Hydrodynamical models for the chaotic dripping faucet

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Original Entry by Michelle Borkin, AP225 Fall 2009

Overview

"Hydrodynamical models for the chaotic dripping faucet."

P. Coullet, L. Mahadevan and C. Riera, Journal of Fluid Mechanics, 526, 1-17, 2005.

Keywords

surface tension, capillary force, chaos, Hydrodynamics

Summary

This paper presents a series of mathematical models describing the chaotic dripping of a facet. First, a static droplet of water hanging at the end of a facet is discussed, then time dependence is examined with a lubrication model, chaotic behavior is investigated using a proper orthogonal decomposition (POD) resulting in a simplified model, and finally a mechanical description encompassing the main features of these detailed fluid models is presented. The mathematical models presented importantly include and explain features of the dripping faucet such as the time between drips, chaotic dripping (i.e. drip-drop behavior), and the critical point between dripping and jetting. In the section below, the basic concepts and equations behind these models are presented.

Soft Matter

Figure 1.
Figure 2.
Figure 9.

Before examining chaotic dripping, the authors first model a static drop and investigate the possible stable shapes. A small flow rate is assumed thus the drop will not detach form the facet unless its volume exceeds a critical volume (<math>V < V_{c}</math>), and it is also assumed with this condition that the drop will be axisymmetric. The shape of the drop (see Figure 1(a)) is determined by the minimization of the gravitational energy and surface energy. This can be written in terms of when the forces perpendicular to the interface balance (nondimensionalized forms):

<math>\frac{d\theta}{ds} - \frac{cos(\theta)}{r} = -z</math>

The interfaces (r and z) can be written as:

<math>\frac{dz}{ds} = -cos(\theta)</math>

<math>\frac{dr}{ds} = sin(\theta)</math>

The boundary conditions at the bottom of the drop are:

<math>r(0) = 0</math>

<math>\theta(0) = \frac{\pi}{2}</math>

<math>z(0) = \frac{P_{b}}{\rho g}</math>

where <math>P_{b}</math> is the hydrostatic pressure - this unknown is the main determinant to the different family of solutions. As shown in Figure 1(b), up to <math>V_{c}</math> the capillary forces are able to support the weight of the drop, after which the drop will be pinched-off.


In order to incorporate time dependence and examine the motion of the droplet, the authors created a model based on a lubrication approximation to describe the fluid dynamics. The result is a hydrodynamical set of linearized equations around a stationary solution. Their Lagrangian strategy assumes the drop remains axisymmetric and the fluid velocity is negligible in the radial direction. Thus the Lagrangian of the system is:

<math>L = E_{kinetic} - U_{g} - U_{surface tension}</math>

where <math>E_{kinetic}</math> is the kinetic energy, <math>U_{g}</math> is the potential energy, and <math>U_{surface tension}</math> is the surface tension energy. The discretized Lagrangian gives <math>N</math> equations of motion:

<math>\frac{d}{dt}\frac{\partial L}{\partial z_{i}} = \frac{\partial L}{\partial z} + \frac{1}{2}\frac{\partial \dot{E}_{kinetic}}{\partial \nu_{i}}</math>, <math>i = 1,N</math>

The result of integrating this equation is shown in Figure 2.


Finally, the authors make a "simplified" model that encompasses the main characteristics of the more rigorous models. The time scale for formation of a droplet is <math>\tau_{f} \sim \frac{R}{\nu_{0}}</math>, and the pinch-off time once the drop reaches <math>V_{c}</math> is <math>\tau_{n} \sim R\sqrt{\frac{\rho R}{\Gamma}}</math>. The oscillation frequency after the remaining liquid from a pinched-off droplet recoils due to capillary forces is <math>f \sim \sqrt{\frac{\Gamma}{\rho V}}</math> where <math>\Gamma</math> is the surface tension. For small flow rates, viscosity damps these oscillations thus you have a steady drip. For a visual illustration of the relation between the gravitational and capillary forces, see Figure 9.