Surface hydrodynamic theory

General hydrodynamic theory for liquid penetrant testing (PT) has been worked out in [1], Basic principles of the theory were described in details in [2,3], This theory enables, for example, to calculate the minimum crack s width that can be detected by prescribed product family (penetrant, excess penetrant remover and developer), when dry powder is used as the developer. One needs for that such characteristics as surface tension of penetrant a and some characteristics of developer s layer, thickness h, effective radius of pores and porosity TI. One more characteristic is the residual depth of defect s filling with penetrant before the application of a developer. The methods for experimental determination of these characteristics were worked out in [4]. 

Hydrodynamic theory shows that the thickness, 8, of the boundary layer is not constant but increases with increasing distance y from the flow s stagnation point at the surface (Fig. 4.4) it also depends on the flow velocity ... 

In a hydrodynamic theory of the free, clean, surface of a turbulent liquid, Levich 19a) postulates that there exists an upper zone of liquid, of thickness X, in which the turbulent regime is so altered by the surface tension (which opposes local deformations) that within this zone the turbulence is severely damped. Right in the plane of the surface (at... 

As we have indicated, Sbl represents an average thickness of the unstirred air layer adjacent to a leaf (or to leaflets for a compound leaf). The main factors affecting Sbl are the ambient (local) wind speed and the leaf size, with leaf shape exerting a secondary influence. Partly for convenience, but mainly because it has proved experimentally justifiable, we will handle the effect of leaf size and shape on boundary layer thickness by the characteristic dimension /, which is the mean length of a leaf in the direction of the wind. Based on hydrodynamic theory for laminar flow adj acent to a flat surface as modified by actual observations under field conditions, an approximate expression for the average thickness of the boundary layer next to a flat leaf is... 

Instead of the numerical factor 4.0 in Equation 7.10, hydrodynamic theory predicts a factor near 6.0 for the effective boundary layer thickness adjacent to a flat plate (both numbers increase about 3% per 10°C Schlichting and Gersten, 2003). However, wind tunnel measurements under an appropriate turbulence intensity, as well as field measurements, indicate that 4.0 is more suitable for leaves. This divergence from theory relates to the relatively small size of leaves, their irregular shape, leaf curl, leaf flutter, and, most important, the high turbulence intensity under field conditions. Moreover, the dependency of 6bl on /° 5, which applies to large flat surfaces, does... 

The earliest available hydrodynamic theory of water wave damping by elastic surface films was published by Lamb (1895). He refers to Reynolds (1880) and the experiments by Aitken (see Scott 1979, Giles and Forrester 1970), but prior publication of the detailed theory is not indicated. All but the outline of the theory was omitted from later editions of this book, and it is likely that Lamb assumed that damping was greatest with an inextensible film, and that intermediate elasticities, therefore, had less effect (cited after Scott 1979). This conclusion was shown by Dorrestein (1951) to be incorrect. The paper by Levich (1940) was the first to present in detail the linearised hydrodynamics of waves on a water surface with surface dilational elasticity. The only cases considered in detail concern insoluble films, and represent the clean and incompressible-film-covered surface. A detailed treatment of the hydrodynamic theory of surface waves, including the effect of an elastic surface film, was published by Levich in 1962. In addition, the damping caused by dissolved surface-active material was considered. Further laboratory experiments performed until 1978 were briefly reviewed by Scott (1979). 

If the idealized concept of fluid film failure proposed above is to have any significance in the world of experimental mechanics and engineering, we must find a basis for its validity and utility. The study of fluid film failure in a practical sense then becomes the study of the behavior of the boundary surfaces of the solids and of the intervening fluid lubricant as the thickness of the lubricant film approaches zero. An important aspect is the reliability of the measurement technique for very thin films. We must be careful not to think of fluid film failure as rupture or breakdown by exceeding the intrinsic strength of the lubricant material. Bulk liquid films do not behave in that way. We know by hydrodynamic theory that the pressure a film of fluid is able to... 

Even in this extreme condition, a very thin film still remains under flow to keep the surfaces apart. This means that something more exists in addition to hydrodynamic theory. EHD film formation depends on the coupled effects of physical property changes of the lubricants at high pressure and elastic changes in the shape of the solids about the contact area, which fect the pressure distribution. The high pressure in the EHD contact area increases the viscosity of lubricants enormously. 

In other words, at the interface the charge transfer reaction of cupric ion accounts for the entire current flow, but outside the double layer the cupric ions contribute in a negligible marmer to current flow. This immediately raises the question of how copper ion will get to the electrode surface in sufficient quantity to account for the faradic current. Obviously, additional transport mechanisms are needed for this, namely, diffusion and convection. From hydrodynamic theory, it is known that the fluid velocity is zero at a solid surface. The convective flux therefore is zero also at the surface and diffusion alone must account... 

Other hand, where a solid surface is involved, the film theory and the boundary layer theory are the most frequently used (although several other hydrodynamic theories have also been proposed). However, it has been shown (Vieth et al., 1963) that almost the same values of enhancement are obtained even when the turbulent boundary layer theory is used for solid-liquid reactions. All theories give exactly the same result ... 

Obviously, at very high contact pressures, the lubricating liquid between the two surfaces rapidly increases in viscosity until it must attain the consistency of a solid or wax rather than a liquid. In such a case, it is easy to see why some lubricating oils that exhibit such thickening behavior show better performance than would be predicted for classic hydrodynamic theories. It also helps explain why other materials (e.g., sihcone oils), which have less dramatic viscosity increases with pressure, do not perform as well under extreme conditions. In the viscosity range where elastohydrodynamic lubrication occurs, fluids may begin to exhibit non-Newtonian behavior leading to a more complicated relationship in terms of lubricant effectiveness. 

The Maxwell boundary condition [Eq. (122)] also gives rise to hydro-dynamic boundary conditions of the form (125) with o>, t, and x roughly prop)ortional to So, as long as a is of order unity (more specifically a IJ/h) the corrections to stick boundary conditions remain small, and only if a becomes of order boundary conditions differing appreciably from stick are obtained.This is an indication why stick boundary conditions are for most purposes a very good approximation in hydrodynamic theory a reflection mechanism that is almost specular is not very likely to occur in nature, due to irregularities in surface structures and thermal motion of the surface molecules. 

Besides this confusion over v and F, it is further incoirect to confuse a thermodynamic quantity ifi) with a hydrodynamic one (F,). The quantity Fe was determined from the hydrodynamic theory of rigid, impermeable ellipsoids. However, the protein may not be ellipsoidal in shape, it may not be rigid in a hydrodynamic field, and it may not be impermeable to the flow of solvent. In addition, the hydrodynamic boundary condition of no slippage on the. surface of the particle may not be satisfied, and the... 

---