Hydrodynamic methods current distribution

In the quasi-steady-state approximation, which is also known as the step method [9], it is assumed that the rate of variation in the WP shape, that is, the anodic dissolution rate, is small compared with the rates of transfer processes in the gap therefore, for calculating the distribution of the current density, the WP surface can be considered as being immobile. This approximation can be used at not very high current densities. At very high current densities, ignoring the WP surface motion during anodic dissolution and the hydrodynamic flow induced by this motion causes a considerable error in the calculated distribution of current density [33]. 

There are a number of different formulations of the time correlation function method, all of which lead to the same results for the linearized hydrodynamic equations. One way is to generalize the Chapman-Enskog normal solution method so as to apply it to the Liouville equations, and obtain the N-particle distribution function for a system near a local equilibrium state. " Expressions for the heat current and pressure tensor for a general fluid system can be obtained, which have the form of the macroscopic linear laws, with explicit expressions for the various transport coefficients. These expressions for the transport coefficients have the form of time integrals of equilibrium correlation functions of microscopic currents, viz., a transport coefficient t is given by... 

Currently, analytical approaches are still the most preferred tools for model reduction in microfluidic research community. While it is impossible to enumerate all of them in this chapter, we will discuss one particular technique - the Method of Moments, which has been systematically investigated for species dispersion modeling [9, 10]. The Method of Moments was originally proposed to study Taylor dispersion in a circular tube under hydrodynamic flow. Later it was successfully applied to investigate the analyte band dispersion in microfluidic chips (in particular electrophoresis chip). Essentially, the Method of Moments is employed to reduce the transient convection-diffusion equation that contains non-uniform transverse species velocity into a system of simple PDEs governing the spatial moments of the species concentration. Such moments are capable of describing typical characteristics of the species band (such as transverse mass distribution, skew, and variance). 

Currently, the only practical way of accurately determining an MWD is via a separation technique. The most widely used methods include SEC, field flow fractionation (FFF), capillary hydrodynamic fractionation, and gel electrophoresis. Since knowledge of C M) furnishes the most complete description of a polymer distribution, there is intensive development and application of these techniques currently in progress. SEC is covered in great detail in Chapter 9. 

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