Hydrodynamic analogy

The causal interpretation of quantum theory as proposed by De Broglie and Bohm is an extension of the hydrodynamic model originally proposed by Madelung and further developed by Takabayasi [36]. In Madelung s original proposal R2 was interpreted as the density p(x) of a continuous fluid with stream velocity v= VS/rri. Equation (5) then expresses conservation of fluid, while (6) determines changes of the velocity potential S in terms of the classical potential V, and the quantum potential 

The quantum potential therefore arises in the effects of an internal stress in the fluid and depends on derivatives of the fluid density rather than on external factors. 

This model is not adequate in itself as it contains nothing to describe the actual location x(t) of the particle which is required for a causal interpretation of quantum theory. It is therefore necessary to postulate a particle that takes the form of a highly localized inhomogeneity that moves with the local fluid velocity v(x,t). The inhomogeneity could be of density close to that of the fluid, which is simply being carried along with the local velocity of the fluid. As in any macroscopic fluid random fluctuations are assumed [37] to occur in the Madelung fluid. It is shown that such fluctuations may lead to the statistical result, P = 2. 

The one-particle model is readily extended to the case of many particles. The wave function ip(xi,X2,. , Xpf,t) is defined in 3iV-dimensional configuration space. On writing ip — R exp(iS/h) a set of 3N velocity fields, 

---

The hydrodynamical analogy now follows by comparing Eq. (B.6) to the conservation law for a classical fluid... 

The significance of this theorem is obvious on the basis of the above-mentioned hydrodynamical analogy. In fact, condition (a) requires that there must be a sink (source) in D and (b) states that this sink (source) cannot be a point hence, it must be a curvilinear one, i.e., a stable (unstable) cycle. 

The second law as it left the hands of Carnot required no explanation. On the caloric theory then prevalent, it was a necessary consequence of a hydrodynamical analogy—the mechanical explanation was in fact, as Carnot s words show, the source of the principle. When the caloric theory was thrown down, the analogy and explanation fell with it, and the reconstruction of Carnot s principle by Clausius and Kelvin resulted in a law of experience. 

Explosive Bonding, Mechanism of. An aero-hydrodynamic analogy is used to construct a description of the formation of wavy interfaces during expl bonding, which is described by J.T. Kowalick D.R. Hay in Metallurgical Transactions, Vol 2, July 1971, pp 1953-58... 

Here v is the matrix of the Horiuti (stoichiometric) numbers and v and w the vector-columns of the rates along basic routes and of the step rates, respectively. Thus the rate of every step is represented as a linear combination of the rates along the basic routes. Here it is recommended that a simple hydrodynamic analogy be used. The total liquid flow along the tube (step) is the reaction rate. This flow consists of individual streams which are the rates along the routes. 

Madelung made the important observation that, irrespective of the overall validity of the hydrodynamic analogy, all linear combinations of the stationary solutions of the amplitude equation... 

However, there exists a way to employ the rigorous equations of continuum mechanics even for the cases, in which real phase boundaries cannot be exactly localized. This way is associated with the idea of hydrodynamic analogy between complex and simpler flow phenomena. More precisely, some particular similarities are meant between complex flow patterns encountered in industrial separations and geometrically simpler flows like planar films, cylindrical jets, spherical drops, etc., as well as their combinations (Kenig, 1997). These similarities are used in the hydrodynamic analogy approach by which the complex hydrodynamics established in a real column is replaced with an appropriate combination of simpler flow patterns. Such a replacement occurs on the basis of experimental observations which are very important for the successful... 

To build up a hydrodynamic analogy for columns equipped with SP, substantial features of the fluid flow have to be revealed and captured. Let us consider an example of a gas/vapor-liquid (reactive) separation unit filled with a non-catalytic structured packing (Shilkin and Kenig, 2005a). 

This paper gives a comprehensive review of the up-to-date modeling of reactive separation processes in columns equipped with structured packings and consider in detail two different modeling ways. The first approach is based on the application of CFD, whereas the second one employs the idea of hydrodynamic analogy between complex and simple flow patterns. 

Another innovative way to modeling of columns equipped with corrugated sheet structured packings is based on the hydrodynamic analogy between complex process hydrodynamics present in real industrial separations and geometrically simpler flow patterns. 

Shilkin, A., Kenig, E.Y., Olujic, Z. A hydrodynamic-analogy-based model for efficiency of structured packing columns. AIChE Journal, Vol. 52,... 

KEYWORDS reactive separations, structured packings, mass transfer, CFD, hydrodynamic analogies... 

Figure 4. A hydrodynamic analogy to transport through bulk solution and a membrane. Left, without solubilization right, with solubilization. Lower part shows distribution of pressure drop which is increased across membrane on the right by reduced resistance in rest of the system...

In the middle of cells and in faces that are perpendicular to the flowing direction, the borders are branched, which means that the effective number of borders, equivalent to that in a real system, is different from five. The number of independent borders with constant by height radius and length L can be determined by the electro-hydrodynamic analogy between current intensity and liquid flow rate through borders, both being directly proportional to the cross-sectional areas [6,35]. This analogy indicates that the proportionality coefficients (structural coefficients B = 3) in the dependences border hydroconductivity vs. foam expansion ratio and foam electrical conductivity vs. foam expansion ratio, are identical [10]. From the electrical conductivity data about foam expansion ratio it follows... 

Here we are drawing a hydrodynamic analogy between multi-tube reptation and polymer branching. deGennes s work indicates that a branched chain frictional coefficient should increase exponentially with the size or number of branches. 

Fig. 3. These diagrams (A, B) are intended to illustrate the regulation and operation of these mitochondrial fatty acid synthetic systems using a hydrodynamic analogy. Electron pressure and flow are depicted as fluid pressure and flow under gravitational influence. Electron flow down the electron transport chain is ultimately controlled by (ADP Pi) ATP ratio. When the latter ratio is high (A, left) electron flow rate is maximal and the steady-state NADHiNAD" ratio is low (State 3 Chance and Williams, 1956), On the other hand, (B, right) when either the (ADP + P,) ATP ratio is low or oxygen is lacking, substrate reduces NAD+ faster than it can be oxidized. The elevated NADHiNAD ratio reverses the usual flow of electrons from fatty acid oxidation. Acetate now becomes incorporated into fatty acids with the consequent oxidation of NADH and, therefore, perhaps permits some ATP to be synthesized via other substrate-level energy conserving steps.

An overview of this field would visualize a hierarchy of controls (ADP + Pj) ATP ratio controls the flow of electrons down the cytochrome chain or respiratory assembly. Using hydrodynamic analogies, the (ADP -1- P,) ATP ratio would control the major locks in the... 

---