Rear stagnation point

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. 

The presence of adverse pressure gradients induces separation towards the rear stagnation point (Figure Ic). Whereas its influence on the gas-phase motion may not be critical, its effect on the liquid-phase motion may be quite significant because of the closed streamlines. If this is the case, then entirely different formulations may be required to account for the effects from separation and the wake. Such a detailed analysis has not been performed for either the gas or the liquid flow. 

As discussed in the previous section, the convective droplet vaporization case has yet to be analyzed completely. The major difficulty lies with describing the fluid mechanical aspect of the phenomena, particularly for large Reynolds number flow when separation and reverse flow occur towards the rear stagnation point in both the gas and liquid phases. The difficulty is further compounded when the components in the droplet are not completely miscible, as is the case for emulsified fuels. The drop-... 

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). 

This solution is finite for all , except for the rear-stagnation point ati = l,providedci = 0. Thus,... 

Let us consider a solid spherical particle of radius o in a translational Stokes flow with velocity U and dynamic viscosity /i (Figure 2.1). We assume that the fluid has a dynamic viscosity /z. We use the spherical coordinate system. R, 9, ip with origin at the center of the particle and with angle 0 measured from the direction of the incoming flow (that is, from the rear stagnation point on the particle surface). In view of the axial symmetry, only two components of the fluid velocity, namely, Vr and Vg, are nonzero, and all the unknowns are independent of the third coordinate [Pg.58]

Since the boundary condition for the normal stresses must be satisfied at the front and rear stagnation points as well as along the boundary of the midsection of the bubble, we have the following relationship between the Weber number We and the ratio x of the major to the minor semiaxis of the ellipsoid [291] ... 

The condition for the stagnant cup formation, the surface concentration variation along the bubble surface, and the adsorption saturation near the rear stagnation point is determined in terms of Marangoni niunber. If the compressive viscous shear force exceeds the characteristic linear surface pressure force, i.e. at small Marangoni numbers, the adsorption layer is compressed considerably and no saturation near the rear stagnation point can result. 

An initial and important result is that the accumulation of surfactant molecules at the rear stagnation point of a translating droplet imposes a surface stress which opposes the outer flow (see Fig. 2). A direct implication is that for a gas bubble rising in a quiescent liquid, the surface is immobilized and the rise velocity is more similar to the rise of a rigid disc than to an inviscid bubble. 

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... 

Fig. 4.63 Stem diameter of a trapped bubble just before detachment from the rear stagnation point of a cylinder...

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