Incompressibility assumption

As will be outlined below, the computation of compressible flow is significantly more challenging than the corresponding problem for incompressible flow. In order to reduce the computational effort, within a CED model a fluid medium should be treated as incompressible whenever possible. A rule of thumb often found in the literature and used as a criterion for the incompressibility assumption to be valid is based on the Mach number of the flow. The Mach number is defined as the ratio of the local flow velocity and the speed of sound. The rule states that if the Mach number is below 0.3 in the whole flow domain, the flow may be treated as incompressible [84], In practice, this rule has to be supplemented by a few additional criteria [3], Especially for micro flows it is important to consider also the total pressure drop as a criterion for incompressibility. In a long micro channel the Mach number may be well below 0.3, but owing to the small hydraulic diameter of the channel a large pressure drop may be obtained. A pressure drop of a few atmospheres for a gas flow clearly indicates that compressibility effects should be taken into account. 

A storage tank is shown in Figure 4-5. A hole develops at a height hL below the fluid level. The flow of liquid through this hole is represented by the mechanical energy balance (Equation 4-1) and the incompressible assumption, as shown in Equation 4-2. 

Finally, tensile deformations provide the same information as shear deformation as long as the incompressibility assumption is not violated. In this case, the tensile stress relaxation modulus E(t) is directly related to the shear modulus E(t) = 3G(f), and all other relationships follow accordingly. 

The fuel flow in the gas channels is modeled by applying the equation of state and the principles of mass and momentum conservation. From Equation (3.27), considering that no reaction takes place within the gas channel (at the anode side only humidified hydrogen is provided), and that the fluid flow is regarded as incompressible (assumption (3)), the mass conservation equation becomes ... 

In fluid dynamics we frequently invoke the incompressibility assumption, even though fluid densities change with pressure and temperature, and these may vary in time and space. If the density change cannot be neglected, then an appropriate equation of state of the form p = p(T, P) must be used in conjunction with the balance equations. 

Figure 3.2 (Case 1) shows a simple uniaxial extensional flow created by the uniform stretching of a rectangular or a thin filament in the 1 direction. For this flow, 22 = - n /2, and because of the incompressibility assumption, 22 = 33. Thus, in Eq. 3.1-1, m = —0.5, giving the following rate of deformation matrix...

Note that without the incompressibility assumption (and without using the Lagrange multiplier X), one would arrive to an identical equation for nxn matrices but with W replacing V(Q). Note also that V(Q) has the dimension of inverse volume. 

We have also shown that the Edmonds-Ogston expression and its extension by King, et are valid only if one assumes that the fluids are incompressible, that the solvent is non-interacting, and that Equations 28 and 29 are valid. The incompressibility assumption seems reasonable, but the lack of interaction between solvent and solutes seems less reasonable. We are currently investigating the consequences to the theory if this assumption is dropped. The assumption of the validity of Equations 28 and 29 is not a problem since these equations are valid when both species are flexible coils they are also valid in the case when one species is rigid and the other is a flexible coil. 

The intensity therefore approaches zero as q 0, and this is a consequence of the incompressibility assumption. For qRg >> 1, on the other hand, the structure factor S(q) can be approximated by... 

An even simpler relationship is obtained when the incompressibility assumption is justified. The scattering intensity from a sample containing only a hydrogenous or deuterated polymer, due solely to density fluctuations, is weak for q in the small-angle region and can be completely neglected under the incompressibility assumption. This means that in (6.63) we can set I(q) = 0 and obtain... 

The Flory-Huggins (FH) lattice theory (8,9) has been customarily employed to establish phase diagram of binary amorphous-amorphous polymer mixtures based on the incompressible assumption that reads,... 

While the term a ln(A A2A3) is not important in the mechanical response, because of the incompressibility assumption, it may be important in swelling [61]. We also note that some of the molecular models include this logarithmic term. Then, the principal stresses o- in any deformation can be related through the strain energy function and deformations as follows ... 

Here, ( ) = Pm Vm/tj is the volume fraction of sites of type M, Vm is the volume of a site of type M, Vq is a reference volume , and Sm is the total neutron scattering length of a site of species M. Note that this chi-parameter will generally diverge as < )m - 0 due to the unrealistic incompressibility assumption. The SANS chi is, by construction, equivalent to the incompressible Flory value at the spinodal... 

The conditions required for accuracy of an incompressibility assumption at the level of the scattering functions and spinodal condition are easily derived within the PRISM formalism [67]. From Eq. (6.7) a small isothermal compressibility implies that — oCw -fOl P 1, which is generally true for any dense fluid. If the related wavevector-dependent condition... 

Poisson s ratio, v, for rubber is approximately 0.5 (incompressibility assumption), so... 

Throughout this text we will assume that polymer melts are incompressible liquids, by which we mean that the density never changes with position or time. This is clearly an approximation that must be relaxed in some applications - injection molding, for example, where the compressibility of the melt becomes important because of the extremely high pressures - but the incompressibility assumption will suffice for our purpose here. If the density never changes in time or space, rates of change with respect to these variables (i.e., derivatives) must be zero dp/dt = 0, dp/dx = 0, etc.), and the continuity equation simplifies to... 

The corresponding dimensionless form of the continuity equation for incompressibility assumption, V - i = 0, is quite simple. Using V = 2-V and v = VoV, we have... 

The basic idea of these theories is to look at the distribution of conformations of a chain molecule attached to the surface. The conformational probability distribution function is written in terms of the non-local interaction field induced by the other chain molecules. This field is anisotropic, i.e., it depends on the direction perpendicular to the surface, because the presence of the surface and the inhomogeneous variation of the density of polymer segments and solvent molecules as a function of the distance from the surface. The non-local mean-field is determined by packing constraints that take into account the fact that the volume (at all distances from the surface) must be filled by polymer segments or solvent molecules. These self-consistent criteria represent the incompressibility assumption at all distances from the surface. 

In fact, as a student, 1 had been very concerned about the relationship of the Flory theory of polymer solutions and the Onsager theory. Whereas the Onsager theory is about compressible liquids, the Flory theory is about incompressible liquids, and thus the so-called lattice theory. Eater, it has been shown that the extension of the Onsager theory under incompressible assumptions results in the Flory theory of polymer solutions [41]. 

Hand-calculated pressure drop value is 17.59 kPa, and according to Hysys, PRO/II, and Aspen it was found to be 17 kPa, 1714 kPa, and 17.16 kPa, respectively. Simulation software results were very close to each other on the contrary, hand calculation is greater than all the other results, and this is attributed to the incompressibility assumption made during hand calculation. It is clear from the solution of this example that the density of gases is a function of both temperature and pressure. 

The smooth specimens are tested using standard methods in which a noncontacting video extensometer is used to gather strain data in addition to the load cell data [3]. The same incompressibility assumptions are also made in the analysis of the smooth specimens. 

Since PRISM theory accounts for density fluctuations, it can be used to calculate of local structure and physical properties of polymer liquids. This is not possible, for example, in self-consistent field theory [2] where density fluctuations are neglected due to the incompressibility assumption. 

Note here that the Aj s are stretches and not relaxation times as used previously. Also note that for polymer melts and solutions, the shear response function (shear modulus) is several orders of magnitude smaller than is the bulk modulus. Therefore, these materials can be treated as incompressible. For the linear case, the incompressibility assumption has A" oo or v 0.5. Here, /a = 1 and the constitutive law for the incompressible material becomes... 

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