Diffusion boundary layer unstable

Figure 3.6. Schlieren photographs showing the changes in thickness of the diffusion boundary layer and the behavior of buoyancy-driven convection shown in relation to bulk supersaturation [1], [2]. The figure shows the (111) faceofaBa(N03)2 crystal from an aqueous solution. In region I, only the thickness of the diffusion boundary layer increases in region II, we see unstable lateral convection (HA) and intermittently rising plumes (IIB) and in region III we see steady buoyancy-driven convection.

As we saw in Chapter 4, the left-hand and right-hand branches of the curve in Figure 6.4 are stable, while the middle one is unstable. We define two functions, h+(v) and h (v), that express, respectively, the functional dependence of m on n on the left and right branches of the curve. If we now imagine a one- or two-dimensional medium in which the chemistry and the diffusion are described by eqs. (6.28), we will find, after any transients have decayed, that either (1) the whole system is in one of the stable states described by h (v) and h (v), or (2) part of the system is in one of these states and part is in the other state. In case (2), the area that separates regions in the positive and negative states is called a boundary layer. For small e, this layer is extremely narrow—in the limit as e 0, it is a point or a... 

A number of other models for unstable conditions have been used, notably the formulations of O Brien (1970) and of Myrup and Ranzieri (1976). O Brien s model defines a cubic polynomial variation of above the surface layer. Boundary conditions are established by matching a similarity solution at the top of the surface layer and fixing profile gradients at z = L and Z. The expression for the diffusivity is... 

If local equilibrium is assumed, then the reaction kinetics can be calculated from the flux equation for A ions in the product layer and from the solution of Pick s second law in the alloy. The continuity condition at the phase boundary must also be observed. The calculations are analogous to those in sections 7.2.1 and 8.1.4. However, one additional important point must be considered. If diffusion in the alloy is the rate-controlling step in the overall process, then a slight disturbance in the planar metal/oxide phase boundary will be unstable. That is, if the phase boundary bows into the metal phase at some spot, then the growth rate of AO is increased at this point, and the disturbance increases in magnitude. The result is a fissured phase boundary. The morphology of this phase boundary depends upon a variety of factors such as the ratio of molar volumes pAo/ aiioy plastic behaviour of the oxide and the metal, and the adherency between oxide and alloy. Examples of reactions in which strongly fissured phase boundaries form are the reactions of Ag-Au and Cu-Au alloys with sulphur at 400 °C [31 ]. 

Non-monotonic density profiles are unstable. Since, however, the influence of the wall decays exponentially with the distance, the dynamics is practically frozen whenever the interphase boundary is separated from the wall by a layer thick compared to the characteristic width of the diffuse interface. A static solution with a fixed h exists only at a certain fixed value of i, which can be determined using a solvability condition of the first-order equation as in Section 1.3. In a wider context, an appropriate solvability condition serves to obtain an evolution equation for the nominal position h of the interphase boundary. The technique of derivation of solvability conditions for a problem involving a semi-infinite region and exponentially decaying interactions is non-standard and therefore deserves some attention. 

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