Interdisciplinary Applied Mathematics

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Y n x dx,

where n is the    normal    vector    to    the    surface    at    the    point    described by

2 = Z(x,y). Upon substituting n = VZ, we obtain


where V2 denotes the two-dimensional Laplacian operator. We see that the curvature of the interface determines the value of the force, which is equal to a pressure of magnitude 7V2 Z acting on the infinitesimal surface SA. We can rewrite the above expression in terms of reference-independent variables by introducing the principal radii of curvature, i.e.,


1    1

R~i+ W and hence we can express the pressure jump at the interface induced by the surface tension as

=7 +£) ■    (s 2)

This equation is usually referred to as the Laplace-Young equation.

We note here that in the absence of equilibrium, i.e., in moving interfaces, the full stress balance equation should be employed instead, which has the form

Uh, — ТЧ = -ij (-Щ + w,    (8.3)

where aj denotes the stress tensor in medium (n), and n is the normal unit vector.    Also,    the    radius R is    positive    if its    center    of    curvature is    on

the side toward the direction defined by %. Some known examples in which the equilibrium equation is realizable are bubbles and droplets, which are spherical. In this case R = R2 = R, and thus Ap = 2y/R.

ls = yjFor pure water bubbles (at room temperature) we get ls

We have assumed here that the effect of gravity is negligible; this is valid if the radius of the microbubble is less than the characteristic length scale -H- Frtr П11ГР Wtltpir bllbbllPQ (tit rnnm tpirnniPrtltliriU WP1 (TPit / CS 2.5

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