Hydrodynamic fields

In a concentrated solution, characterized by an effective medium viscosity r e "Hs, the hydrodynamic field decays much faster due to the shielding effect of the encountered polymer segments ... 

The equations of motion (75) can also be solved for polymers in good solvents. Averaging the Oseen tensor over the equilibrium segment distribution then gives = l/ n — m Y t 1 = p3v/rz and Dz kBT/r sNY are obtained for the relaxation times and the diffusion constant. The same relations as (80) and (82) follow as a function of the end-to-end distance with slightly altered numerical factors. In the same way, a solution of equations of motion (75), without any orientational averaging of the hydrodynamic field, merely leads to slightly modified numerical factors [35], In conclusion, Table 4 summarizes the essential assertions for the Zimm and Rouse model and compares them. 

In principle, one should solve the Boltzmann equation Eq. (65) in order to arrive at explicit expressions for the pressure tensor p and heat flux q, which proves not possible, not even for the simple BGK equation Eq. (11). However, one can arrive at an approximate expression via the Chapman Enskog expansion, in which the distribution function is expanded about the equilibrium distribution function fseq, where the expansion parameter is a measure of the variation of the hydrodynamic fields in time and space. To second order, one arrives at the familiar expression for p and q... 

Hypothetical sphere, impenetrable to the surrounding medium, displaying in a hydrodynamic field the same frictional effect as an actual polymer molecule. 

Adjective describing a chain macromolecule that behaves in a hydrodynamic field as though the solvent within the domain of the macromolecule were virtually immobilized with respect to the macromolecule. 

The investigation by Lynn and Huff201 was the first one in which the true velocity profiles during polymerization in a tubular reactor were determined. The idea of the dependence of the hydrodynamic field on the varying rheokinetics during a chemical reaction is quite fruitful and has... 

When a low frequency AC electric field is imposed, the particle oscillates around its mean position and platy particles may become optimally aligned with the field. At high frequencies, neither particle shift nor alignment takes place. However, translational movement of dispersed particles can be attained in an asymmetric AC field (without a DC component). The observed drift is attributed to the velocity-dependent viscous drag force in relation to double layer polarization as sketched in Figure 2 for reference, bacteria swim at 0.02-1 mm/s. For more details see Palomino [2], The field frequency co must be low enough such that ionic concentrations and hydrodynamic fields may adjust to... 

In two-dimensional electrophoresis the charged particle migrates in a field of two forces which act perpendicularly to one another. The first force F creates a vertical hydrodynamic field. A flow of liquid runs by gravity down a vertical curtainlike supporting medium to which we shall refer as the substrate. The liquid is a buffer solution which through its pH and ionic strength determines the mobility of the particle. 

T. Takabayashi, Remarks on the Formulation of Quantum Mechanics with Classical Pictures and on Relations between Linear Scalar Fields and Hydrodynamic Fields, Prog. Theor. Phys., 1953 (9) 187-220. 

The state variables (called hydrodynamic fields) chosen in fluid mechanics are... 

Complex fluids are the fluids for which the classical fluid mechanics discussed in Section 3.1.4 is found to be inadequate. This is because the internal structure in them evolves on the same time scale as the hydro-dynamic fields (85). The role of state variables in the extended fluid mechanics that is suitable for complex fluids play the hydrodynamic fields supplemented with additional fields or distribution functions that are chosen to characterize the internal structure. In general, a different internal structure requires a different choice of the additional fields. The necessity to deal with the time evolution of complex fluids was the main motivation for developing the framework of dynamics and thermodynamics discussed in this review. There is now a large amount of papers in which the framework is used to investigate complex fluids. In this review we shall list only a few among them. The list below is limited to recent papers and to the papers in which I was involved. 

We note that the transformation (xj, x2) —> x defined in (103) is one-to-one. We interpret physically 77, j>. u as overall (i.e., large-scale) hydrodynamic fields on which there are superposed "/ -fluctuations" (i.e., small-scale quantities) characterized by the np-particle distribution function f p. 

Sjoberg, B., and Osterberg, R. Small-angle X-ray scattering of chain molecules in a hydrodynamic field. The internal rigidity of double-stranded DNA. J. Appl. Cryst. 16, 349-353 (1983). 

