Gravity-driven draining of a thin rivulet with constant width down a slowly varying substrate

The locally unidirectional gravity-driven draining of a thin rivulet with constant width but slowly varying contact angle down a slowly varying substrate is considered. Specifically, the flow of a rivulet in the azimuthal direction from the top to the bottom of a large horizontal cylinder is investigated. In particular, it is shown that, despite behaving the same locally, this flow has qualitatively different global behaviour from that of a rivulet with constant contact angle but slowly varying width. For example, whereas in the case of constant contact angle there is always a rivulet that runs all the way from the top to the bottom of the cylinder, in the case of constant width this is possible only for sufficiently narrow rivulets. Wider rivulets with constant width are possible only between the top of the cylinder and a critical azimuthal angle on the lower half of the cylinder. Assuming that the contact lines de-pin at this critical angle (where the contact angle is zero) the rivulet runs from the critical angle to the bottom of the cylinder with zero contact angle, monotonically decreasing width and monotonically increasing maximum thickness. The total mass of fluid on the cylinder is found to be a monotonically increasing function of the value of the constant width

A Kinematic Evolution Equation for the Dynamic Contact Angle and some Consequences

We investigate the moving contact line problem for two-phase incompressible flows with a kinematic approach. The key idea is to derive an evolution equation for the contact angle in terms of the transporting velocity field. It turns out that the resulting equation has a simple structure and expresses the time derivative of the contact angle in terms of the velocity gradient at the solid wall. Together with the additionally imposed boundary conditions for the velocity, it yields a more specific form of the contact angle evolution. Thus, the kinematic evolution equation is a tool to analyze the evolution of the contact angle. Since the transporting velocity field is required only on the moving interface, the kinematic evolution equation also applies when the interface moves with its own velocity independent of the fluid velocity. We apply the developed tool to a class of moving contact line models which employ the Navier slip boundary condition. We derive an explicit form of the contact angle evolution for sufficiently regular solutions, showing that such solutions are unphysical. Within the simplest model, this rigorously shows that the contact angle can only relax to equilibrium if some kind of singularity is present at the contact line. Moreover, we analyze more general models including surface tension gradients at the contact line, slip at the fluid-fluid interface and mass transfer across the fluid-fluid interface.Comment: 25 pages, 6 figures; accepted manuscript

Contact Angle Hysteresis on Superhydrophobic Stripes

We study experimentally and discuss quantitatively the contact angle hysteresis on striped superhydrophobic surfaces as a function of a solid fraction, $\phi_S$. It is shown that the receding regime is determined by a longitudinal sliding motion the deformed contact line. Despite an anisotropy of the texture the receding contact angle remains isotropic, i.e. is practically the same in the longitudinal and transverse directions. The cosine of the receding angle grows nonlinearly with $\phi_S$, in contrast to predictions of the Cassie equation. To interpret this we develop a simple theoretical model, which shows that the value of the receding angle depends both on weak defects at smooth solid areas and on the elastic energy of strong defects at the borders of stripes, which scales as $\phi_S^2 \ln \phi_S$. The advancing contact angle was found to be anisotropic, except as in a dilute regime, and its value is determined by the rolling motion of the drop. The cosine of the longitudinal advancing angle depends linearly on $\phi_S$, but a satisfactory fit to the data can only be provided if we generalize the Cassie equation to account for weak defects. The cosine of the transverse advancing angle is much smaller and is maximized at $\phi_S\simeq 0.5$. An explanation of its value can be obtained if we invoke an additional energy due to strong defects in this direction, which is shown to be proportional to $\phi_S^2$. Finally, the contact angle hysteresis is found to be quite large and generally anisotropic, but it becomes isotropic when $\phi_S\leq 0.2$.Comment: 17 pages, 8 figure

The effect of die half angle in tube nosing with relieved die

This research concentrates on rotary nosing using cone shape dies, in particular using “relieved dies”. “Relieved die” is a cone-shape die with contact surfaces and grounddown relieved surfaces. The objective of this research is to improve the forming limit by reducing the necessary force during the process. The present research focuses attention on the effect of die half angle. Die half angle is important parameter because the angle of nosing depends on the angle of the die in press and rotary nosing. According to the previous research, the forming limit is highest when the number of contact areas is three [1]. Therefore, this paper researched on the effect of die half angle by experiment and calculation, FEM, under the condition that the number of contact areas is three. It is revealed that the optimum contact angle changes depending on the die half angle

The contact angle in inviscid fluid mechanics

We show that in general, the specification of a contact angle condition at the contact line in inviscid fluid motions is incompatible with the classical field equations and boundary conditions generally applicable to them. The limited conditions under which such a specification is permissible are derived; however, these include cases where the static meniscus is not flat. In view of this situation, the status of the many `solutions' in the literature which prescribe a contact angle in potential flows comes into question. We suggest that these solutions which attempt to incorporate a phenomenological, but incompatible, condition are in some, imprecise sense `weak-type solutions'; they satisfy or are likely to satisfy, at least in the limit, the governing equations and boundary conditions everywhere except in the neighbourhood of the contact line. We discuss the implications of the result for the analysis of inviscid flows with free surfaces.Comment: 13 pages, no figures, no table