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55 if one replaces u(v) with ρ(v) ϕ, u(a) with ρ(a)ϕ, u(l) with ρ(l)ϕ, and U(0) with M(0)ϕ (cf. 29). The three subsidiary conditions denoted by the subscripted quantities above are necessary if one requires that the calculus of variations problem remain equivalent to the problem stated by Gibbs for an isolated composite system. In other words, an isolated system does not permit the transfer of heat, mass, or work across its outer boundary. If these restrictions are imposed on our system and on the formulation of the calculus of variations problem that accompanies the system, then we must force all dissipation processes to vanish, restrict the total mass of each species in the system to remain fixed and require that all outer boundary variations that would perform work be zero.

64 is equivalent to two orthogonal scalar relations. One of these relations may be shown to be equivalent to the Young equation of capillarity (cf. derivation in Chapter 2 for axisymmetric capillary systems), while the second relation represents the mechanical equilibrium condition in a direc tion that is orthogonal to the first relation. 2. 64 as two scalar relations; one resulting from projecting the vectors into the horizontal plane and another from a corresponding projection into the vertical plane [36].

110) where R1 is the radius of the curvature c1 of the area along side x; that is, side BD. 110 can be simplified to © 2011 by Taylor and Francis Group, LLC δx = x δz = xc1δz. 112) where c2 is the curvature of the area along side y. 113) which represents the change in area of the planar patch in terms of the mean curvature, the original area and the normal displacement. 114) where δJ can be evaluated as 1 1 1 1 δJ = δ ( c1 + c2 ) = δ + = − 2 + 2 δz = ( c12 + c22 ) δz = ( 2 K − J 2 ) δz.