Surface, equations mean curvature

Considering capillary dynamics, the pressure drop term is often described by the Laplace equation, AP = 2yH, where y represents liquid surface tension and H represents the mean curvature of the liquid-gas interface associated with aU curves, C, passing through the surface. Furthermore, the character of a sufficiently smooth surface is through the invariant from differential geometry, the principal curvature of each curve, kj. The radii of curvature are the inverse of each principal curvature, A , = 1/r,. Considering the maximum and minimum radii of curvature at a point on a three-dimensional surface, the mean curvature can be calculated explicitly (see Appendix for more thorough derivation of the mean curvature parameter) ... 

This equation was derived by Bruns, and it establishes a relation between the derivative of the field along the vertical and the mean curvature of the level surface. 

A more complicated situation emerges in motion along nonintersecting surfaces with variable curvatures. If the distance between these surfaces remains finite everywhere, then the field lines do not expand infinitely in the directions normal to the surfaces. In the absence of dissipation this means that there is no unbounded growth of the normal field component. However, introduction of the finite conductivity yields an equation for the normal component which is not decoupled it contains the contribution of the Laplacian of the remaining components. At the same time, it is possible for all other components to increase exponentially with an increment which depends on the conductivity and vanishes for infinite conductivity. The authors called this mechanism of field amplification a slow dynamo, in contrast to the fast dynamo feasible in the three-dimensional case, i.e., the mechanism related only to infinite expansion of the field lines as, for example, in motion with magnetic field loop doubling. In a fast dynamo the characteristic time of the field increase must be of the same order as the characteristic period of the motion s fundamental scale. 

The basic equation defining a capillary surface when gravity can be neglected is quite simple - the liquid surface has constant mean curvature However, the application of this equation... 

In the general form of the Young-Laplace equation, Ap = Pa — Pb (A above and B below), and the local mean curvature is positive for a meniscus that is concave upward. The Young-Laplace equation is more often formulated in terms of the hydrostatic pressures developed in a uniform body force field, such as that of gravity near the earth s surface ... 

A rigorous mathematical existence proof for a periodic surface of small, nonzero constant mean curvature can be obtained with the methods of the theory of nonlinear elliptic differential equations. The resulting surface would be a perturbation of a known periodic minimal surface, but the intent of chapter is rather to exhibit numerical solutions that extend over wide ranges in mean curvature. 

Figure 12-13. Mean curvature of the peaks of a ground surface as a function of the interval at which the profilometric trace is sampled. The full line is calculated from a theoretical equation. The experimental points are derived from digital analysis of the profile shown in Fig. 12-11. Data by Whitehouse and Archard [10].

This equation, related to Poisson s equation of Section 13.6.1, states that the mean curvature of the pressure surface is zero. Figure 5.26 shows the predicted isobars for a flow into a mould with a cut-out (computed using the steady-state heat flow analogue). The velocity vectors are perpendicular to the isobars the circular arc isobars near the gate show there is radial flow, but in sections with parallel side walls, the velocity is parallel to the walls. 

This equation means that, the vapor pressure of a given system under isothermal conditions increases with increasing curvature of the surface. 

Comparing with Equations 3.32 and 3.33, we see that n i is equal to the electrical force per unit area which acts on the surface of a film of uniform thickness. Substituting Equations 5.108, 5.109, 5.111, and 5.112 into Equation 5.107 and evaluating the mean curvature H as in Section 2.2, we obtain... 

According to Equation 5.127, interfacial curvature changes for two reasons. The first, represented by the term ( -VjTi ) is the nonuniformity (e.g., waviness) of the perturbation itself. The second, represented by the term (2K - 4tP) iq, is displacement of an initially curved surface normal to itself. For instance, uniform outward displacement of a spherical surface produces an increase in the radii of curvature and hence a decrease in the magnitude in mean curvature H. Whereas the first term tends to stabilize the system, the second term tends to destabilize it. However, in the case of a spherical liquid drop, the critical wavelength turns out to be so large that disturbances of such large wavelengths cannot be accommodated and drops are stable to infinitesimal perturbations. This constraint does... 

Here, rj and r2 are the two principal radii of curvature of the surface and k is the constant mean curvature. For homogeneous substrates, this means that droplets will adopt the shape of a spherical cap in mechanical equilibrium. The second condition is given by Young s equation... 

A curved interface is an indicator of a pressure jump across the interface with higher pressure on the concave side. This can be easily seen in the case of a spherical droplet or bubble. For example consider the free body diagram of a droplet with radius R cut in half, as depicted in Fig. 4. The uniform surface tension along the circumference of the droplet is balanced by the pressure acting on the projected area nR. The balance of forces in the horizontal direction results in (2itR)y = nR )AP, or AP = Pi — Po = y/R where Pi and Po are the equilibrium pressure inside and outside of the droplet, respectively. In the case of a bubble, one obtains AP = Pi Po = 4y/R since there are two layers of surface tensions one in contact with the outside gas and one in contact with the gas inside the bubble. This simple relation can be extended to any surface with a mean curvature K = l/Ri + 1/R2 where R and R2 are the principal radii of curvature. The resulting equation is known as the Young-Laplace equation... 

These concepts were formalized by Laplace who derived mathematical expressions for the mean curvature, and developed a differential equation that must be satisfied on S (Laplace, 1805). In modern notation (Gostick et al., 2010) the resulting Young-Laplace equation relates the pressure differential across the surface Ap to the mean curvature H and provides a definition for the surface tension a ... 

When a liquid-fluid interface is curved, the pressure on both sides is not the same it is higher in the fluid situated on the concave side. The mean curvature of the interface (C) is related with the pressure difference (AP) and the surface tension (y) through Laplace equation... 

The value of P is called the capillary pressure, and the last equation is the common Laplace equation. According to the well-known theorem [3,5], the value of is equal to the mean curvature of the interface. For highly curved surfaces, the value of a may depend on K and the Gaussian curvature [2,26,27]. We do not consider this dependence in the present work. 

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