43 Spherical Interfaces Generally speaking the equilibrium shape of an

# 43 spherical interfaces generally speaking the

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4.3 Spherical Interfaces Generally speaking, the equilibrium shape of an interface between two immiscible fluids is determined by solving the force balance equation (3.1) in each fluid, and then applying the Young-Laplace equation to the interface. However, in situations in which a mass of one fluid is completely immersed in a second fluid— e.g. , a mist droplet in air, or a gas bubble in water—the shape of the interface is fairly obvious. Provided that either the size of the droplet or bubble, or the di ff erence in densities on the two sides of the interface, is su ffi ciently small, we can safely ignore the e ff ect of gravity. This implies that the pressure is uniform in each fluid, and consequently that the pressure jump Δ p is constant over the interface. Hence, from (4.12), the mean curvature ∇ · n of the interface is also constant . Since a sphere is the only closed surface which possesses a constant mean curvature, we conclude that the interface is spherical . This result also follows from the argument that a stable equilibrium state is one which minimizes the free energy of the interface, subject to the constraint that the enclosed volume be constant . In other words, the equilibrium shape of the interface is that which has the least surface area for a given volume: i.e. , a sphere. Suppose that the interface corresponds to the spherical surface r = R , where r is a spherical coordinate. (See Section C.4.) It follows that n = e r | r = R . (Note, for future reference, that n points away from the center of curvature of the interface.) Hence, from (C.65), ∇ · n = 1 r 2 r 2 r vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle r = R = 2 R . (4.13) The Young-Laplace equation, (4.12), then gives Δ p = 2 γ R . (4.14) Thus, given that Δ p is the pressure jump seen crossing the interface in the opposite direction to n , we conclude that the pressure inside a droplet or bubble exceeds that outside by an amount proportional to the surface tension, and inversely proportional to the droplet or bubble radius. This explains why small bubbles are louder that large ones when they burst at a free surface: e.g. , champagne fizzes louder than beer. Note that soap bubbles in air have two interfaces defining the inner and outer extents of the soap film. Consequently, the net pressure di ff erence is twice that across a single interface. 4.4 Capillary Length Consider an interface separating the atmosphere from a liquid of uniform density ρ that is at rest on the surface of the Earth. Neglecting the density of air compared to that of the liquid, the pressure in the atmosphere can be regarded as
64 FLUID MECHANICS δr 2 1 3 θ Figure 4.2: Interface between a liquid (1), a gas (2), and a solid (3). constant. On the other hand, the pressure in the liquid varies as p = p 0 ρg z (see Chapter 3), where p 0 is the pressure of the atmosphere, g the acceleration due to gravity, and z measures vertical height (relative to the equilibrium height of the interface in the absence of surface tension). Note that z increases upward. In this situation, the Young-Laplace

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