Coupling of length scales

Realistic mixing problems are inherently difficult owing to the complexity of the flow fields, to the fact that the fluids themselves are Theologically complex, and to the coupling of length scales. For this very reason, mixing problems have been attacked traditionally on a case-by-case basis. Modeling becomes intractable if one wants to incorporate all details at once. Nevertheless it appears important to focus on common features and to take a broad view. 

Concurrent Coupling of Length Scales Methodology and Application. 

Figure 6. Hypothetical coupling of length scales in the attachment of two hydroxylated oxide nanoparticles in aqueous solution. The crystal cores are represented by continuum finite elements with elastic moduli Xy and dielectric tensor sy. The far-field continuum solvent has viscosity p, dielectric constant s, and exerts random boundary forces Fstoch on the fluid inside the large sphere modeled using particle methods. SPC is a simple point charge model for water.

Rudd RE, Broughton JQ (2000) Concurrent coupling of length scales in solid state systems. Physica Status Solidi B-Basic Res 217 251-291... 

Rudd, R.E. and Broughton, J.Q. (2000) Concurrent Coupling of Length Scales in Solid State Systems. Phys. Status Solidi B -Basic Res., 217, 251-291. 

There are numerous coupled atomistic-continuum modeling strategies, for example, peridynamics, a family of quasi-continuum (QC) methods, finite element adaptive techniques (FEAt), coupling-of-length scales (CLS) method, and coupled atomistic-and-discrete-dislocalion (CADD) method. The grand challenge of multiscale modeling is to connect atomic-scale processes to mesoscale and bulk continuum models. 

The coupling-of-length scales (CLS) method was developed to study the dynamic response of the stmcture and its nanoscale contributions to dissipation. In the CLS technique, further progress in coupling methods was made by embedding a quantum-mechanically based TB region inside the empirical atomistic region. 

Coupling of Length Scales Method Atomistic/Continuum Part The FE/MD/ TB coupling of length scales (CLS) method, " " " "° "" also known as MAAD (macroscopic, atomistic, ab initio dynamics), is one of the few methodologies to provide a dynamical coupling of three different regimes the macroscopic... 

Coupling of Length Scales Method Atomistic/Quantum Part... 

Element Modeling in MEMS Coupling of Length Scales. 

Monterey, CA. Coupling of Length Scales in Cracks and Micromachines. 

Clara, CA, Computational Publications, Boston 1998, p. 287. Coupling of Length Scales and Atomistic Simulation of a MEMS Device. 

At the heart of the mode coupling theory of liquids is the assumption that a separation of time scale exists between different dynamical events. While the time scale separation between the fast collisional events and the slower collective relaxation is explicitly exploited in the formulation of the theory, there is also an underlying assumption of the separation of length scales between different relaxation modes. Much of the success of MCT depends on the validity of this separation of length and time scales. 

The experimental observation that viscoelastic behavior can be shifted and superimposed, at least to a first approximation, means that all the relaxation processes involved have the same (or nearly the same) temperature dependence. Because the relaxation processes that occur in polymers involve a whole range of length scales, by which we mean they involve coupled motions that range from local rearrangements of just a few chain segments to motions of the chain as a whole, this implies that frictional forces encountered by a chain act in the same or a very similar manner on all the segments. This is important if you intend to be in the business of developing theories of polymer dynamics, but concerns us no further here. 

Yet another complication is the property demand on the structure of the hydrocarbon chain, which is totally different when the oleochemical is used as an energy source from when it is used as the hydrophobic part of a surfactant. To produce energy through combustion you just need a certain amount of carbon material, but for a surfactant the behaviour is mostly determined by the length and structure of the hydrophobe. This means that, for example, tallow oil cannot be easily substituted by, for instance, palm oil to get the same surfactant properties. Therefore, if a couple of major power plants choose to use tallow oil for their combustion, they could easily consume the total amount produced in Europe. This would be a rather attractive option for the tallow producers, having to deal with only a couple of large-scale customers prepared to pay premium... 

On short length scales the coarse-grained description breaks down, because the fluctuations which build up the (smooth) intrinsic profile and the fluctuations of the local interface position are strongly coupled and camiot be distinguished. The effective interface Flamiltonian can describe the properties only on length scales large compared with the width w of the intrinsic profile. The absolute value of the cut-off is difficult... 

Other more exotic types of calamitic liquid crystal molecules include those having chiral components. This molecular modification leads to the formation of chiral nematic phases in which the director adopts a natural helical twist which may range from sub-micron to macroscopic length scales. Chirality coupled with smectic ordering may also lead to the formation of ferroelectric phases [20]. 

The size-dependence of the intensity of single shake-up lines is dictated by the squares of the coupling amplitudes between the Ih and 2h-lp manifolds, which by definition (22) scale like bielectron integrals. Upon a development based on Bloch functions ((t>n(k)), a LCAO expansion over atomic primitives (y) and lattice summations over cell indices (p), these, in the limit of a stereoregular polymer chain consisting of a large number (Nq) of cells of length ao, take the form (31) ... 

When the length scale approaches molecular dimensions, the inner spinning" of molecules will contribute to the lubrication performance. It should be borne in mind that it is not considered in the conventional theory of lubrication. The continuum fluid theories with microstructure were studied in the early 1960s by Stokes [22]. Two concepts were introduced couple stress and microstructure. The notion of couple stress stems from the assumption that the mechanical interaction between two parts of one body is composed of a force distribution and a moment distribution. And the microstructure is a kinematic one. The velocity field is no longer sufficient to determine the kinematic parameters the spin tensor and vorticity will appear. One simplified model of polar fluids is the micropolar theory, which assumes that the fluid particles are rigid and randomly ordered in viscous media. Thus, the viscous action, the effect of couple stress, and... 

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