Region rear stagnation

In the rear stagnation region W(3), the mass transfer equation can be somewhat simplified. In doing so, one must take into account convective terms as well as the radial and tangential molecular diffusion components. 

In the case of a drop (or a bubble), the concentration distribution was obtained in a closed form in all diffusion wake regions Wh) [166, 167], and in the case of solid sphere, closed-form analytical expressions were obtained in all regions except for the rear stagnation region [166,446], The concentration field in W(3) for the cases of a solid sphere and a circular cylinder was analyzed by numerical methods in [309]. 

In region III near the tube center, viscous stresses scale by the tube radius and for small capillary numbers do not significantly distort the bubble shape from a spherical segment. Thus, even though surfactant collects near the front stagnation point (and depletes near the rear stagnation point), the bubble ends are treated as spherical caps at the equilibrium tension, aQ. Region... 

Interestingly, the shape of the wake is similar to that developed behind a hypersonic blunt body where the flow converges to form a narrow recompression neck region several body diameters downstream of the rear stagnation point due to strong lateral pressure gradients. The liquid material, that is continuously stripped off from the droplet surface, is accelerated almost instantaneously to the particle velocity behind the wave front and follows the streamline pattern of the wake, suggesting that the droplet is reduced to a fine micromist. 

As indicated by Fig. 23 and Fig. 24, the source function can be highly asymmetrical. For the liquid droplet corresponding to Fig. 23, one would expect the internal temperature to be higher near the back and front of the sphere because of the spikes in the source function in those regions. As a result, the evaporation rate should be enhanced at the rear stagnation point and the front of the sphere. To calculate the evaporation rate when internal heating occurs, one must solve the full problem of conduction within the sphere coupled with convective heat and mass transport in the surrounding gas. 

Using these equations, the numerical procedure outlined above can be used to find the variation of the local heat transfer rate in the form of Nup/Pe 2 around the cylinder. The solution procedure does not really apply near the rear stagnation point (R in Fig. 10.17) because the effective boundary thickness becomes very large in this area as indicated in Fig. 10.17. This is however, of little practical importance because the heat transfer rate is very low in the region of the rear stagnation point. 

Marangonl Convection, Figure 2 Schematic of a stationary drop subjected to the fiow of a surfactant-iaden externai fiuid. The surfactant molecules which adsorb on the interface are transported by the flow on this surface towards the rear stagnation point. This leads to a strong concentration of surfactant molecules in the rear region... 

Figure 5.15 shows streamlines and concentration contours calculated by Masliyah and Epstein (M6). Even in creeping flow, Fig. 5.15a, the concentration contours are not symmetrical. The concentration gradient at the surface, and thus Shjoc is largest at the front stagnation point and decreases with polar angle see also Fig. 3.11. The diffusing species is convected downstream forming a region of high concentration at the rear (often referred to as a concentration wake ) which becomes narrower at higher Peclet number.

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