Effective mean curvature

For cylindrical geometry, similar statements can be made as in the first paragraph of sec. 3.5e. We consider infinitely long rods, Ignoring end effects for which it is noted that the mean curvature J, defined in [1.2.23.21, is the same as that for a sphere with twice the radius. For such a cylinder, the potential only depends on r. Taking for the Laplacian the first term on the r.h.s. of 1I.A7.46), we arrive at... 

The net effects of the immersed boundary technique are that in the interface transition zone the fluid properties change smoothly from the value on one side of the interface to the value on the other side. The artificial interface thickness is fixed as a function of the mesh size applied to provide stability and smoothness, and it does not change during the calculations. Therefore no numerical diffusion is present. The finite thickness also serves to position the interface more accurately on the grid. The surface tension force (normal contribution only) is then calculated estimating the mean curvature based on geometric information obtained from the restructured interface. 

Assume that Biot numbers for the inner and outer surfaces satisfy the conditions Ajii/ki < 0.1 and hz jjkt < 0.1, and lumped conditions prevail. Also, in view of the fact that Si and So are small compared with the mean radius R of the brake system, neglect the effect of curvature. For the lumped system shown in Fig. 3.6, the first law of thermodynamics combined with Newton s law for the outer and inner surfaces results in the governing equation for the brake system,... 

Here k is the bending modulus, and the subscripts of xx and yy represent two derivatives of the membrane position variable — i.e., h + h yy) is the mean curvature of the nth membrane. This expression is correct for membranes with gentle undulations (Vh 1) otherwise the simple expression for the curvature is incorrect and the area constraints must be reconsidered as well. The compressional elastic constant, B, represents an effective repulsion between the membranes and will be computed self-consistently. Note that this Hamiltonian is unchanged if the positions of all the membranes are uniformly shifted, representing a trivial translation of the system. Fourier transforming in both the z direction (Fourier wavevector Q with an upper cutoff of nlD due to the periodicity) and the a — y plane (Fourier wavevector q — qy)) we have... 

Swelling effect. Oil molecules are solubilized in the aggregate s core and expand its volume. In this case, as is almost constant. Since the volume of the surfactant s hydrophobic tail is increased, the surfactant mean curvature tends to be more positive (i.e., more convex toward water) in order to maintain Us constant. Thus, for the hydrophilic Ci2(EO)7 (whose curvature tends to be positive), the swelling effect will be dominant, leading to the Hi-Ii transition. However, when an aromatic oil like w-xylene is used instead of decane, it penetrates into the surfactant palisade layer and makes the curvature negative, leading to the Hi-L, transition [296]. 

Calculations [33] of self-diffusion in ordered bicontinuous structures have, as illustrated in Fig. 21, reproduced the main features of the experimental studies for a large number of microemulsion systems. (For others, a quantitative comparison is difficult because of large influences of effects other than obstruction, such as a high surfactant film concentration or incomplete segregation between domains.) Furthermore, the symmetry of the self-diffusion pattern around the crossover of the oil and water curves implies a symmetry also in structural changes. This symmetry is easy to understand in terms of structures that have a constant mean curvature surface, where changes in spontaneous curvature away from zero in the two directions should be equivalent except for the two solvents changing place. 

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