Systems with Regular Hydrodynamic Dissipative Structures

Models of Systems with Regular Hydrodynamic Dissipative Structures 

However, it is to be noted, that the results of the electrochemical experiments are not sufficient to explain the appearance of Benard cells in the system. In fact, these results are sometimes misinterpreted. For example, in Ref. [51] the monotonic development of the current-versus-time curves, similar to that of the curves a and b in Fig. 5, was explained as a fact indicative of the absence of any circulational cell. However, optical measurements show that the circulational cell structure of the flux takes place even for a low current density (which possesses certain features connected with the temporal evolution of the velocity field and determining the monotonic 

The flux between two lateral walls caused by the nonuniformity of the ion concentration profiles in the layers adjacent to the electrodes is of the same nature as the heat convection arising while the bottom wall is heated [2]. In the latter case a disturbance of the steady state occurs if the Rayleigh number reachs a certain (critical) value (Ra = gPd AT/vx, where P is the coefficient of bulk heat expansion, d is the distance between the walls, AT the increment of temperature, v the dynamic viscosity, and X the thermal diffusivity) the liquid transforms into a new state with a periodic cell structure in such a way that the circulation in the interior of each cell has an opposite direction compared to that of the adjacent one. According to previous evaluations [53] the critical Rayleigh number in the case of lateral rigid walls is about 1700. 

In view of the convection induced by the ion mass transfer, the force responsible for the momentum transfer is proportional to the product of the Grashoff and Schmidt numbers  

In this diffusion analogy of the Rayleigh number we have p = Q dQ/dc, and the values Cg and Cq which are the ion concentrations participating in the interfacial mass exchange at the anode surface and in the interior of the solution, respectively. 

---