Incompressibility constraint

Level of enforcement of the incompressibility condition depends on the magnitude of the penalty parameter. If this parameter is chosen to be excessively large then the working equations of the scheme will be dominated by the incompressibility constraint and may become singular. On the other hand, if the selected penalty parameter is too small then the mass conservation will not be assured. In non-Newtonian flow problems, where shear-dependent viscosity varies locally, to enforce the continuity at the right level it is necessary to maintain a balance between the viscosity and the penalty parameter. To achieve this the penalty parameter should be related to the viscosity as A = Xorj (Nakazawa et al, 1982) where Ao is a large dimensionless parameter and tj is the local viscosity. The recommended value for Ao in typical polymer flow problems is about 10. ... 

Lattice-Boltzmann is an inherently time-dependent approach. Using LB for steady flows, however, and letting the flow develop in time from some starting condition toward a steady-state is not a very good idea, since the LB time steps need to be small (compared to, e.g., FV time steps) in order to meet the incompressibility constraint. 

Here a is the Kuhn or segment length. The delta functional selects out only those configurations satisfying the incompressibility constraint. The Flory-Huggins parameter, measures the incompatibility between A and B monomers. The dimensionless A monomer-density operator is given by... 

Here also, we use the nx(n — 1) matrix P to apply the incompressibility constraint and obtain ... 

The actuation of DEs can be approximated as the lateral electrostatic compression and planar expansion of an incompressible Unearly elastic material where the electrical component is treated as a parallel plate capacitor [141], The incompressibility constraint can be expressed as ... 

The two-fluid formulation consists of solving the governing equations in both fluids independently and then matching the interfacial boundary conditions at the interface, which usually requires an iterative algorithm. This approach keeps the interface as a discontinuity, consistent with the continuum mechanics concept. For each phase, we can write the following momentum equation along with the incompressibility constraint ... 

We assume that there is no relative slippage at the crystalline/amorphous interface. Then the interface compatibility condition demands velocity continuity across the crystalline/amorphous interface. These compatibility conditions in conjunction with incompressibility in both phases require definite continuity conditions on strain-rate and spin components in the inclusion between the crystalline and amorphous components. Moreover, the crystalline/amorphous interface also enforces shear-traction equilibrium across the interface. More complete statements of the compatibility, continuity, and incompressibility constraints necessary for the full implementation of the model can be found elsewhere (Lee et al. 1993a). 

In the following we introduce a minimal transformation to decouple the interactions between polymer chains. Two different schemes have to be employed for the thermal interactions between the monomers and the incompressibility constraint. For the thermal interactions which give rise to the term (0a - in the Hamiltonian, we use the Hubbard-Stratonovich formula... 

To rewrite the incompressibility constraint, we use the Fourier representation of the 5-function ... 

We note that this complex term cannot be avoided easily. Replacing the incompressibility constraint by a finite compressibihty... 

Note that the last contribution in the equation above is similar to the single chain correlations in Eq. 31. The incompressibility constraint is enforced on the microscopic density a + 0b. At this stage, 0 and 0g are only airx-iliary fimctionals of the fields U, W, which are proportional to the density distribution of a single chain in the corresponding external fields. 

The fluctuations described by the two fields, U and W, are qualitatively different. This is already apparent from the fact that one, U, gives rise to a complex contribution to the field that acts on a chain, while the other, W, corresponds to a real one. The field U couples to the total density 0 + < b and has been introduced to decouple the incompressibility constraint. Qualitatively, it con-... 

Instead of enforcing the incompressibility constraint on each of the microscopic conformations, we thus only require that the single chain averages in the external field obey the constraint. We recall that the 4>a functionals of Wa = (iU + W)/2 and Wb = iU - W)/2 (cf. Eq. 28), thus Eq. 40 implicitly defines a functional U [W. Substituting the saddle point value into the free energy functional (Eq. 11), we obtain an approximate partition function... 

This demonstrates that the saddle point approximation in U enforces the incompressibility constraint only on average, but the literal fluctuations of the total density in the EP theory do not vanish. 

The composition dependence accounts for the fact that currents of A- and B-densities have to exactly cancel in order to fulfill the incompressibility constraint. This local Onsager coefficient completely neglects the propagation of forces along the backbone of the chain and monomers move independently. Such a local Onsager coefficient is often used in calculations of dynamic models based on Ginzburg-Landau type energy functionals for reasons of simplicity [85-87]. 

We have briefly reviewed methods which extend the self-consistent mean-field theory in order to investigate the statics and dynamics of collective composition fluctuations in polymer blends. Within the standard model of the self-consistent field theory, the blend is described as an ensemble of Gaussian threads of extension Rg. There are two types of interactions zero-ranged repulsions between threads of different species with strength /AT and an incompressibility constraint for the local density. 

In the EP theory one regards fluctuation of the composition but still invokes a mean-field approximation for fluctuations of the total density. This approach is accurate for dense mixtures of long molecules, because composition fluctuations decouple from fluctuations of the density. The energy per monomer due to composition fluctuations is typically on the order of UbT/N, and it is therefore much smaller than the energy of repulsive interactions (on the order of 1 8 ) in the polymer fluid that give rise to the incompressibility constraint. If there were a couphng between composition and density fluctuations, a better description of the repulsive hard-core interactions in the compressible mixture woidd be required in the first place. 

The explkat compressibility contribution is unimportant if the inequality — NpCj fX — T ) 1 is obeyed. Adopting the latter condition can be viewed as enforcing an effective incompressibility constraint in a thermo-dynamicaUy post facto manner It differs enormously from the spinodal predicted based on the literal IRPA approach of Eqs. (6.17) and (6.18) which is given by [67]... 

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