Incompressibility Conditions

Continuum theory has also been applied to analyse tire dynamics of flow of nematics [77, 80, 81 and 82]. The equations provide tire time-dependent velocity, director and pressure fields. These can be detennined from equations for tire fluid acceleration (in tenns of tire total stress tensor split into reversible and viscous parts), tire rate of change of director in tenns of tire velocity gradients and tire molecular field and tire incompressibility condition [20]. 

The penalty method is based on the expression of pressure in terms of the incompressibility condition (i.e. the continuity equation) as... 

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. ... 

Using a known solution at the inlet. To provide an example for tins option, let us consider the finite element scheme described in Section 2.1. Assuming a fully developed flow at the inlet to the domain shown in Figure 3.3, v, (dvy/dy) = 0 and by the incompressibility condition (dvx/dx) - 0, x derivatives of all stress components are also zero. Therefore at the inlet the components of the equation of motion (3.25) are reduced to... 

The incompressible Navier-Stokes equations are obtained by substituting the above form for into the generalized Euler equation (equation 9.9) and by using the incompressibility condition (5 ) dvijdxi = 0 equation 9.4) and Euler s equation dvijdt = -Y k Vkidvi/dxk) - dp/dxi) equation 9.7) ... 

The system of equations (64)-(68) can be solved numerically by eliminating pressure from the incompressibility condition... 

Equilibrium values of numbers Na of monomer molecules M in a globule can obviously be found from the condition of the equilibrium of the values of the chemical potentials of these molecules inside, iia, and outside, // , this globule. The explicit expressions for //1 and /x2 are obtained by differentiation of the relationship (Eq. 65), complemented by the incompressibility condition ip + cp = i, with respect to Mi and M2, respectively. With these expressions at hand, the set of two equations, (/j,i = //p /j2 = pu ), for the calculation of Mi and N2 has been derived [51]. Having these quantities calculated, it is easy... 

In these equations, r/ is the shear viscosity of the fluid and P the pressure tensor moreover, as they are only valid at large distances, the incompressibility condition has been used (see footnote to page 255). 

However, one difference exists with classical theory in this latter case, the Navier-Stokes equation (443) and the incompressibility condition (444) are assumed to be valid for all distances rict. In this case, it is an easy matter to calculate explicitly the higher-order terms in Eq. (445), and the boundary condition at the B-particle (assumed to be spherical) imposes the condition... 

Furthermore, the incompressibility condition allows It and /2 to be expressed only in terms of Xt and X2 as follows ... 

It may be noted that, denoting the fraction of A monomers in the corona belonging to the copolymer by t], and the micellar association number by p, incompressibility conditions give simple relations between R, RB, p and t) (de Gennes 1978 Leibler el al. 1983) ... 

Simplest model. The incompressibility condition is not obeyed, the velocity perturbation is stationary (the time derivative in (66) is neglected) and has only a radial direction,... 

The effective hard core potential u (z) is a Lagrange multiplier that enforces the incompressibility condition... 

Consider a number n of stiff polymer components (here stiff is used to mean semiflexible ) and define orientation-dependent ideal and interacting response (n x n) matrices X0(Q, u, u ) and X(Q, u, u ) respectively. In this case, orientational correlations have to be included in addition to the usual isotropic ones. Doi et al. [36-38] have developed the theory for solutions of stiff homopolymers. Their formalism is applied in Appendices C and D to multicomponent blend mixtures of stiff polymers without and with the incompressibility condition respectively. The interaction potentials comprise anisotropic (also called nematic) contributions as well as the usual isotropic ones ... 

In the case of a binary incompressible mixture of stiff homopolymers (components are named A and B), the above equations simplify. Assuming that component A is flexible (freely-jointed chains) and B is rigid (rigid rod polymers) and imposing the incompressibility condition, the following result can be obtained ... 

The main difficulty is to conveniently satisfy the incompressibility condition. In the following we will first recall the continuous mathematical formulation of the Stokes problem. Then it is recalled that a compatibility condition between pressure and velocity elements is necessary to prove convergence. Finally several possible strategies to solve the discretized system are developed. 

Equation (10) holds for any function V vanishing on Fi. The last temi of the augmented Lagragian (for r=0, Lr is a Lagrangian) introduces a penalty of the incompressibility condition and the Uzawa algorithm allows us to satisfy equation (3) as precisely as we wish using moderate values of r. 

Equations related to the velocity components verify the incompressibility condition. The function / is to be determined firom the governing equations, by considering domain D as computational domain. 

Equation [16] is known as the Liouville equation and is, in fact, a statement of the conservation of the phase space probability density. Indeed, it can be seen that the Liouville equation takes the form of a continuity equation for a flow field on the phase space satisfying the incompressibility condition dIdT F = 0. Thus, given an initial phase space distribution function /(F, 0) and some appropriate boundary conditions on the phase space satisfied by f, Eq. [16] can be used to determine /(F, t) at any time t later. 

Fig. 2 Left . Simulated monomer density profile (distribution of intrastar density around the center of mass) for a melt of stars of varying arm number f (from left star to right open square . 2 (linear chain), 4, 8, 16, 24, 36, 48, 64) [41]. The low intrastar density at low functionalities indicates penetrability by other stars (to satisfy incompressibility condition), whereas at high functionality the star core is formed with constant density. Inset . Cartoon illustration of a multiarm star with the three areas in different colors melt-like inner black core, theta-like intermediate blue unswollen and excluded-volume outer red swollen region. Right . The predicted Daoud-Cotton [28] monomer density distribution. The horizontal dashed line indicates the average solution concentration...

The adjoint Smoluchowski operator was obtained using in the partial integrations over the particle positions the incompressibility condition, Trace K = 0, which should always holds for the solvents of interest in this review. It takes the explicit form (where boundary contributions are neglected throughout, simplifying the partial integrations) ... 

The momentum equation (1.8) together with the additional constraint of incompressibility (1.3) fully define the motion of an incompressible fluid. They form a system of four scalar equations for four unknown functions the three components of the velocity field and the pressure field. Note that there is no evolution equation for the pressure field, which is given implicitly through the incompressibility condition. This somewhat complicates the analysis and numerical solution of the Navier-Stokes equation. 

Thus, in the absence of the forcing and viscous dissipation the vorticity is conserved along the trajectories of fluid elements. Equation (1.30) describes the evolution of the vorticity distribution in a given velocity field. However, u and v are obviously not independent of each other, but are two alternative representations of the instantaneous flow field. The velocity field is a vector, but it is subjected to the incompressibility condition. This constraint can be eliminated... 

Similarly substituting (2) and (3) into the incompressible condition (1) yields... 

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