Sharp-interface approximation

It is clear from Figs. 4, 6, 8 that, for all samples, a finite interphase must be presumed in order to achieve even order-of-magnitude agreement between model and G data. The sharp-interface approximation is extremely bad. Sensitivity of G" to c()j(x) is reflected in obvious deficiencies of the linear profile in most cases. Interestingly, the storage modulus G is insensitive to interphase character Figs. 3,5,7 show the plateau level and shape to be almost independent of and (()j(x), respectively. Thus, the G plateau is determined primarily by this permits G data to be used to find while G data... 

Computation of density can be avoided if one assumes that it changes abruptly across the interphase boundary, i.e. jumps between both equilibrium values p/,Pv over a molecular-scale distance d. This is the sharp interface approximation. The interfacial energy is contributed then only by the second term in Eq. (9). We consider a flat boundary at 2 = 0 and neglect the vapor density compared to the liquid density, i.e. set p = pi = const at 2 < 0 and p = 0 at 2 > 0. Then a short computation using the hard- core interaction potential (5) yields the interfacial energy per unit area... 

In this sub-section the embedded interface method (frequently referred to as a front tracking method) developed for direct numerical simulations of viscous multi-fluid flows is outlined and discussed. The unsteady model is based on the whole held formulation in which a sharp interface separates immiscible fluids, thus the different phases are treated as one fluid with variable material properties. Therefore, equations (3.14) and (3.15) account for both the differences in the material properties of the different phases as well as surface tension effects at the phase boundary. The bulk fluids are incompressible. The numerical surface tension force approximation used is consistent with the VOF and LS techniques [222] [32], hence the major novelty of the embedded interface method is in the way the density and viscosity fields are updated when the fluids and the interface evolve in time and space. 

Fig. 6. Coarse-grained description of a liquid-gas interface, where the intrinsic profile and local structure of the interface is disregarded, and one rather treats the interface as an elastic membrane at position z h(x, v) ( sharp kink -approximation for the interfacial profile).

This coarse-graining can also be used to justify the "drumhead model of an interface, cf. fig. 6, where on a more macroscopic scale the internal structure of the interface is disregarded, and one is more interested in large scale fluctuations of the local position z = h(x,y) of this interface. In this sharp kink approximation the interface is described similarly to an elastically deformable membrane. 

Suppose now that the interface is weakly curved, so that isodensity levels do not coincide anymore with planes 2 = const. The nominal location of a curved diffuse interface (e.g. the Gibbs equimolar surface) can be used to describe it in the language of differential geometry commonly applied to sharp interfaces. Its spatial position can be defined in a most general way as a vector function X(4) of surface coordinates 4- A curved interface can be approximated locally by an ellipsoid with the half-axes equal to the principal curvature radii. If both radii far exceed the characteristic interface thickness, all isodensity levels are approximated by ellipsoidal segments equidistant from the interface. The density changes along the direction 2 normal to isodensity surfaces, and the... 

X. Feng and A. Prohl. Analysis of a fully discrete finite element method for the phase field model and approximation of Its sharp interface limits. Math. Comput, 73 541-567, 2004. 

Macroscopic vessels represent only a small fraction of the circulatory system. Approximately lO blood vessels are the capillaries whose diameters are comparable with the dimensions of the red blood cells, i.e., 5-10 p,m [18]. In the smallest capillaries with diameters 10 m, where the pressure drops to 1 kPa, blood flow represents a composite system with sharp interface between liquid and solid phases. In this spatial scale, the red blood cells (RBC) describe the phase volume with distinct elastic properties. In contrast to the macroscale, at the microscopic scale blood flow can be viewed at as the collective motion of an ensemble of microscopically interacting discrete particles. Unlike in large blood arteries, in the blood capillaries the wall consists of a layer of endothelial cells [18] responding to the shear flow. 

In the rest of this section, discussion and elaboration of concepts are presented based on the existence of sharp interfaces. Note also that the following discussion assumes that the considerations presented for unfilled polymer blends also apply to filled polymer blends. This assumption is based on the continuum approximation, which in turn is based on the approximation that the presence of fillers, when used in small quantities, does not significantly change the flow field. The continuum approximation is considered to be valid in view of the relatively small impact of fillers on the flow field, especially at low loading levels and at relatively higher shear rates that are common in processing flows [139]. 

FIG. 4. Loss modulus of TR 41-1649, with 0.482 polystyrene. Data symbols and lines are the same as in Fig. 3. Major differences are apparent between the three types of interphases incorporated in the model. The linear profile is approximately correct in the mid-range but deficient as the peaks are approached, while the sharp interface case is an extremely poor predictor over the entire interpeak region. 

The integral can be computed numerically using here f p) given by the second expression in Eq. (6). The result (Fig. 4) is more informative than the sharp-interface computation, as it also gives the dependence of surface tension on temperature. The surface tension decreases with growing temperature and vanishes at the critical point a/ bTc) = bpc = The low-temperature limit 7 oc pfy/aK/b oc pfAi/(fi is qualitatively the same as in the sharp interface model, and the interface thickness reduces to y/Kja a d away from the critical point. The density profile and surface energy can be computed analytically in the vicinity of a critical point [at Tc-T = 0(e ) and p - Pc = 0( )] where g p) can be approximated by a cubic polynomial. 

Mass transport across isodensity lines should become particularly important when the lubrication approximation breaks down. This should happen near the contact line in the case when two alternative fluid densities near the solid wall are possible. If. say. the boundary densities are Psv 1 and / 5/ = 1 - a, a 1, the three-phase contact line can be viewed as a sharp transition between 0(1) positive and negative values of the nominal thickness h, such that < 1 on either side. This can be treated as a shock of Eq. (91) or (93). The Hugoniot condition, which should ensure zero net flux through the shock, is the equality of chemical potentials on both sides. Unfortunately, this condition cannot be formulated precisely, since the sharp-interface limit of the surface tension term is inapplicable in the shock region. Moreover, our test computations of the profile of the dense layer using Eq. (94) with different boundary conditions imposed on the shock at h = ho showed that the spreading velocity is very sensitive to the conditions on the shock. 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

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