Hadamard-Rybczynski velocity field

Section II shows that the dimensionless external velocity field uJU, UqIU) is a function of dimensionless position r/a, 0) and k for creeping flow. The dimensionless concentration defined in Eq. (1-45) is a function of these quantities and of the Peclet number, Pe = 2aU/. Hence the Sherwood number, Sh = is a function of k and Pe (with additional dependence on Re unless the creeping flow approximation is valid). The exact solution of Eqs. (3-39) to (3-42) with the Hadamard-Rybczynski velocity field is not available for all values of Pe and K, but several special cases have been treated. 

For fluid spheres with k = 0, Eq. (3-39) has been solved numerically with the Hadamard-Rybczynski velocity field (Ol), and the resulting variation of Sh with Pe is shown in Fig. 3.10. The values are approximated within 6% for all Pe by... 

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,... 

This result is known as the Hadamard-Rybczynski solution. It describes the velocity field for a spherical drop that is translating through an otherwise motionless fluid. A plot of the streamlines is shown in Fig. 7-15 for several different ratios of the internal to external viscosity. Obviously, for A. = 10, the interior fluid is moving slowly, and this is reflected in the small number of streamlines inside the drop. In addition, the exterior fluid is required to come almost to a stop on the drop surface, and this results in rather large velocity gradients (the streamlines are close together). As k is decreased, on the other hand, the interior fluid moves more freely, and the velocity gradients in the exterior fluid are reduced. 

Mass transfer inside a drop is described by Eq. (4.12.1) and the first two conditions (4.12.2). The fluid velocity field v = (vr,vg) at low Reynolds numbers is given by the Hadamard-Rybczynski stream function and, in the dimensionless variables, has the form... 

Let us consider a transient solute concentration field in a liquid outside and inside a spherical drop of radius a moving at a constant velocity U in an infinite fluid medium. We assume that the fluid velocity fields for the continuous and disperse phases are determined by the Hadamard-Rybczynski solution [177, 420], obtained for low Reynolds numbers. The concentration far from the drop is maintained constant and equal to C,. At the initial time f = 0, the concentration outside the drop is everywhere uniform and is equal to C inside the drop, it is also uniform, but is equal to Co-... 

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),... 

---

See also in sourсe #XX -- [ ]

---