Similarity solutions stagnation point

Vertical CVD Reactors. Models of vertical reactors fall into two broad groups. In the first group, the flow field is assumed to be described by the one-dimensional similarity solution to one of the classical axisymmetric flows rotating-disk flow, impinging-jet flow, or stagnation point flow (222). A detailed chemical mechanism is included in the model. In the second category, the finite dimension of the susceptor and the presence of the reactor walls are included in a detailed treatment of axisymmetric flow phenomena, including inertia- and buoyancy-driven recirculations, whereas the chemical mechanism is simplified to a few surface and gas-phase reactions. 

Therefore, near the stagnation point, u is proportional to x so that the similarity solution with m equal to 1 will apply in this region. 

As another example of a situation in which a similarity-type solution can be obtained, consider flow in the region of a stagnation point of an isothermal body as shown in Fig. 10.13. 

Now, Eq. (9-260) is identical to equation (9 234), which was found earlier for the sphere, and we have already seen that it can be solved subject to the conditions (9 261). The solution for 9 is given in (9 240). The existence of a similarity solution to (9 257) thus rests with Eq. (9 259). Specifically, for a similarity solution to exist, it must be possible to obtain a solution of (9-259) for g(q2), which remains finite for all q2 except possibly at a stagnation point where a = 0, from which a thermal wake may emanate. 

With this definition, g(q2) remains finite for all q2 (other than the rear-stagnation point), and we claim to have constructed a similarity solution for the complete class of smooth 2D solid bodies (with no closed-streamlines). 

The only changes required in these solutions are due to the fact that a(q2) may be more complex than for uniform streaming flows. For example, a qualitative sketch of the flow structure for a nonrotating cylinder in simple shear flow at low Reynolds number is shown in Fig. 9-16.23 It is evident in this case that there are fom stagnation points on the cylinder smface rather than two, as in the streaming-flow problem. Two of the streamlines that lead to the stagnation points A and C are lines of inflow, and two from B and D are lines of outflow, where we should expect a thin thermal wake. In the limit as these outflow points are approached, we thus expect a breakdown of the similarity solution with g -> 00. At the inflow stagnation points, on the other hand, we require that g be finite. To accommodate... 

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