4-vector interfacial material

Performing macro-scale experiments it has been observed that the normal surface tension force induces higher normal stresses in the fluid on the concave side of the interface than on the other fluid on the convex side of the interface. In a micro-scale view we may say that this interfacial tension force is exerted by the interfacial material lying on the convex side of the surface upon the material lying on the concave side. The normal component of the surface force is thus frequently (not always ) defined positive into the mean curvature of the surface, in line with the physical observations. The direction of the normal component of the interface force given by (3.9) is determined by two factors, the interface normal unit vector n/ which we have defined positive into the curvature, and the mean curvature variable which we have chosen to define as an absolute value. That is, the variable used here determining the mean curvature of the surface Hi = ( i + K2)/ 2) is consistent with the definition... 

The interfacial material velocity 4-vector has the same definition as in the case of any given continurun (Chapter 1) ... 

A few words of explanation are not useless in order to understand this formalism. As a consequence of mixing, the medium is assumed to have a lamellar structure and n is a unit vector which remains normal to the material slices undergoing deformations in the velocity field, n n denotes a dyadic product (the dyadic product of vectors a and b is the tensor a.jbj) and 13 n n denotes the scalar product of the two tensors (the scalar product of tensors i = Tij and W = is the scalar quantity T W = E Z T j wji)- Assume that we start with two miscible fluids A J and B (having for instance different colors). Upon mixing, we obtain a lamellar marbled structure characterized by a striation thickness 6 and a specific "interfacial" area av. If the fluid is incompressible, avS = 1. Then,by application of (7-1)... 

Table 9.9 Interfacial Tension (y) and Spreading Coefficient (A) of Different Materials Evaluated as a Vector Fluid, Comparing with Degree of PS Grafting on PE ([PS g), (Calculated for 200°C, Y(ps/pe) = 4.91 x 10 N/cm). From Y. Sun et al., The Canadian Journal of Chemical Engineering (1997) 75, p. 1-6...

Here, A,okl1 is the total interfadal area in the material of volume V, n is the unit normal vector of the interface (cf. Figure 4), and is the dyadic of n averaged over the whole interfadal area. Equation [19] holds for blends having dther a discontinuous (droplet/matrix-typ>e) morphology or a co-continuous morphology. As dearly noted from eqn [19], Oint reflects anisotropy of the shape of the phase-separated domains. Thus, the relaxation of detects recovery of isotropic shape of the domains driven by the interfadal tension. (In this relaxed state, the interfacial tension works isotropically to balance the isotropic pressure.)... 

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