Hydrodynamic variables

We shall discuss here the macroscopic dynamics of liquid crystals that is an area of hydrodynamics or macroscopic properties related to elasticity and viscosity. With respect to the molecular dynamics, which deals, for example, with NMR, molecular diffusion or dipolar relaxation of molecules, the area of hydrodynamics is a long scale, both in space and time. The molecular dynamics deals with distances of about molecular size, a 10 A, i.e., with wavevectors about 10 cm , however, in the vicinity of phase transitions, due to critical behaviour, characteristic lengths of short-range correlations can be one or two orders of magnitude larger. Therefore, as a limit of the hydrodynamic approach we may safely take the range of wavevectors q 10 cm and corresponding frequencies (O c q 10 - 10 = 10 s (c is sound velocity). 

In the hydrodynamic limit one considers only those variables whose relaxation times decrease with increasing wavevector of the corresponding visco-elastic modes. For instance, a small vortex made by a spoon in a glass of tea relaxes faster than a whirl in a river, or, after a tempest, short waves at the sea surface disappear faster then waves with a large period. The relaxation of cyclones in atmosphere takes days or weeks. As a rule, the hydrodynamic relaxation times follow the law x Aq. The strings of a guitar also obey the same law. 

For the isotropic liquid, one introduces five variables related to the corresponding conservation laws. The variables are density of mass p, three components of the vector of linear momentum density m, and density of energy E. When electric charges enter the problem, the conservation of charge must be taken into account. Then we are in the realm of electrohydrodynamics. 

For nematic liquid crystals, the synunetry is reduced and we need additional variables. The nematic is degenerate in the sense that all equilibrium orientations of the director are equivalent. According to the Goldstone theorem the parameter of degeneracy is also a hydrodynamic variable for a long distance process 0 and the relaxation time should diverge, x— oo. In nematics, this parameter is the director n(r), the orientational part of the order parameter tensor. For a finite distortion of the director over a large distance (L— oo), the distortion wavevector 0 and the 

For discussion of dynamics of lamellar smectic phases it is important to include another variable, the layer displacement u (r) [3] or, more generally, the phase of the density wave [4]. This variable is also hydrodynamic for a weak compression or dilatation of a very thick stack of smectic layers (L oo) the relaxation would require infinite time. On the other hand, the director in the smectic A phase is no longer independent variable because it must always be perpendicular to the smectic layers. Therefore, total number of hydrodynamic variables for a SmA is six. For the smectic C phase, the director acquires a degree of freedom for rotation about the normal to the layers and the number of variables again becomes seven. 

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The tliree conservation laws of mass, momentum and energy play a central role in the hydrodynamic description. For a one-component system, these are the only hydrodynamic variables. The mass density has an interesting feature in the associated continuity equation the mass current (flux) is the momentum density and thus itself is conserved, in the absence of external forces. The mass density p(r,0 satisfies a continuity equation which can be expressed in the fonn (see, for example, the book on fluid mechanics by Landau and Lifshitz, cited in the Furtlier Reading)... 

The intrinsic rejection and maximum obtainable water flux of different membranes can be easily evaluated in a stirred batch system. A typical batch unit (42) is shown in Figure 5. A continuous system is needed for full-scale system design and to determine the effects of hydrodynamic variables and fouling in different module configurations. A typical laboratory/pilot-scale continuous unit using computer control and on-line data acquisition is shown in Figure 6. 

Here q is chosen to be along the z-axis. cy is the specific heat per particle at constant volume and pq, Tq, and gq are the spatial Fourier components at t = 0. a = A/pcy. The above set of equations is used to obtain the correlation functions of the hydrodynamic variables. 

As discussed in Section II, the five hydrodynamic variables are the number, energy, and momenta densities. Instead of considering the number and energy density, it turns out to be more convenient to consider the entropy density, which can be expressed as a combination of energy and number density. The second variable is now constructed so that it is orthogonal to the entropy density. To explicitly write the states we start with writing a, which is related to the entropy density, s0p>... 

In the above discussion, only a single variable was considered. This can be extended to evaluate the time evolution of many coupled variables—for example, the five hydrodynamical variables. In that case, A is not a single variable but a column matrix and C(q, f) = (A(q, r) A+(q)) is now the correlation matrix. O(q) and T(q, x) are the frequency and the memory function matrices, respectively [20]. 

In Sections V and VI, a brief history of the developments of the MCT from the hydrodynamic approach (Critical Phenomena) and the renormalized kinetic theory approach has been presented. The basic concept of MCT is to use the product of the slow (hydrodynamic) variables to span the orthogonal subspace of the fast variables. 

