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Extra resources for Four Hamiltonian Systems
Of the order of one hundredth of a micron. Therefore, for convenience, we will assume that the interface is a zero-thickness surface, where the properties of the system, such as density and composition, may jump discontinuously. , ½r ¼ NmÀ1 ¼ JmÀ2 : To better understand the meaning of surface tension, consider a ﬁlm with a rectangular boundary, so constructed that one of the sides is mobile (see Fig. 4). Here, the surface tension r is the force per unit length directed along the normal to the inner contour and tending to decrease the surface of the ﬁlm.
As in this case there are no diffusive shear stresses, the characteristics of this motion depend only on the pressure ﬁeld. This is the so-called inviscid flow, which has been studied in depth, starting in the 19th century. The inviscid flow model is completely described by Newtonian mechanics, following the laws of conservation of mass and momentum, as it does not present any energy dissipation, that is there is no conversion of the mechanical energy into heat. Obviously, although this type of movement can also exist close to the boundaries, the fluid velocity changes very rapidly in that region, thus producing an energy dissipation (proportional to the velocity gradient, as we saw in Sect.
At any gas-liquid interface, since the pressure in the gas phase is constant, the condition for equilibrium at any point of the interface is, 1 1 ðDqÞgy À r þ ¼ const: R1 R2 ð2:5:3Þ In our case, the constant on the RHS is zero, because far from the wall the interface becomes plane. In addition, our problem is two-dimensional, so that R1 = R and R2 ! 1. Then, considering that the curvature of a given curve y(x) is RÀ1 ð xÞ ¼ y00 =ð1 þ y0 Þ3=2 , where we have denoted y0 ¼ dy=dx, Eq. 2). Integrating once, we obtain: 1 y 1 ¼C þ 2 kr ð1 þ y0 Þ1=2 ð2:5:5Þ Applying again the boundary conditions far from the walls, we ﬁnd C = 1.
Four Hamiltonian Systems