Scaling argument

Boundary layer flows are a special class of flows in which the flow far from the surface of an object is inviscid, and the effects of viscosity are manifest only in a thin region near the surface where steep velocity gradients occur to satisfy the no-slip condition at the solid surface. The thin layer where the velocity decreases from the inviscid, potential flow velocity to zero (relative velocity) at the sohd surface is called the boundary layer The thickness of the boundary layer is indefinite because the velocity asymptotically approaches the free-stream velocity at the outer edge. The boundaiy layer thickness is conventionally t en to be the distance for which the velocity equals 0.99 times the free-stream velocity. The boundary layer may be either laminar or turbulent. Particularly in the former case, the equations of motion may be simphfied by scaling arguments. Schhchting Boundary Layer Theory, 8th ed., McGraw-HiU, New York, 1987) is the most comprehensive source for information on boundary layer flows. 

On the basis of general scaling arguments it can be shown that the stochastic evolution of a driven interface along a strip of width L is characterized by long wavelength fluctuations (w(L, t)) which have the following time and finite-size behavior [60]... 

The scaling arguments given here for two-dimensional growth patterns can be extended formally in a straightforward fashion to three dimensions. For dendritic structures this seems to be perfectly permissible since the basic growth laws are rather similar in two and three dimensions [117,118] ... 

This estimate, which is based on simple scaling arguments, may be substantiated by calculating the free energy of a disordered Peierls chain, as we did in Ref. [31]. Treating the lattice vibrations classically, the final result for the kink... 

A second approach [7] allows for the effects of excluded volume correlations and self-avoidance by use of scaling arguments. In this picture, the layer is viewed... 

Numerical SCF calculations give theoretical predictions for the chain length dependences of the characteristic dimensions in reasonable agreement with these economical scaling arguments [63],... 

The dynamic scaling argument supposes that when the geometrical parameters of the chain (i.e. N and lp) are changed from N into N/A and lp into lpV, any physical quantity (A), either static or dynamic, related to the molecular size will be transformed into XXA. The parameter v is the exponent in Eq. (9) and is equal to 1/2 in 0-solvent and 3/5 in good solvent. 

The scaling argument provides only the exponent but not the absolute numerical value for the constant. Therefore, for quantitative results, it should be completed by some more refined technique like the afore-mentioned renormalization group method [49],... 

Given that for a rubber compound, F = 3 ji, comparison of Eqs. (11) and (14) shows that the simple scaling argument is in agreement with the more exact expression to within a factor of about 2. 

On the basis of scaling arguments, general functional dependencies can also be derived. For example, dimensional analysis shows that the center of mass diffusion coefficient DG for Zimm relaxation has the form... 

This is different for the star core. Figure 57 provides a comparison of the spectra at two Q-values with those from an equivalent full star (sample 3). Over short periods of time, both sets of spectra nearly coincide. However, over longer periods of time, the relaxation of the star core is strongly retarded and seems to reach a plateau level. This effect may be explained by the occurrence of interarm entanglements as recently proposed by scaling arguments [135]. 

In the first part to follow, the equations of motion of a soft solid are written in the harmonic approximation. The matrices that describe the potential, and hence the structure, of the material are then considered in a general way, and their properties under a normal mode transformation are discussed. The same treatment is given to the dissipation terms. The long wavelength end of the spectral density is of interest, and here it seems that detailed matrix calculations can be replaced by simple scaling arguments. This shows how the inertial term, usually absent in molecular problems, is magnified to become important in the continuum limit. 

One may wonder to what extent our predictions for hard spheres apply to a system of soft particles in a polymer solution. A definite answer to this question cannot be given at the moment since numerical data for the depletion of free polymer chains in the neighbourhood of a surface with terminally attached chains are not yet available. Some qualitative features for such a system have been discussed using scaling arguments (24). We may expect that the depleted amount of polymer is, at least in some cases, less than near a hard surface, giving rise to weaker attraction. Both the destabilization concentration (J) and the restabilisation concentration (<(> ) could be much lower. Experimental observations support this qualitative conclusion (1-5). 

For this reason, we will restrict our subsequent approach to planar configurations of the two electrons and of the nucleus, with the polarization axis within this plane. This presents the most accurate quantum treatment of the driven three body Coulomb problem to date, valid in the entire nonrelativistic parameter range, without any adjustable parameter, and with no further approximation beyond the confinement of the accessible configuration space to two dimensions. Whilst this latter approximation certainly does restrict the generality of our model, semiclassical scaling arguments suggest that the unperturbed three... 

First of all it is seen that the SCF results are free of any noise, whereas there is plenty of noise in the MD profiles (note, however, that the density profiles on both halves of the bilayer are in this case not averaged the close resemblance between the profiles on both halves thus indicates that the membranes are well equilibrated). Apart from this, inspection of Figure 18 shows a remarkable resemblance between the two set of predictions. Many details are in semi-quantitative agreement. Moreover, many of the features of membranes composed of SOPC resemble those of DMPC discussed above. For example, the width of the membrane-water interface is about 1 nm, i.e. the size of just two to three water molecules. This width is consistent with the scaling arguments mentioned at the beginning of this chapter. A more accurate comparison... 

Chapter 2 reviews the statistical theory of turbulent flows. The emphasis, however, is on collecting in one place all of the necessary concepts and formulae needed in subsequent chapters. The discussion of these concepts is necessarily brief, and the reader is referred to Pope (2000) for further details. It is, nonetheless, essential that the reader become familiar with the basic scaling arguments and length/time scales needed to describe high-Reynolds-number turbulent flows. Likewise, the transport equations for important one-point statistics in inhomogeneous turbulent flows are derived in Chapter 2 for future reference. 

The mixing parameter Q must be chosen to yield the correct mixture-fraction-variance dissipation rate. However, inertial-range scaling arguments suggest that its value should be near unity.165... 

A scaling argument does not describe the full functional form of the dependence of the chain dimensions on concentration. We would expect the dimensions of the polymer to be a smoothly changing function between c and the concentration J where the coils reach their limiting radius equivalent to the 6 dimensions. However, to a first approximation we can suppose that Equation (5.73) applies up to the concentration ct ... 

For this molecule one needs, in addition to the H-H and D-D interactions, the H-D interactions. These can be obtained from experiments, if available, or can be estimated from scaling arguments. Introducing the reduced masses, p, of the C-H and C-D bonds, and defining p by... 

The distribution of the center-to-end distance, F(R(,), in a star can also be predicted from scaling theory. For EV chains, it is expected to be close to Gaussian [26], except for small R. Applying scaling arguments and RG theory, Ohno and Binder [27] obtained a power-law behavior for small R, F(Rj,)=(Rj,/) with the exponent value 0(f)=l/2 for high f. They also considered the case of a star center adsorbed on a planar surface, evaluating the bead density profiles and the distribution of center-to-end distance in the directions perpendicular and parallel to the surface in terms of similar power-laws. 

These results can readily be obtained by a scaling argument, Dc. -g... 

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