New Fluid-Dynamic Slip Law Subsumes 40 Years of Research

Picture a drop of liquid running down a smooth incline. How does it move? It’s a seemingly simple question, but the mechanisms that control the behavior of the contact line, the curve that is the intersection of the liquid, solid, and gas phases at the edge of the drop, are very complex.

Controversy arises around the concept of slip, the relative motion between, say, a stationary solid and a liquid. Classical fluid dynamics denies the existence of slip, maintaining that liquid points in direct contact with a solid do not move, instead sticking to the surface while other particles in the droplet moved overhead. Today most theorists believe a relative motion exists between fluid and solid in the neighborhood of a contact line.

Northwestern University researchers have recently developed a law governing slip that encompasses all previous slip laws — some 40 years of research. The application of this law may also shed light on properties of fluid coatings that have not yet been observed by experimentalists.

The findings could lead to improved manufacturing processes, such as coatings, adhesives, and lubricants, as well as ink-jet printing, DNA screening, and other modern applications.

“With this single law, not only can we explain past formulations of slip, but we may be able to predict fundamental physical effects that experiments may observe in the future,” said Stephen Davis, Walter P. Murphy Professor of engineering sciences and applied mathematics and of mechanical engineering at Northwestern’s McCormick School of Engineering and one of the paper’s authors.

A paper about the research, “Hydrodynamic Theory of Liquid Slippage on a Solid Substrate Near a Moving Contact Line,” was published June 7 in Physical Review Letters.

Previous research has determined that slip is vital in order for a contact line to move, as it does when a drop is rushing down a surface. The microscopic physics that determine this motion remains unknown, but on a continuum level simulations have shown that it is caused by slip only in the region of the moving contact line.

Using this key observation, the Northwestern researchers developed a hydrodynamic theory of liquid slippage on a solid surface in which the slip is only active in the vicinity of the contact line, and the no-slip properties are present elsewhere in the droplet.

When the fluid-dynamic equations are solved using the slip law as a boundary condition, the presence of multiple inner circulations — swirls of liquid “eddies” within the droplet — are seen close to the contact line. These “corner circulations” could affect transport of heat and other properties. The law can also help determine how the spreading radius of a droplet increases with time even when the liquid is composed of complex materials such as polymers.

Eleftherios Kirkinis, Alexander Golovin Assistant Professor of Applied Mathematics at McCormick, was the paper’s lead author.