of gravity and surface tension forces. The drops may be formed under conditions of constant volume, as when they hang from the horizontal lower edge of a vertical plate. Alternatively, equilibrium may be maintained under conditions of constant pressure, as, for example, when a drop is formed in a narrow gap between two plane parallel vertical plates when the fluid between the plates is connected to a reservoir of large free surface area. Equilibrium profiles, symmetric about a vertical centre plane, are calcu­lated for drops hanging from horizontal apertures of varying widths. It is shown that the volume of the suspended drop achieves a maximum as a function of, say, its depth, for each given aperture. For apertures of width less than a critical value π in capillary units of length, it is found that the pressure, as measured by the internal pressure at the aperture in excess of the external pressure, also achieves a positive maximum. If the volume and depth of the drop are gradually raised from zero the pressure maximum occurs before the volume maximum is reached. For an aperture of width π the pressure maximum is reduced to zero and the maximum point occurs at the initial position where the volume and depth are zero. For apertures greater than π the pressure decreases monotonically from zero as the depth is raised from zero. In the discussion of the stability we distinguish between two and three dimensional disturbances, and between disturbances which are symmetric or asymmetric about the centre plane of symmetry of the equilibrium profile. Drops maintained at constant volume are shown to have a limit point instability to two dimensional weak disturbances symmetric about the centre plane, which occurs at the point of maximum volume. Likewise, drops maintained at constant pressure in an aperture of width less than π, have a limit point instability to disturbances of the same form, which occurs at the point of maximum pressure. The analysis shows that for apertures greater than π drops cannot be formed under conditions of constant pressure, since they are unstable from the initial position of zero depth and volume. A brief discussion is given for the instability to plane asymmetric disturbances, which we find to be unimportant in the sense that these instabilities cannot occur before the equilibrium has already become unstable to disturbances of symmetric type. The most important source of instability of the two dimensional drop is the three dimensional disturbance, that is, one having a sinuous variation along the length of the drop. The stability to such displacements is studied in the last main section of the paper. It is shown there that instability of this form, with disturbances symmetric about the vertical centre plane, will begin to arise for apertures less than π when the equilibrium profile reaches the pressure maximum, for drops maintained both at constant volume and at constant pressure. For drops maintained at constant pressure this does not make any change in the position of the first onset of instability, since, then, two and three dimensional instabilities arise at the same point. However, for drops maintained at constant volume there is a marked change in behaviour. Since the pressure maximum occurs at a lower volume than the volume maximum, and at a lower depth, the drop maintained at constant volume will always become unstable to three dimensional instability first. The nature of the equilibrium for apertures greater than π is such that the three dimensional instability sets in at the outset, in this case for drops of both kinds. This is in agreement with a result of Plateau (1873) and Maxwell (1875), who studied the stability of a plane horizontal interface in a slit, and showed that for a slit of width greater than π the plane interface is not stable to small displacements which conserve the volume. Finally, we have considered three dimensional disturbances which are asymmetric about the centre plane, and we have shown that instabilities of this form cannot arise.