The first five components are related to the conserved hydrodynamic variables, the density, the longitudinal and transverse current, and the temperature, respectively. 

Let us briefly review the essential ingredients to this procedure (for more details of the method see [30] and for our model [42]). For a given system the hydrodynamic variables can be split up into two categories variables reflecting conserved quantities (e.g., the linear momentum density, the mass density, etc.) and variables due to spontaneously broken continuous symmetries (e.g., the nematic director or the layer displacements of the smectic layers). Additionally, in some cases non-hydrodynamic variables (e.g., the strength of the order parameter [48]) can show slow dynamics which can be described within this framework (see, e.g., [30,47]). 

The rigorous approach to a kinetic-theory derivation of the fluid-dynamical conservation equations, which begins with the Liouville equation and involves a number of subtle assumptions, will be omitted here because of its complexity. The same result will be obtained in a simpler manner from a physical derivation of the Boltzmann equation, followed by the identification of the hydrodynamic variables and the development of the equations of change. For additional details the reader may consult [1] and [2]. 

To test the reliability of the hydrodynamic variable q/T, we plots the rotational correlation times for a heavy water molecule in various solvents against q/T over a wide range of temperature in Figure 2. The plots are approximately linear and expressed by... 

In order to derive hydrodynamic equations one has to define the microscopic basic set of the slowest (hydrodynamic) variables, which for a mixture may be introduced as follows P yd = A k. Jk Tk. where TVk = hk)CJ is a... 

In general for a /-component fluid mixture one has a (z/ + 3 l 1)-component set p yd of dynamic variables, containing //-component of the number densities / k,a, the three components of the total current density Jk, and the total energy density E. However, as follows from the symmetric properties, the number ttk.a and energy densities are coupled only with the longitudinal component of Jk, directed along k. This is due to the space isotropy of the system. As a result, one may split the set of the hydrodynamic variables into two separate subsets ... 

In general, one may choose as the set of longitudinal hydrodynamic variables any other (u + 2) variables, constructed as linearly independent combinations of the variables from P yd. 

Moment methods come in many different variations, but the general idea is to increase the number of transported moments (beyond the hydrodynamic variables) in order to improve the description of non-equilibrium behavior. As noted earlier, the moment-transport equations are usually not closed in terms of any finite set of moments. Thus, the first step in any moment method is to apply a closure procedure to the truncated set of moment equations. Broadly speaking, this can be done in one of two ways. 

Note that this equation is closed in terms of the hydrodynamic variables, and corresponds to a compressible granular gas where plays the role of the density. 

Note that this equation is closed in terms of the hydrodynamic variables a and Uj. However, if the mass-average velocity, defined by U = (pioriUi -i- piaiUi)/p and p = piOTi + P2CX2, were used as the reference velocity then it would be necessary to provide a closure for Ui in terms of U. This is typically accomplished by introducing a gradient-diffusion term involving the spatial gradient of a. ... 

Physical hydrodynamic variables that can affect the flotation, evert though not to as great an extent as the chemical variables listed above, include gas How rate, bubble size distribotion, agitation, feed rate, foam height, and reegent addition modes. 

Now Saxy(q) is odd with respect to reflections through the xz and yz planes, whereas none of the other components of the polarizability tensor and none of the other hydrodynamic variables p(q), v (q), vy(q), v2(q), and the energy density have this symmetry. It follows from Theorems 9-11 of Section 11.5 that Saxy(q) is not coupled to these other variables, and is consequently the only slow variable with this symmetry, so that the relaxation equation describing this variable is... 

The second example is electrolytic plating of copper films. Contrary to some expectations growth close to equilibrium potential conditions does not result in the highest quality deposit. In general, the composition and structure of the deposit depends on the detailed combination of transport processes towards and away from the electrode surface. Adding the local hydrodynamic variables provides the system with an extremely broad structural chemical range. 

The popular problems of kinetics theory is the derivation of hydrodynamic equations, in certain conditions, solution of f (r, v,t) transport equation is similar the form that can relate directly to continuous or hydrodynamic description. In certain conditions the transport process is like hydrodynamic limit. In 1911 David Hilbert was who ptropwsed the existence Boltzmann equations solutions (named normal solutions), and these are determinate by initial values of hydrodynamic variables it return to collision invariant (mass, momentum and kinetics energy), Sydney Chapman and David Enskog in 1917 were whose imroUed a systematic process for derivate the hydrodynamic equations (and their corrections of superior order) for these variables. 

These are called slip boundary conditions, since the gas slips along the walls. tAdditional boundary conditions are needed for these higher-order equations, since they involve third- and higher-order spatial gradients of the hydrodynamic variables. 

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