Interface Shape at Equilibrium
From ThermalFluidsPedia
Surface tension effects on the shape of liquidvapor interfaces can be demonstrated by considering a free liquid surface of a completely wetting liquid meeting a planar vertical wall (see Fig. 1). Since the interface is twodimensional, the second principal radius of curvature is infinite. For the first principal radius of curvature RI, it follows from analytical geometry that

The YoungLaplace equation, eq. (13), becomes

At a point on the interface that is far away from the wall (), the liquid and vapor pressures are the same, i.e.,

The liquid and vapor pressures near the wall (z > 0) are


The relationship between the pressures in two phases can be obtained by combining eqs. (3) – (5), i.e.,

Combining eqs. (2) and (6), the equation for the interface shape becomes

Multiplying this equation by dz/dy and integrating gives

Since at both z and dz/dy equal zero, C1 = 1. The boundary condition for equation (8) is

Equations (8) and (9) can then be solved for z at y=0:

Using eq. (10) as a boundary condition, integrating eq. (8) yields the following relation for the shape of the interface:

where

is a characteristic length for the capillary scale. For small tubes with , the interfacial radius of curvature is approximately constant along the interface and equal to R / cosθ, where θ is the contact angle.
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Additional Equations

References
Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.