Solving Fluid Dynamics Question on Droplet Shape

[mdergance3]

Hi everyone. I've been out of college for some time and forgot how to determine the shape of a droplet. I know how to determine the contact angle based on the interracial energies of three mediums, but I'm not sure how to calculate the actual shape of the hanging drop. In my scenario I have a drop hanging form a flat surface. I know the derivative (dh/dr) of the droplet profile is zero at the maximum hanging height, and is equal to the tangent of the contact angle at the h=0. How do I determine the actual profile of the droplet? I originally thought that surface tension would be equal across the surface but then I realized that is not the case. I also know that the droplet wants to assume the smallest surface area to volume ratio possible.

Any ideas as to where I can get some more bounding conditions for the profile shape of the water droplet?

 

[Curl]

Sounds tough, are you sure there is an analytic solution to this?

 

[Andy Resnick]

There is, but it's not a simple one: it's the Laplace equation $\Delta P = -2 \gamma\kappa$ where P is the pressure jump across the boundary, $\gamma$ the interfacial energy, and $\kappa$ the curvature. The pressure jump is given by the hydrostatic pressure difference due to gravity and different densities.

For a hanging drop or sessile drop:

http://resources.metapress.com/pdf-preview.axd?code=n510973p8675h585&size=largest

But there are several parameters that are required to fit a profile- the contact angle, for example.

http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHR-45PV3X9-3&_user=10&_coverDate=12%2F15%2F1997&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1647833937&_rerunOrigin=google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=afccbc3992579948e48ac243e38edd16&searchtype=a