G The Boundary-Layer on a Spherical Bubble

By now the reader may have come to associate the momentum boundary layer as a thin region of ()(Re l/2) adjacent to a body surface across which the dimensionless tangential velocity changes by 0(1) in order to satisfy the no-slip condition at the body surface. To be sure, many boundary layers do exhibit this structure, but it is overly restrictive as a general description. A more accurate and generally applicable description is that the boundary layer is a thin region of ()(Re l/2) adjacent to a body surface inside of which the vorticity generated at the body surface is confined. 

In this regard, it is of interest to contrast the two problems of the streaming motion of a fluid at large Reynolds number past a solid sphere and a spherical bubble. In the case of a solid sphere, the potential-flow solution (10 155)—(10—156) does not satisfy the no-slip condition at the sphere surface, and the necessity for a boundary layer in which viscous forces are important is transparent. For the spherical bubble, on the other hand, the noslip condition is replaced with the condition of zero tangential stress, Tr = 0, and it may not be immediately obvious that a boundary layer is needed. However, in this case, the potential-flow solution does not satisfy the zero-tangential-stress condition (as we shall see shortly), and a boundary-layer in which viscous forces are important still must exist. We shall see that the detailed features of the boundary layer are different from those of a no-shp, sohd body. However, in both cases, the surface of the body acts as a source of vorticity, and this vorticity is confined at high Reynolds number to a thin 0(Re x/2) region near the surface. 

The source of vorticity at a solid, no-slip surface is the velocity gradient that is generated in satisfying the no-shp condition. This mechanism yields vorticity of 0(Rel/2) at the body surface. At an interface where the tangential velocity is not zero, on the other hand, vorticity is produced by rotation of fluid elements caused by the surface curvature. This latter mechanism generates vorticity of magnitude proportional to the local curvature of the surface in the direction of the motion of the fluid. As an example of vorticity production in the latter case, we may consider the condition of zero tangential stress at the surface of a bubble whose shape we assume, for simplicity, to be spherical. In this case, for an axisymmetric motion, 

The corresponding condition for motion past an axisymmetric bubble of arbitrary shape is 

Regardless of the source of the vorticity, it remains confined near the body surface for Re I because the time scale for radial diffusion is limited by convection around the body 

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