Levich

V. G. Levich, Physicochemical Hydrodyruunics, translated by Scripta Technica, Inc., Prentice-Hall, Englewood Cliffs, NJ, 1962. 

Levich V G 1962 Physiooohemioal Hydrodynamios (Englewood Cliffs, NJ Prentice-Flall)... 

B, G. Levich, V. A. Myanlin, and Y, A. Vdovin, Theoretical Physics. An Advanced Text, North-Holland, Amsterdam, 1973, p, 418. 

In work for which Marcus and Levich separately received the Nobel Prize, it was shown from first principles (37) and ia different contexts that... 

David W. Taylor Model Basin, Washington, September 1953 Jackson, loc. cit. Valentin, op. cit.. Chap. 2 Soo, op. cit.. Chap. 3 Calderbank, loc. cit., p. CE220 and Levich, op. cit.. Chap. 8). A comprehensive and apparently accurate predictive method has been publisned [Jami-alahamadi et al., Trans ICE, 72, part A, 119-122 (1994)]. Small bubbles (below 0.2 mm in diameter) are essentially rigid spheres and rise at terminal velocities that place them clearly in the laminar-flow region hence their rising velocity may be calculated from Stokes law. As bubble size increases to about 2 mm, the spherical shape is retained, and the Reynolds number is still sufficiently small (<10) that Stokes law should be nearly obeyed. 

The dissolution of the refractoty by die first mechanism is described by an equation due to Levich (1962), where the flux, j, of the material away from the wall is calculated tlrrough the equation... 

Levich, V.G., 1962. Physicochemical Hydrodynamics. New York Prentice-Hall Inc. 

Levich Physiochemical Hydrodynamics, Prentice Hall, New Jersey (1962)... 

The final result is very well known in experimental studies and is usually used to evaluate a32 from known and s values. Other studies in this field include the work by Shinnar and Church (S7) who used the Kolmogoroff theory of local isotropy to predict particle size in agitated dispersions, and an analysis by Levich (L3) on the breakup of bubbles. Levich derived an expression for the critical bubble radius (the radius at which breakup begins) ... 

Studying the steady motion of a single medium-size bubble rising in a liquid medium under the influence of gravity, Levich (L3, L4) solved the continuity equation simultaneously with the equations of motion by introducing the concept of a boundary layer for the case of a bubble. This boundary layer accounts for the zero, or extremely low, shear stress at the interface. Despite some errors in deriving the equations, his result was later confirmed with minor improvements (A4, M3, M10). 

Assuming that the velocity distribution for flow past a gas bubble differs relatively little from the velocity distribution in an ideal liquid, and neglecting the curvature of the boundary layer, Levich finds that... 

An evaluation of the retardation effects of surfactants on the steady velocity of a single drop (or bubble) under the influence of gravity has been made by Levich (L3) and extended recently by Newman (Nl). A further generalization to the domain of flow around an ensemble of many drops or bubbles in the presence of surfactants has been completed most recently by Waslo and Gal-Or (Wl). The terminal velocity of the ensemble is expressed in terms of the dispersed-phase holdup fraction and reduces to Levich s solution for a single particle when approaches zero. The basic theoretical principles governing these retardation effects will be demonstrated here for the case of a single drop or bubble. Thermodynamically, this is a case where coupling effects between the diffusion of surfactants (first-order tensorial transfer) and viscous flow (second-order tensorial transfer) takes place. Subject to the Curie principle, it demonstrates that this retardation effect occurs on a nonisotropic interface. Therefore, it is necessary to express the concentration of surfactants T, as it varies from point to point on the interface, in terms of the coordinates of the interface, i.e.,... 

The flux of surface-active agents from the surface into the bulk of the liquid may be controlled by the slower of the following processes 1) adsorption or desorption of surfactants at the surface or 2) diffusion of surfactants from the liquid bulk to the surface. Consequently, Levich evaluated the solution for a single drop... 

Waslo and Gal-Or (Wl) recently generalized the Levich solution [Eq. (65)] by evaluating the effect of and y on the terminal velocity of an ensemble of spherical drops of bubbles. Their solution is... 

Levich (L3, L4) has demonstrated that for NKt < 1, the local diffusional flux to the surface of the bubble is given by... 

According to Levich (L3) Eq. (142) can approximate the convective dif-fusional flux even in the domain 1 < jVRc < 600. Hence the application of the Marrucci correction [Eq. (44)] to Eq. (38) before combining it with Eq. (142) gives... 

This case can also be approached using Kolmogoroff s (K9, H15) theory of local isotropic turbulence to predict the velocity of suspended particles relative to a homogeneous and isotropic turbulent flow. By examining this situation for spherical particles moving with a constant relative velocity, varying randomly in direction, Levich, (L3) has demonstrated that... 

Solution From the Levich equation, (4-5), one can calculate first the disk current under the new conditions ... 

Derive the Levich equation for the limiting current at the rotating disk electrode [based on combining equations (4-4) and (1-12)]. 

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