Kapitsa theory

More recent film thickness measurements in the laminar wavy regime obtained by improved techniques (F2, F7, P3) have shown that there are appreciable reductions in the mean film thickness in the wavy regime for both vertical and sloped surfaces, as predicted by the Kapitsa theory. 

Hence, the trend predicted by the Kapitsa theory is supported by the recent, more accurate, film thickness measurements. This does not indicate, however, that the Kapitsa theory will apply in detail over the whole wavy laminar regime of film flow, since Kapitsa (K7) pointed out that such a reduction in the mean thickness should result for other types of wavy flow besides the particular case considered in his theory. 

It is seen that in general the various theories are in reasonable agreement with each other, especially the Benjamin and the Ishihara theories. The simple condition Nvt = 1 shows a similar trend with channel slope, but the values lie slightly higher at all slopes. The Kapitsa theory predicts that the values of Nr<>, will change little with slope except at very small slopes. 

Tailby and Portalski (T4) have reported measurements of the wavelengths near the point of wave inception on vertical films of various liquids. Even in this case, it was found that the wavelengths were considerably greater than predicted by the Kapitsa theory, Eq. (67), even in the cases in which the conditions of the theory were satisfied. Similar results have been obtained for water films on walls of various slopes (F7). 

Although there are numerous published investigations in which records of the wavy surface profile have been obtained, e.g. (H9, D16, Sll), not many of these have been analyzed for information on wavelengths, most being concerned with wave-size (height) distributions. However, it may be noted that the experimental wavelengths of Kapitsa and Kapitsa (K10) show a trend in the direction of the data reported above, even at very small Reynolds numbers (lVa < 25). It seems, therefore, that the Kapitsa theory is applicable only at very small flow rates, as far as wave characteristics are concerned, in the case of the free flow of wavy films. Allen (A3) has reported a similar conclusion. 

Portalski (T2) has extended Kapitsa s treatment of wavy film flow to obtain an expression for the increase in interfacial area due to the waves [Eq. (68)]. For mobile liquids this relationship predicts that the increase in interfacial area will be very large, reaching 150% for 2-propanol at NRe = 175, for example, though the applicability of the Kapitsa theory at such large Reynolds numbers is in doubt. Experimental values of the... 

Kapitsa and Kapitsa (K10), 1949 Wavy flow of water and alcohol films on outside of tube of diameter 2 cm., NRe < 100, studied photographically and stroboscopically. Experimental data at low flow rates in agreement with Kapitsa theory waves become random at large flow rates. 

Semenov (S7), 1950 Extension of earlier work to wavy film flow. Kapitsa theory simplified by omitting inertia terms, and applied to wavy film flow with co- or counter-flow of gas to give thickness, velocity, wavelength, wave velocity, stability, onset of flooding, etc. 

Tailby and Portalski (T2), 1960 Reports on extension of Kapitsa theory (K7) to give increase in interfacial area due to waves experimental measurements of Am, for wave inception, entry length, and increase in interfacial area. 

Allen (A3), 1962 Investigation of characteristics of liquid films on vertical surface, with emphasis on surface features. Kapitsa theory shown to be applicable only at low flow rates. Increase in interfacial area reported to be smaller than predicted by Portalski theory. 

Recently, Kasimov and Zigmund (K12) have published the first part of a new theoretical treatment of wavy film flow, extending their recent work on smooth laminar film flow (Section III, B, 5) to this case also. It is shown that, with appropriate assumptions, the new theory reduces to the Nusselt solution for smooth films, or to a result similar to the corrected Kapitsa solution. The most interesting conclusions to be drawn from the part of the theory so far published are ... 

Fig. 5. The lines c/u = 3 and c/u = 2.4 corresponding to the theories of Benjamin and Hanratty and Hershman and of Kapitsa are shown, together with the line given by Eq. (115), using 6 = 7 °. The remaining lines represent smoothed experimental wave velocity data (F7) for water films on wetted walls of slopes 7, 62, and 90°.

It can be seen immediately that the experimental values of c/U reach a value of 3 only at very small flow rates, near the flow rate for the onset of rippling, which is the zone for which Benjamin s theory is strictly applicable. The experimental values fall below the Kapitsa value of 2.4 at NRC = 30. The theoretical relationship by Ishihara et al. predicts that c/u will decrease as NRe increases, but less rapidly than observed experimentally. However, this theory is strictly applicable only at very small channel slopes and for waves of negligibly small amplitude, so that exact agreement cannot be expected. 

Kapitsa (K7, K8), 1948 Theoretical treatment of wavy flow of thin films of viscous liquids, including capillary effects. Only regular waves considered. Wavy flow shown to be more stable than smooth film, and about 7% thinner than smooth film at same flow rate. Also calculates wave amplitudes, wavelengths, etc., onset of wavy flow, effects of countercurrent gas stream, heat transfer. Theory applicable only if wavelength exceeds 14 film thicknesses. Error in treatment pointed out by Levich (L9). 

Levich (L9), 1959 Final chapter deals with film flow theory (smooth, wavy laminar, turbulent) with and without gas flow. Also considers mass transfer to such films. Correction to theory of Kapitsa (K7). 

Portalski (P4), 1963 Theories of film flow and methods of measuring film thickness are reviewed. Film thicknesses on vertical plate (zero gas flow) reported for glycerol solutions, methanol, isopropanol, water, and aqueous solutions of surfactants. Results compared with values calculated by Nusselt, Kapitsa, and corrected Dukler and Bergelin treatments. 

Portalski (P5), 1964 From Kapitsa s theory of wavy film flow, it is shown that regions of reversed flow exist under the wave troughs, leading to the generation of circulating eddies which may explain the increased rates of heat and mass transfer to wavy laminar films. 

The superfluidity phenomenon was discovered by P. L. Kapitsa in 1938 and in just three years it was interpreted in theory proposed by L. D. Landau. This theory (supported by quantum-mechanical premises) has gained recognition, since it has satisfactorily explained the results of experiments with superfluid helium 2. 

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