Hydrodynamic paradox

On a molecular scale there is no sharp boundary between hydrodynamically stagnant and movable solvent molecules. As discussed In sec. 2.2, the, say tangential, diffusion coefficient of water near many surfaces may be somewhat lower than in bulk, but it is not zero. The very existence of ionic conduction In the layer(s) adjacent to surfaces also points to non-zero mobility. Yet, phenomenologically such layers behave as immobilized. This looks like a paradox, but the phenomenon is encountered in other places as well. For Instance, a few percent of gelatin added to water may hydrodynamically immobilize the liquid completely, without markedly impairing ionic conduction or self-diffusion of dissolved ions. Macroscopic immobilization of a fluid is not in conflict with mobility on a molecular sceile. 

Hanshaw B. B. and Hill G. A. (1969) Geochemistry and hydrodynamics of the Paradox Basin Region, Utah, Colorado and New Mexico. Chem. Geol. 4, 264- 294. 

For the UF of proteins, the concentration polarization model has been found to predict the filtration performance reasonably well [56]. However, this model is inherently weak in describing the two-dimensional mass transport mechanism during crossflow filtration and does not take into account the solute-solute interactions on mass transport that occur extensively in colloids, especially during MF [21,44,158,159]. The diffusion coefficient, which is inversely proportional to the particle radius, is low and underestimates the movement of particles away from the membrane [56]. This results to the well-known flux paradox problem where the predicted permeate flux is as much as two orders of magnitude lower than the observed flux during MF of colloidal suspensions [56,58,158]. This problem has then been underlined by the experimental finding of a critical flux for colloids, which demonstrates the specificity of colloidal suspension filtration wherein just a small variation in physicochemical or hydrodynamic conditions induces important changes in the way the process has to be operated [21]. 

An attempt to solve the hydrodynamic problem on the flow past a cylinder by using the linear Stokes equations (2.1.1) leads to the Stokes paradox [179, 485],... 

Abstract. The lectures review the statics and dynamics of the gas-liquid-solid contact line, with the emphasis on the role of intermolecular forces and mesoscopic dynamics in the immediate vicinity of the three-phase boundary. We discuss paradoxes of the existing hydrodynamic theories and ways to resoluve them by taking account of intermoleculr forces, activated slip in the first molecular layer, diffuse character of the gas-liquid interface and interphase transport. 

The observed power laws can be found theoretically by hydrodynamical calculations in the S = 0 case. For strictly positive values of S and volatile liquids, one might argue that the drop spreads on a preexisting liquid film and that the true value of S is zero. But the paradox remains in the case of nonvolatile liquids. Why is the dynamics of spreading independent of S ... 

This paradox has its counterpart in the theoretical calculations when the hydrodynamic equations for spreading are solved, one obtains a divergence of the viscous dissipation at the edge of the drop. Ad hoc cut-off lengths or modified limit conditions must be introduced to remove this divergence. 

Table 20.8 contains a compilation of literature entries on the voltammetry of conducting polymer films. The scope of these studies is similar to that of the transient experiments discussed in Section V.A in terms of the types of electrodes and media employed. Both cyclic and hydrodynamic voltammetry have been used as shown in Table 20.8. Other aspects under discussion include the mathematic modeling of cyclic voltammo-grams [277,278], the occurrence and origin of prewaves in the cyclic voltammograms [319], the use of very fast scan rates [220], structural relaxation effects and their manifestation in voltammetry [304,317,320], the inactivation of polymer electroactivity when driven to extreme potentials, and the so-called polythiophene paradox [225,226,306,321]. Unusual media and cryogenic temperatures have also been employed for the volta-mmetric observation of doping phenomena [322-325]. Dual-electrode voltammetry (Section II.1) has been performed on derivatized polypyrrole [290] in an attempt to deconvolute the electronic and ionic contributions to the overall conductivity of the sample as a function of electrode potential. Finally, voltammetry has been carried out in the solid state , i.e., in the absence of electrolyte solutions [215,323].

Young equation. This has remained an unsolved theoretical problem for a long time because of the apparent contradiction between the advancing motion of the contact line and the no slip hydrodynamic boundary condition for the liquid at the solid surface this paradox is solved by a rolling motion of the spreading liquid on the solid surface. ... 

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