Stokes Number for Inviscid Flows

By analyzing the motion of a small panicle in the region near the stagnation point, it can be shown that for an inviscid How, theory predicts that impaction does not occur until a critical Stokes number is reached. For an inviscid flow, the first term in an expansion of the velocity along the streamline in the plane of symmetry which leads to the stagnation point is 

For Stk 1 /46. the roots are real and both are negative. The u versus x diagram has a nodal point that corresponds to zero particle velocity at the forward stagnation point and zero impaction efficiency (Fig, 4,6b). For Stk = I/4h, the roots are equal and the system 

This analysis provides a lower anchor point for curves of impaction efficiency as a function of Stokes number. It applies also to non-Stokesian particles, discussed in the next section, because the point of vanishing efficiency corresponds to zero relative velocity between particle and gas. Hence Stokes law can be used to approximate the particle motion near the stagnation point. This is one of the few impaction problems for which an analytical solution is possible. 

The analysis neglects boundary layer effects and is probably best applied when the particle diameter is larger than, or of the order of, the boundary layer thickness. The change in the drag law as the particle approaches the surface is also not taken into account. Hence the criterion provides only a rough estimate of the range in which the impaction efficiency becomes small. 

For most real (viscous) flow.s, U f (ai + 1) in the region near the stagnation point because of the no-slip boundary condition and the continuity relation. In this case, (4.29) does not apply, the equation of particle motion cannot be put into the form of (4,30), and the analysis developed above is not valid. Instead, numerical calculations for the viscous flow regime ( / (x i + 1) ) indicate that the collection efficiency is finite for all nonzero values of Stk, vanishing for Stk - 0 (Ingham et al., 1990). 

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