Boundary layer thickness creeping flow

When the fluid approaches the sphere from above, the fluid initially contacts the sphere at 0 = 0 (i.e., the stagnation point) because polar angle 6 is defined relative to the positive z axis. This is convenient because the mass transfer boundary layer thickness Sc is a function of 6, and 5c = 0 at 0 = 0. In the laminar and creeping flow regimes, the two-dimensional fluid dynamics problem is axisymmetric (i.e., about the z axis) with... 

Creeping flow of an incompressible Newtonian fluid around a solid sphere corresponds to g (0) = I sin0. For any flow regime that does not include turbulent transport mechanisms, the dimensionless boundary layer thickness is... 

Hence, an equivalent form of the previous scaling law, 5c A.mix ", is Sc l/(Re Sc) where m = for boundary layer theory adjacent to a solid-liquid interface in the creeping flow regime, and m = for gas-liquid interfaces. As expected, the boundary layer thickness at any position along the interface decreases at higher flow rates and increases when the diffusivity is larger. Since... 

J9A,mix in the expressions for 5c and Sc represents a diffusivity instead of a molecular transport property, one must replace a, mix by the thermal diffusivity 0 (= kidpCp, where p = density, Cp = specific heat, and kjc = thermal conductivity) to calculate the analogous heat transfer boundary layer thickness Sj and the Prandtl number [i.e., Pr = d/p)ja. In the creeping flow regime, where g 9) = I sine. 

Obtain analytical expressions for the liquid-phase mass transfer boundary layer thickness for (a) creeping flow, and (b) potential flow around a gas bubble of radius R. In which case will the boundary layer thickness be larger at the same relative position along the surface of the bubble ... 

Compare mass transfer boundary layer thicknesses for creeping flow of identical fluids around (a) a sohd sphere, and (b) a gas bubble at the same value of the Reynolds number. In which case is the boundary layer thickness greater ... 

In Chap. 9, we considered the solution of this equation in the limit Re 1, where the velocity distribution could be approximated by means of solutions of the creeping-flow equations. When Pe 1, we found that the fluid was heated (or cooled) significantly in only a very thin thermal boundary layer of 0(Pe l/3) in thickness, immediately adjacent to the surface of a no-slip body, or () Pe l/2) in thickness if the surface were a slip surface with finite interfacial velocities. We may recall that the governing convection di ffusion equation for mass transfer of a single solute in a solvent takes the same form as (111) except that 6 now stands for a dimensionless solute concentration, and the Peclet number is now the product of Reynolds number and Schmidt number,... 

Figure 11-1 Thickness of the mass transfer boundary layer around a solid sphere, primarily in the creeping flow regime. This graph in polar coordinates illustrates 8c 9) divided by the sphere diameter vs. polar angle 9, and the fluid approaches the solid sphere horizontally from the right. No data are plotted at the stagnation point, where 9=0.

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