Continuum hydrodynamics

Most theoretical studies of osmosis and reverse osmosis have been carried out using macroscopic continuum hydrodynamics [5,8-13]. The models used include those that treat the wall as either nonporous or porous. In the nonporous models the membrane surface is assumed homogeneous and nonporous. Transport occurs by the molecules dissolving in the membrane phase and then diffusing through the membrane. Mass transfer across the membrane in these models is usually described using the solution-diffusion... 

The remaining four chapters discuss theoretical approaches and considerations which have been suggested to include the effects of many-body complications, to use approximate techniques, to use more realistic continuum hydrodynamic equations than the diffusion equation, and to use more satisfactory statistical mechanical descriptions of liquid structure. This work is still in a comparatively early stage of its development. There is a growing need for more detailed experiments which might probe the effects anticipated by these studies. 

All of the above was based on continuum hydrodynamics and is strictly applicable only to shocks in perfectly homogenous media. 

We anticipate that as these mesoscale approaches are developed and their shortcomings are overcome, their application in chemical engineering will become more and more widespread. Eventually, it is our expectation that the use of these models will become as common as the use of continuum hydrodynamics and quantum chemistry models is today. 

For particles of a size similar to that of the bath molecules, the Knudsen number is insufficient to characterize the transport processes, but nonetheless it is found exp>erimentally in many cases that the transport properties of single particles and pairs of particles are surprisingly well modeled by continuum (hydrodynamic) theories at high densities and molecular theories at low densities. 

Experimental results to support both theoretical models have been reported. McAdam et al. (122) and Carlson et al. (123,124) found nonexponential decays of the correlation function in studies of cross-linked PS gels swollen in tetralin and cross-linked aqueous polyacrylamide gels, respectively. On the other hand, the data of Tanaka et al. (122) for aqueous polyacrylamide gels agree with the continuum hydrodynamic... 

This spatial averaging over different length-scales transforms the partial differential equations of continuum hydrodynamics into a set of nonlinear ordinary differential equations for the particle velocity v and internal energy e... 

Finite-element calcutaions Continuum hydrodynamics Dissipative particle dynamics MD-MC-Lattice Boltzmann... 

Delgado-Buscalitmi R, Kremer K, Praprotnik M (2009) Coupling atomistic and continuum hydrodynamics through a mesoscopic model application to liquid water. J Chem Phys 131(24) 244107... 

The value of A is a measure of the ion-solvent interaction and, within the scope of the continuum hydrodynamic model, is related to the bulk viscosity of the solvent, t], through the Nemst-Einstein equation or its empirical equivalent, Walden s law (t .A = constant). However, it is well known that the simple continuum model is not valid in aqueous systems even at room temperature. 

Transport coefficients occur in all forms of continuum, hydrodynamic equations concerned with mass, momentum and energy conservation once constitutive equations for the fluids of interest are introduced. Such equations are frequently encountered in trying to model mathematically technological processes with a view to their refinement. Attempts to model such processes mathematically (usually numerically) are frequently limited by a lack of knowledge of the physical properties of the materials involved including the transport coefficients of the fluids. 

Since lijima identified carbon nanotubes (CNTs) in 1991, CNTs have been investigated in various fields and become extremely desirable for a wide range of applications. CNTs, with diameters in nanometer scale and a smooth surface may offer a very unique molecular transport through their pores. In fact, several studies in recent years suggest that the water transport through single-walled carbon nanotubes (SWNT) would become much faster than the transport rate that the continuum hydrodynamic theory would predict. This was attributed by Molecular Dynamic (MD) simulation to the smoothness of the nano-tube wall [1,2]. 

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