Equilibrium Shapes of Fluid Interfaces

As indicated above, the Young-Laplace equation (Equation 1.22) is one fundamental result obtained from the theory of interfaces. Because this equation relates interfadal tension to the pressure difference between fluids at each point along an interface, it can be used with the equations of hydrostatics to calculate the shape of a static interface. Or, if interfadal shape can be determined 

According to the usual equations of hydrostatics, we have in bulk fluids A and B 

We now want to apply Eqnation 1.22 to a graieral point P on the drop where the radii of curvature are r and r. Using Equations 1.48 through 1.50 with Equation 1.22, we have 

The principal radii of curvature for an axisymmetric drop are given by (Adamson and Giast, 1997) 

Invoking an identity from trigonometry, we can rewrite this equation in the following form  

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The problem also occurs in other places in physics. One example is finding the energetically most favourable path that a moving object follows to get from one point to another under the influence of (conservative) external variables. An example from surface science is to find the spatial equilibrium shape of fluid interfaces under the constraint that the Interface is fixed at its extremities. 

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