Hadamard-Rybczynski equation

However, if the bubble interface is assumed deformable and completely free from contaminants, the bubble rise velocity is given by the Hadamard-Rybczynski equation. When the viscosity ratio ( cg/p-l) is zero, the terminal bubble rise velocity is given by... 

For the limiting case of (j)=0 (no surfactant), equation (8.151) is transformed into the Hadamard-Rybczynski equation. For ())=7i (completely stagnant interface), a solid sphere behaviour results. 

As constant interfacial tension is no guarantee of absence of contamination, measurement of the upward velocity of a bubble in solution seems a much more sensitive criterion. Thus, the verification of the Hadamard-Rybczynski equation could be considered as a criterion, indicating that the solution has reached a purity beyond which there is no longer any contamination [22]. 

Internal circulation patterns have been observed experimentally for drops by observing striae caused by the shearing of viscous solutions (S7) or by photographing non-surface-active aluminum particles or dyes dispersed in the drop fluid [e.g. (G2, G3, J2, L5, Ml, SI)]. A photograph of a fully circulating falling drop is shown in Fig. 3.5a. Since the internal flow pattern for the Hadamard-Rybczynski analysis satisfies the complete Navier-Stokes equation... 

The formula of Hadamard and Rybczynski are also valid for the "moving bubble" problem with Ti Ti. Using the Hadamard-Rybczynski velocity field, it is easy to show that the difference between the tangential component of the velocity in the diffusion boundary layer and the surface velocity field is negligible. This is the reason why the reduction of equation (8.8) to variables 0, P leads to a coefficient on the right hand side which independent of T = xsin 0,... 

The main distinction of the theory of a dynamic adsorption layer formed under weak and strong retardation arises when formulating the convective diffusion equation. At weak retardation the Hadamard-Rybczynski hydrodynamic velocity field is used while at strong retardation the Stokes velocity field. Different formulas for the dependence of the diffusion layer thickness on Peclet numbers are obtained. The problem of convective diffusion in the neighbourhood of a spherical particle with an immobile surface at small Reynolds numbers and condition (8.74) is solved, so that the well-known expression for the density distribution of the diffusion flow along the surface can be used. As a result, Eq. (8.10) takes the form (Dukhin, 1982),... 

A theoretical analysis of the Stokes flow problem for a noimeutraUy buoyant droplet is clearly called for. Germane to this problem is the theoretical analysis of Haberman (H3), dealing with axially symmetric Stokes flow relative to a liquid droplet at the axis of a circular tube, and Taylor and Acrivos (T2c) extension of the classical Hadamard-Rybczynski liquid droplet problem to the case of nonzero Reynolds numbers. In particular, Haberman shows that the assumption of a spherical shape for the droplet in a tube is incompatible with the differential equations and boundary conditions. Taylor and Acrivos (T2c) point out that, though Hadamard (H3a) and Rybczynski (RIO) were able to solve the Stokes flow problem by assuming a spherical shape for a liquid droplet, irrespective of the magnitude of the interfacial tension, the correctness of their a priori assumption was, to some extent, fortuitous. These remarks are undoubtedly pertinent to the resolution of Haberman s paradox and, ultimately, to the solution of the nonaxially symmetric droplet problem. 

For creeping flow (0 < Re < 1), the solutions of the conduction-convection equation with flow held of the Hadamard-Rybczynski or Stokes are given by numerical integration [1]. The numerical results show that the concentration contours are not symmetrical (Figure 5.1 and Figure 5.2) and that the how inside and outside the sphere largely inhuences heat or mass transfer. In the case of a sphere with weak viscosity ratio, the heat or mass transfer is facilitated. 

For fluid spheres in creeping flow Re < 1) with k = 0, the conduction-convection equation with the Hadamard-Rybczynski flow field has been solved numerically by several authors. Clift et al. [1] have correlated the available numerical data in the form ... 

For a fluid sphere with Fe —> oo, the thin boundary layer approximation and the flow field of Hadamard-Rybczynski give the following equation [1] ... 

Redlich-Kwong equation, 181 Rybczynski-Hadamard formula, 318, 332, 348... 

There are three factors that would tend to cause the drag data of Fig. 8 to deviate from Stokes s law. The first is internal circulation however, based on the equation due to Hadamard (1911) and Rybczynski (1911), one would not expect motion within the droplet. The drag force determined by Hadamard and Rybczynski is... 

Hadamard and Rybczynski developed a terminal velocity equation for the creaming of bubbles with a mobile surface ... 

The equations of fluid motion inside and outside a circulating drop under viscous flow regime were solved by Hadamard (H2) and Rybczynski (R9) in 1911, and are quoted in hydrodynamics textbooks (L2). The complete derivation is also repeated by Levich (L8). Although Hadamard s stream functions are strictly applicable to the viscous region only, visual observations (GIO, S18) indicated that the function approximates actual flows (E2, H3). Hadamard s stream function inside the drop, as given in polar coordinates with the origin at the center of the drop (K5), is... 

With their lower viscosity, bubbles will deform more readily than emulsion droplets and, therefore, be relatively more prone to depart from Stokes law behaviour. Hadamard and Rybczynski developed a terminal velocity equation for the creaming of bubbles with a mobile surface ... 

In a more recent study, which is an extension of the previous works, Saboni et al. [15] proposed a predictive equation for drag coefficients covering Reynolds numbers in the range 0.01 < Re < 400 and viscosity ratio from 0 to 1000. This correlation, which is reduced to the solution of Hadamard [4] and Rybczynski [5] for Re 0, is as follows ... 

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