Diffusion Boundary Layer Near the Surface of a Particle

Diffusion Boundary Layer Near the Surface of a Particle 

Following [270], we first consider steady-state diffusion to the surface of a solid spherical particle in a translational Stokes flow (Re - 0) at high Peclet numbers. In the dimensionless variables, the mathematical statement of the corresponding problem for the concentration distribution is given by Eq. (4.4.3) with the boundary conditions (4.4.4) and (4.4.5), where the stream function is determined by (4.4.2). 

As Pe increases, a diffusion boundary layer is formed near the surface of the sphere. The ratio of the thickness of this layer to the radius of the particle is of the order of Pe 1//3. In this region, the radial component of molecular diffusion to the surface of the particle is essential, and tangential diffusion may be neglected. Convective mass transfer due to the motion of the fluid must also be taken into account. 

Assuming that e = Pe 1//3 is a small parameter, we introduce a stretched coordinate y in the diffusion boundary layer region according to the formula r = 1 + ey. By substituting this coordinate into (4.2.2) and (4.4.3) and by retaining the leading terms of the expansion with respect to e, we obtain 

Further, by passing from 8, y to the von Mises variables 8, ip, we obtain The transformation 

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