Diffusion wake

Marino, J. Cudeiro, J. (2003). Nitric oxide-mediated cortical activation a diffuse wake-up system. J. Neurosci. 23, 4299-307. 

Figure 4.5. Flow pattern inside the drop and the concentration field structure here d and di are regions of the diffusion boundary layer, W and Wi are regions of the diffusion wake, ei and ej are regions of the stream core (when the resistance of the disperse phase is limiting we consider only regions inside the drop)...

Figure 4.6 shows the results of calculation of the mean Sherwood number obtained in [151] by a numerical solution of the corresponding integral equation for various values of dimensionless time and the Peclet number. One can see that, after the inner diffusion wake has been developed, the complete substance flux to the inner surface of the drop decreases rapidly. 

At the intermediate stage of the process, the diffusion wake interacts with the boundary layer and strongly erodes it, producing an increase in the boundary layer thickness (here the boundary layers for the inner and outer problems differ considerably). Gradually, as a result of absorption of the substance dissolved in the liquid on the interface, the diffusion boundary layer spreads over the entire drop and begins to decay. 

Diffusion Wake. Mass Exchange of Liquid With Particles or Drops Arranged in Lines... 

Four subregions Ww (i = 1, 2, 3, 4) lying behind a drop or a solid particle near the flow axis constitute the diffusion wake region (Figure 4.7). 

In the convective-boundary layer region W(1) of the diffusion wake, molecular diffusion can be neglected. The concentration here depends only on the... 

It is worth mentioning that in all diffusion wake regions Ww ( = 1, 2, 3, 4), one must take into account convective mass transfer due to the motion of the fluid. In the regions (i - 2, 3, 4), the molecular diffusion component normal to the streamlines plays an important role. 

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]. 

The order of magnitude of the characteristic dimensions of diffusion wake regions past a spherical drop and a solid particle in a translational flow is shown in Table 4.9. These estimates remain valid at moderate Reynolds numbers, when the stagnation zones past a drop or a solid particle are absent. 

It was shown in [ 166] that in the two-dimensional problem of mass exchange between cylindrical bodies and a viscous flow, the diffusion layer consists of only two subregions W(3) and W(4) of total length L a Pe-1//9 (as Pe -> oo) the regions W(l) and W T> are absent. The diffusion wake in the vicinity of the stagnation lines on the surface of a solid particle has a similar structure. 

Order of magnitude of dimensionless (related to the radius of drops or solid particles) characteristic sizes of regions of diffusion wake in translational Stokes flow at high Peclet number... 

Diffusion wake regions Dimensionless distance from the interface, y = r—l Dimensionless distance from the flow axis, h... 

Mass exchange of the second solid particle is more complicated. The main role is played by the interaction of the diffusion boundary layer of the second particle with the diffusion wake of the first particle. 

The diffusion boundary layer of any given drop in chain interacts with the diffusion wake of the previous drop (located upstream). The concentration field in it is substantially nonuniform and is depleted because of the absorption of the solute at the surfaces of all preceding drops. By virtue of such interaction, the inner mass exchange will be appreciably retarded (the shielding phenomenon) compared with the case of isolated drops. 

In Section 2.9, various aspects were considered of the hydrodynamics of a constrained flow past a system of particles based on the cell model. Here we briefly describe mass and heat transfer in such systems at high Peclet numbers. We investigate either sufficiently rarefied systems of particles or systems with an irregular structure, where the diffusion interaction of isolated particles can be neglected. (Regular disperse systems, where the interaction between diffusion wakes and boundary layers must be taken into account, were investigated in [172, 365].)... 

For large rate constants kv of the volume chemical reaction, a thin diffusion boundary layer is produced near the drop surface its thickness is of the order of ky1//2 at low and moderate Peclet numbers, and the solute in this layer has time to react completely. As the Peclet number is increased further, because of the intensive liquid circulation within the drop, there is not enough time to complete the reaction in the boundary layer. The nonreacted solute begins to get out of the boundary layer and penetrate into the depth of the drop along the streamlines near the flow axis. If the circulation within the drop is well developed, a complete diffusion wake is produced with essentially nonuniform concentration distribution that pierces the entire drop and joins the endpoint and the origin of the diffusion boundary layer. In case of a first-order volume chemical reaction, an appropriate analysis of convective mass transfer within the drop for Pe > 1 and kv > 1 was carried out in [150,151]. It should be said that in this case, in view of the estimate (5.4.8), which is uniform with respect to the Peclet number, the mass transfer intensity within the drop is bounded by the rate of volume chemical reaction. 

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