Senior Thesis

Department of Mathematics
Harvey Mudd College

Benjamin P. Bryant

Harvey Mudd College Mathematics 2003

Thesis Title: Modeling moving droplets: A precursor film approach

Thesis Advisors: Prof. A.J. Bernoff (HMC) and Prof. A.E. Hosoi (MIT)

Modeling moving droplets: A precursor film approach

Laminar flow wings are sometimes used on light aircraft because of their increased efficiency, as compared to traditional wings where the boundary layer becomes turbulent relatively early along the wing. Their use is limited by the problem that small surface disturbances, particularly raindrops and the associated film that forms as they coalesce, induce turbulence in the boundary layer. This drastically reduces the lifting capability of the wing, sometimes by as much as 50 percent, making it dangerous for aircraft with such wings to fly in conditions with any possibility of precipitation developing, and thus limiting their more widespread use. A possible method for overcoming this issue is to attempt to eliminate or minimize the length of the film that forms by guiding the droplets appropriately prior to their coalescence.

Toward this end, we investigate the behavior of moving droplets and rivulets, driven by a combination of gravity and surface shear (wind). We begin with the Stokes equations and use the approximations of lubrication theory to derive the specific thin film equation relevant to our situation:
h_t = \nabla \cdot \left[ \frac{h^3}{3} \nabla \left(Gh -
S\nabla^2 h \right) \right] + \left(\Gamma h^3 + Ch^2 \right)_x,
\end{displaymath} (1)
This fourth-order partial differential equation describing the height of the fluid is then solved numerically from varying initial conditions, using a fully implicit discretization for time stepping, and a precursor film to avoid singularities at the drop contact line. Results describing general features of droplet deformation, limited parameter studies, and the applicability of our implementation to the long-term goal of modeling wings in rain are discussed.

Some color pictures from our simulations can be found below. When looking at them, note that the height and lengths have been scaled out, so the drops would in reality look much flatter.

A standard initial condition for a drop.

A drop after a short period of being subjected to shear.
It is roughly tetrahedral shaped at this stage.

A drop after shearing motion has caused the formation of a crescent depression on the back face.
The same qualitative feature can be seen when blowing on a droplet on a glass surface.

A feature of our model is that as the drop flattens out, shear forces will dominate over gravity forces. In this figure of a droplet profile changing in time, gravity intially sends the droplet to the right, and as it gets spread out surface shear sends it in the opposite direction.

We hope to eventually model the coalescence of many drops. While we have not yet done so, an example of two-drop coalescence is shown here.