Interfaces, Capillarity, and Microdrops - Microfluidics for Biotechnology - page 80

Figure 3.8 Schematic of Marangoni convection induced by a gradient of concentration.

type of convection. Similarly, a concentration gradient results in a gradient of
surface tension and, consequently, to a Marangoni convection (Figure 3.8). Note
that the direction of the motion is always towards the largest value of the surface
tension.

3.3 Laplace Law and Applications

3.3.1 Curvature Radius and Laplace's Law
Laplace's law—sometimes called the Young-Laplace-Gauss law—is fundamental
when dealing with interfaces and microdrops. It relates the pressure inside a droplet
to the curvature of the droplet. Structurally, Laplace's law derives from the Gibbs
approach and is valid for a sufficiently small curvature [5]. Let us first describe the
notion of curvature.

3.3.1.1 Curvature and Radius of Curvature
For a planar curve the radius of curvature R is the radius of the osculating circle, the
circle that is the closest to the curve at the contact point (Figure 3.9). The curvature
of the curve is defined by

(3.7)

κ = 1 R

Note that the curvatures as well as the curvature radii are signed quantities.
Curvature radius can be positive or negative depending on the orientation (convex
or concave) of the curve.
Different expressions exist depending on the expression of the curve. In the case
of a parametric curve c ( t ) = ( x ( t ), y ( t )), the curvature is given by the relation [6]