The radiation of a particle, comparable in size to the light wavelength, leads to induction of dipoles in different parts of the particle that are not in phase (Figure 5.71). The net scattered light received by the detector is a result of the interference of the beams scattered from the different points of the particle. In this case, the function P(9) depends on the particle size and shape. If the particles have an anisodiametrical shape, P(9) could depend on their orientation as well. Typical examples are rodlike particles that are preferentially oriented along a given direction by an electric " or hydrodynamic field. In most systems, however, the particles are randomly oriented, and averaging over all possible orientations is performed to calculate P(9). 

Substantial interest has been raised in the problem of the structure and dynamics of suspensions in shear hydrodynamic fields. ° ° The experiments showed that both shear-induced melting and shear-induced ordering can be observed at different particle volume fractions and shear rates. The nonequilibrium microstructure of the suspension under shear can be investigated in these experiments and compared with the predictions from analytical theories and computer simulations. 

The motion of a particle in infinite fluid creates some velocity and pressure fields. Neighboring particles move in already perturbed hydrodynamic fields. Simultaneously, the first particle itself experiences hydrodynamic interaction with the neighboring particles and neighboring moving or fixed surfaces. Since in the majority of actual disperse systems, the existence of an ensemble of particles and the apparatus walls is inevitable, the consideration of the hydrodynamic interaction between these objects is very important. One of the methods for obtaining the required information about the interaction is based on the construction of exact closed-form solutions. However, even within the framework of Stokes hydrodynamics, to describe the motion of an ensemble of particles is a very complicated problem, which admits an exact closed-form solution only in exceptional cases. 

In LES/FMDF calculations of single-phase flows, the resolved hydrodynamic field is obtained by solving the filtered form of the compressible Navier-Stokes equations [3] ... 

While "kinetics" means only time-dependent, the terminology "adsorption dynamics" includes the coupling of transport by diffusion and hydrodynamic fields. It comprises surface concentration changes, movement in the adsorption layer and correlation between the distribution of surface concentration and velocities along the surface. The adjacent liquid bulk is involved in the diffusion and hydrodynamic flows which exhibit mutual interrelation. The term "dynamic adsorption layer" refers only to the non-equilibrium state of the adsorption layer. 

The theoretical description of a diffusion process of a surfactant to, or from, the surface of a floating bubble is impossible without information on the floating velocity and the hydrodynamic field around the bubble. The first of these quantities can be found comparatively easily experimentally, whereas the Navier-Stokes equation is used to define the hydrodynamic field around the floating bubble. A solution of the equation must satisfy all boundary conditions at the bubble surface. It should be stated that a general analytical solution of this... 

In the dynamic adsorption layer theory of Deqaguin-Dukhin, the hydrodynamic field of a bubble can be assumed to be known as first approximate, while the more difficult stagnant cap problem has still to be solved. For the solution of this hydrodynamic problem unusual and very difficult boundary conditions exist which are very inconvenient even after essential simplifications. The hydrodynamic field of a bubble is studied imder the assumption that the stagnant cap is completely immobilised and any motion of the surface beyond the stagnant cap is ignored. Since the description of the stagnant cap is to a large extent a hydrodynamic problem, it has received less attention (cf. Section 8.7). 

Consider flotation (Chapter 10), dynamic adsorption layers effect all stages of the elementary flotation process, since they affect the hydrodynamic field of a bubble and thus the trajectory of... 

At first the effect of sedimentation on collision efficiency is taken into account since it can strongly decrease the role of the hydrodynamic field of bubble and DAL. This consideration proves to be more obvious when the method of collision efficiency calculation is used as proposed by Dukhin Derjaguin (1958). 

Using the formulas for the hydrodynamic field of a bubble carrying a rear stagnant cap (cf. Section 8.7), we can calculate the effect of the cap on collision efficiency. It is unlikely that such work is of interest since the theory described in Section 8.7 is restricted to small Reynolds numbers. Thus, we caimot expect agreement between this theory and reality since the leading surface must be either completely or strongly retarded, according to experimental data by Okazaki (1964). 

The exactness of Eq. (10.41) is restricted because the stagnant cup can deform the potential flow. However, it is well known that in the theory of the hydrodynamic boundary layer of a solid sphere a potential flow can be used for the description of the hydrodynamic field outside... 

Let us compare the fluxes of the disperse particles and molecular impurities on the bubble surface assuming that the hydrodynamic field is described by the Stokes equation. It enables to use Eq. (8.152) for the description of the impurity flux. This equation can be transformed into an analogue of the capture efficiency. 

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