Interface Shape at Equilibrium

From Thermal-FluidsPedia

Surface tension effects on the shape of liquid-vapor 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 two-dimensional, the second principal radius of curvature is infinite. For the first principal radius of curvature RI, it follows from analytical geometry that

Figure 1 Shape of the liquid-vapor interface near a vertical wall.

$\frac{1}{{{R}_{I}}}=\frac{{{d}^{2}}z/d{{y}^{2}}}{{{\left[ 1+{{(dz/dy)}^{2}} \right]}^{3/2}}}$	(1)

The Young-Laplace equation, eq. (13), becomes

${{p}_{v}}-{{p}_{\ell }}=\sigma /{{R}_{I}}$	(2)

At a point on the interface that is far away from the wall ( $y\to \infty ,\text{ }z=0$ ), the liquid and vapor pressures are the same, i.e.,

${{p}_{v}}(\infty ,0)={{p}_{\ell }}(\infty ,0)$	(3)

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

${{p}_{v}}(z)={{p}_{v}}(\infty ,0)-{{\rho }_{v}}gz$	(4)

${{p}_{\ell }}(z)={{p}_{\ell }}(\infty ,0)-{{\rho }_{\ell }}gz$	(5)

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

${{p}_{v}}-{{p}_{\ell }}=g\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)z$	(6)

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

$\frac{g({{\rho }_{\ell }}-{{\rho }_{v}})z}{\sigma }-{{\left[ 1+{{\left( \frac{dz}{dy} \right)}^{2}} \right]}^{-3/2}}\frac{{{d}^{2}}z}{d{{y}^{2}}}=0$	(7)

Multiplying this equation by dz/dy and integrating gives

$\frac{g({{\rho }_{\ell }}-{{\rho }_{v}}){{z}^{2}}}{\sigma }+{{\left[ 1+{{\left( \frac{dz}{dy} \right)}^{2}} \right]}^{-1/2}}={{C}_{1}}$	(8)

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

${{(dz/dy)}_{y=0}}\to \infty$	(9)

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

${{z}_{0}}=z(0)={{\left[ \frac{2\sigma }{({{\rho }_{\ell }}-{{\rho }_{v}})g} \right]}^{1/2}}$	(10)

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

$\frac{y}{{{L}_{c}}}={{\left[ cos\text{h}\left( \frac{2{{L}_{c}}}{{{z}_{0}}} \right) \right]}^{-1}}-{{\left[ cos\text{h}\left( \frac{2{{L}_{c}}}{z} \right) \right]}^{-1}}+{{\left( 4+\frac{z_{0}^{2}}{L_{c}^{2}} \right)}^{1/2}}-{{\left( 4+\frac{{{z}^{2}}}{L_{c}^{2}} \right)}^{1/2}}$	(11)

where

${{L}_{c}}={{\left[ \frac{\sigma }{({{\rho }_{\ell }}-{{\rho }_{v}})g} \right]}^{1/2}}$	(12)

is a characteristic length for the capillary scale. For small tubes with $R\ll {{L}_{c}}$ , the interfacial radius of curvature is approximately constant along the interface and equal to $R / cosθ$ , where θ is the contact angle.