Reflexive separation

Theorem 1.14. Let V be a reflexive separable Banach space. Assume that an operator A E —> E possesses the following properties ... 

Fig. 3.2. Typical droplet-droplet collision outcome map with four regimes [88] Bouncing (Bo), Coalescence (Co), Stretching Separation (Ss), and Reflexive Separation (Rs). X denotes the impact factor, and TVe is the Weber number.

Abstract We put together the state of knowledge on binary colUsional interactions of droplets in a gaseous environment. Phenomena observed experimentally after drop collisions, such as coalescence, bouncing, reflexive separation and stretching separation, are discussed. Collisions of drops of the same liquid and of different -miscible or immiscible - liquids, as well as collisions of drops of equal and different size are addressed. Collisions of drops of immiscible liquids may lead to an unstable interaction which is not observed with drops of equal or miscible liquids. Regimes characterized by the various phenomena are depicted in nomograms of the Weber number and the non-dimensional impact parameter. The state-of-the-art in the simulation of binary droplet collisions is reviewed. Overall three different methods are represented in the literature on these simulations. We discuss models derived from numerical simulations and from experiments, which are presently in use for simulations of spray flows to account for the influence of coUisional interactions of the spray droplets on the drop size spectrum of the spray. 

Keywords Binary drop colUsions Bouncing Coalescence Collision model Crossing separation Gaseous environment Immiscible liquids Lattice-Boltzmann simulation Miscible liquids Navier-Stokes simulation Reflexive separation Satellite droplets Spray flow simulation SPH simulation Stretching separation... 

The authors postulate that this energy must be greater than 75% of the nominal surface energy of a nominal spherical combined mass formed after the collision for reflexive separation to occur. The equality in (7.1) represents the case where the 75% threshold is just reached. In an analogous way, Ashgriz and Poo [36]... 

Fig. 7.6 Collisions of a water and an ethanol drop (a) head-on collision with coalescence and separation of one satellite (We =20,X = 0) (b) reflexive separation with formation of a small satellite due to Marangoni forces We = 38.5, X = 0.02) (c) stretching separation with formation of three satellite droplets (We = 82.3, X = 0.82). Droplets move from right to left the water droplet coming from above is marked with w [45] (With kind permission from Springer Science+Busi-ness Media Experiments in Fluids [45], Plates 3, 5 6, Copyright Springer-Verlag 2005)...

One point should be made about the identification of a reflexive separation case in paper [49]. Reflexive separation is defined as an unstable post-collision mechanism that separates the droplets collided at near-head-on impact parameters. In this mechanism, the bulk masses of the colliding droplets remain on the sides of the symmetry plane from where they had approached (therefore reflexive separation). In the case shown in Fig. 7.9, however, which is identified as single reflex separation by Chen and Chen [49], the dyed drop changes its side from above (before collision) to below the symmetry plane (after collision), and the transparent drop moves vice versa. The actual mechanism therefore implies a mutual penetration of the liquid portions in the collided complex, which is not reflexive separation. This mechanism was called crossing separation by Planchette et al. [26] and by Planchette and Brenn [50], since the two liquid portions cross the trajectories of their respective collision partners. 

Fig. 7.9 Coiiision mechanism termed single reflex separation in Chen and Chen (2006) at We = 95.3, X = 0. Note that the colmed liquid moves fiom above (before collision) to below the symmetry plane (after collision) [49]. The mechanism is, therefore, not reflexive (With kind permission fiom Springer Science-t-Business Media Experiments in Fluids [49] Fig. 8, Copyright Springer-Verlag 2006)...

Modeling of the complicated phenomena in binary droplet collisions occurring in spray flows is difficult due to the variety of potential outcomes from a collision [69-71]. The first necessity is to predict the stability against stretching or reflexive separation. Then, for unstable drop collisions, the resulting drop sizes need to be predicted. All predictions should follow from algebraic models without the need to solve additional transport equations in the spray flow code to account for the collisions. Needless to say that it is impossible to simulate the full detail of the processes in droplet collisions, as done in the simulations discussed in section Simulations of Droplet Collisions, in the course of a spray flow simulation [72-82]. 

Bouncing, coalescence as well as stretching and reflexive separation were observed for the lower viscosity range. Figure 6.5 shows exemplary the collision maps of K30 solutions, and Fig. 6.6 shows those of sucrose solutions. As the viscosity... 

Fig. 6.6 Collision maps of sucrose solutions with different solids mass fractions (a) 20%, (b) 40 % (c) 50 %, (d) 54 %, (e) 58 %, and (f) 60 % with symbols indicating the collision outcomes red squares coalescence, green triangles separation, and blue diamondshovtncing. Existing models are compared with the experimental data points solid orange lines for the models of Ashgriz and Poo [11] for the boundaries between coalescence and stretching as well as reflexive separation (the boundary line for reflexive separation is omitted when this outcome is not observed experimentally), brown dashed line model of Jiang et al. [13] using C = 2.026 and C2 = 0.556, and blue dashed dotted line adapted model of Estrade et al. [12] (Kuschel and Sommerfeld [9])...

The reflexive separation regime, occurring at near head-on collisions, is shifted to the right in the direction of higher We-numbers, and eventually completely disappears at 20 % soUds mass fraction in accordance with the observation of Jiang et al. [13] and Gotaas et al. [8]. 

The collision maps of sucrose solutions are very similar to the ones of K30, except that stretching separation mostly occurs at higher impact parameters for all mass fractions investigated, and reflexive separation disappears at a lower viscosity already. These differences might arise from the non-Newtonian characteristics of the sucrose solutions. 

The model of Ashgriz and Poo [11] for the boundary line between coalescence and reflexive separation may be however used if the critical We-number of beginning reflexive separation at B = 0 is introduced. It was found that the value of the critical We-number very well correlates with the Capillary-number Ca for all liquids considered, if the parameter K introduced by Naue [19] was applied. Hence, the following new model is proposed [10] ... 

Fig. 6.7 Correlation for the critical We-number of beginning reflexive separation [10]...

In the case of reflexive separation which is only observed for lower viscosities at most the formation of one satellite droplet was observed. However, also events where the ligament formed between the colliding droplets completely retracts again are observed consequently, no satellites are resulting. A higher number of satellites... 

It is found that the existing model for the boundary between coalescence and reflexive separation provided by Ashgriz and Poo [11] works only for liquids of low viscosity. A model for the beginning of the reflexive separation in head-on collision based on Ashgriz and Poo [11] was provided, and it shows excellent agreement with all existing experimental data. 

Hence, only coalescence and stretching separation was discovered, whereas bouncing and reflexive separation were absent. 

Pan and Suga [171], for example, performed 3D dynamic simulations of binary droplets collisions for cases of different Weber numbers and impact parameters. The systems used are water drops in air and tetradecane drops in nitrogen at atmospheric conditions. The bulk fluids are considered incompressible. The simulations cover the four major regimes of binary collision bouncing, coalescence, reflexive separation. 

Figure 12 Schematic illustration of the outcome of droplet collisions (A) bouncing (B) coalescence (C) stretching separation and (D) reflexive separation. Time is running from top to bottom in each schematic illustration. The center plot shows the lines demarcating the different regimes as a function of Weber number (based on the relative velocity between two droplets) and impact parameter for equal sized droplets. For unequal sized droplets, the lines are slightly shifted (see Ko and Ryou, 2005).

Figure 18 Contour plots of the frequencies (number of collisions per second per cm ) of different collision events on a cutting plane at the center of the domain in a spray for 250 bar inlet pressure. A bouncing, B coalescense, C stretching separation, D reflexive separation. Note the logarithmic color scale.

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