Pressure drop across a curved interface

Since we take p to be independent of z within the lubrication approximation, Vz = 0. The pressure drop across a curved interface located at z = h(Xyy) is given by the Laplace condition (see Chapter 2) ... 

The above computation has been carried out for a flat interface, but we know that surface tension affects both equilibrium and dynamics only when the interface is curved. One may ask whether the computed value of 7 is indeed the same surface tension that is responsible for the pressure drop across a curved interface. The answer is positive but qualified yes, approximately, provided the curvature radius is much larger than the characteristic thickness of the interface. 

Figure 5.3 Schematic describing the pressure drop across a curved interface...

To describe a liquid meniscus and hence to obtain the interfacial tension from the drop shape the Laplace equation is used as the mechanical equilibrium condition for two homogeneous fluids separated by an interface [188]. It relates the pressure difference across a curved interface to the surface tension and the curvature of the interface... 

As a consequence of the pressure difference across a curved interface, such an interface will resist deformation by exerting an external force. The larger this pressure difference is, the larger the resistance is. Therefore, smaller drops or bubbles are less easily deformable than larger ones. This phenomenon is relevant for the preparation and stability of emulsions and foams (see Chapter 8). 

Capillarity describes the phenomenon of the rising or depression of a liquid into a capillary, a thin tube with inner diameter less than 1 mm in the case of water. Usually, the pressure drop across a curved liquid-gas interface is given by the Young-Laplace equation ... 

Analogous to the bubble point pressure, the reseal pressure can be defined from a simplification of the general 3D YT.E for the pressure drop across a curved L/V interface embedded within the 3D space of the mesh. Consider the L/V interface formed within the LAD mesh screen as shown in Eigures 3.16 and 3.17. Retaining assumptions 1-4 from the bubble point model in Section 3.2.2, the following additional assumptions are required to solve the reseal pressure ... 

This equation is the governing relationship for the shapes of all bubbles and drops of liquids. It is also the basis for measuring the static surface/interface tensions of fluids. Laplace s equation states that the pressure drop across a curved surface is proportional... 

This equation states that the pressure differential or drop across a curved interface will be directly proportional to the interfacial tension a and inversely proportional to the radius of curvature r. As the drop size decreases, Ap increases for a given value of a. Lowering the value of a helps overcome the increased pressure differential and facilitate the movement of the oil. Although the use of microemulsions in tertiary oil recovery has shown great promise for a number of years, the special conditions of temperature, salinity, and other factors that impact on microemulsion formation and stability place a number of roadblocks in their widespread application. 

It is found that there exists a pressure difference across the curved interfaces of liquids (such as drops or bubbles). For example, if one dips a tube into water (or any fluid) and applies a suitable pressure, then a bubble is formed (Figure 1.13). This means that the pressure inside the bubble is greater than the atmosphere pressure. It thus becomes apparent that curved liquid surfaces induce effects, which need special physicochemical analyses in comparison to flat liquid surfaces. It must be noticed that in this system a mechanical force has induced a change on the surface of a liquid. This phenomenon is also called capillary forces. Then one may ask, does this also require similar consideration in the case of solids The answer is yes, and will be discussed later in detail. For example, in order to remove liquid, which is inside a porous media such as a sponge, one would need force equivalent to these capillary forces. Man has been fascinated with bubbles for many centuries. As seen in Figure 1.13, the bubble is produced by applying a suitable pressure, AP, to obtain a bubble of radius R, where the surface tension of the liquid is y. 

It will be shown here that, due to the presence of surface tension in liquids, a pressure difference exists across the curved interfaces of liquids (such as drops or bubbles). This capillary force will be analyzed later. 

It should be noted that the pressure is always greater on the concave side of the interface irrespective of whether or not this is a condensed phase.) The phenomena due to the presence of curved liquid surfaces are called capillary phenomena, even if no capillaries (tiny cylindrical tubes) are involved. The Young-Laplace equation is the expression that relates the pressure difference, AP, to the curvature of the surface and the surface tension of the liquid. It was derived independently by T. Young and P. S. Laplace around 1805 and relates the surface tension to the curvature of any shape in capillary phenomena. In practice, the pressure drop across curved liquid surfaces should be known from the experimental determination of the surface tension of liquids by the capillary rise method, detailed in Section 6.1. 

The driving force for liquid penetration is governed by the pressure gradient, P, across the curved liquid]gas interface. The pressure drop across the interface in a uniform capillary of radius r is given by ... 

This equation is quite significant in explaining the properties of liquid surfaces and bubbles. First, Eq. 3.61 indicates that, in equilibrium, a pressure difference can be maintained across a curved surface. The pressure inside the liquid drop or gas bubble is higher than the external pressure because of the surface tension. The smaller the droplet or larger the surface tension, the larger the pressure difference that can be maintained. For a flat surface, r = 00, and the pressure difference normal to the interface vanishes. 

Because so many applications of surfactants involve surfaces and interfaces with high degrees of curvature, it is often important to understand the effect of curvature on interfacial properties. What is usually considered to be the most accurate procedure for the determination of the surface tension of liquids, the capillary rise method, depends on a knowledge of the relationship between surface curvature and the pressure drop across curved interfaces. Because of the existence of surface tension effects, there will develop a pressure differential across any curved surface, with the pressure greater on the concave side of the interface that is, the pressure inside a bubble will always be greater than that in the continuous phase. The Young-Laplace equation... 

The equilibrium drop shape is related to the surface tension of the drop through its curvature. In other words, the tendency for the drop to assume a curved interfacial shape is due to its surface tension. A consequence of this is the existence of a pressure difference across the interface... 

When the interface is curved and at equilibrium, interfacial tension, yLA, is equivalent to a pressure drop Ap called the capillary pressure, acting across the interface. Functional dependence can be derived from the equivalence of... 

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