Incompressible Newtonian Fluids in the Creeping Flow Regime

12-2 INCOMPRESSIBLE NEWTONIAN FLUIDS IN THE CREEPING FLOW REGIME 

In the limit of vanishingly small Reynolds numbers, forces due to convective momentum flux are negligible relative to viscous, pressure, and gravity forces. Equation (12-4) is simplified considerably by neglecting the left-hand side in the creeping flow regime. For fluids with constant /r and p, the dimensionless constitutive relation between viscons stress and symmetric linear combinations of velocity gradients is 

Viscous forces in the equation of motion are treated as follows  

Without introducing the stream function or solving this third-order partial differential vector equation for v, it shonld be obvious that the dimensionless velocity vector is a fimction of spatial coordinates and the specific geometry of the flow problem. However, v is not a fimction of the Reynolds number because Re does not appear in (12-9). The generic creeping flow solution is written as 

The dimensionless mass transfer correlation, given by (12-1), reveals the complete dependence of the surface-averaged Sherwood number on the Reynolds and Schmidt nnmbers  

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