Continuum with microstructure

Thus the third principle simply asserts that, in its motion as a whole, a body does not know whether it is a mixture or not but in this paper the skeleton of the medium consists of a continuum with microstructure, as defined by Capriz (1989), and therefore it must satisfy balance equations there proposed. The first and the second principles affirm that the whole is no more than the sum of its parts and that the mixture s constituents, imagined as splitted in geometry, must be considered united in physics by suitable forces or energies, respectively. We should also notice how, unlike balance equations, constitutive proposals for dependent fields are usually affected by microstructural independent variables in addition to gross ones. 

When the length scale approaches molecular dimensions, the inner spinning" of molecules will contribute to the lubrication performance. It should be borne in mind that it is not considered in the conventional theory of lubrication. The continuum fluid theories with microstructure were studied in the early 1960s by Stokes [22]. Two concepts were introduced couple stress and microstructure. The notion of couple stress stems from the assumption that the mechanical interaction between two parts of one body is composed of a force distribution and a moment distribution. And the microstructure is a kinematic one. The velocity field is no longer sufficient to determine the kinematic parameters the spin tensor and vorticity will appear. One simplified model of polar fluids is the micropolar theory, which assumes that the fluid particles are rigid and randomly ordered in viscous media. Thus, the viscous action, the effect of couple stress, and... 

In all but the most basic cases of very dilute systems, with microstructural elements such as rigid particles whose properties can be described simply, the development of a theory in a continuum context to describe the dynamical interactions between structure and flow must involve some degree of modeling. For some systems, such as polymeric solutions, we require modeling to describe both polymer-solvent and polymer-polymer interactions, whereas for suspensions or emulsions we may have an exact basis for describing particle-fluid interactions but require modeling via averaging to describe particle-particle interactions. In any case, the successful development of useful theories of microstructured fluids clearly requires experimental input and a comparison between experimental data and model... 

When fluids can seep through pores, interacting mechanically with the solid skeleton, the material is composed of more than one constituent thus we need to use a mixture theory in which we could clearly make out each part filled by different constituents on a scale which is rather large in comparison with molecular dimensions so we put forward a new continuum theory of an immiscible mixture consisting both of a continuum with ellipsoidal microstructure (the porous elastic solid) and of two classical media (see, also, the conservative case examined by Giovine (2000)). In accordance with Biot (1956), we consider virtual mass effects due to diffusion we also introduce the microinertia associated with the rates of change of the constituents local densities, as well as the one due to the deformation of the pores close to their boundaries. 

In this work we propose an immiscible mixtures consisting of a continuum with ellipsoidal microstructure (the porous elastic solid) and of two classical fluids (a fluid carrier and an adsorbate) to... 

The formulation of extended continua can be traced back to the Cosserat brothers [13], who enriched the standard continuum by rotational degrees of freedom. As a consequence, such a continuum does not only transfer stresses but also couples stresses. The Cosserat or micropolar theory is well established now [17, 22, 32]. It is successfully applied if materials with microstructure are modeled on the macroscale, e.g., the equivalence of the Cosserat model is shown for materials with lattice-Hke microstractures [1, 2, 16, 25, 34, 35, 41] and with granular materials [3-5, 8, 15]. Furthermore, the micropolar approach is taken into account if strain localization arises which in the framework of the standard continuum would lead to a mesh dependence in the numerical solution (see, e.g.. Ref [29]). The Cosserat model introduces a so-called internal length A in the theory which allows for the existence of solutions of the bound-... 

R. Lakes (1995). Experimental methods for study of Cosserat elastic solids and other generalized elastic continua. In Continuum Methods for Materials with Microstructures (Ed. H. Muhlhaus), pp. 1-25. John Wiley Chichester. 

H. Muhlhaus (1995). Continuum Methods for Materials with Microstructures. John Wiley Sons, Chichester. 

The Eulerian description of the instantaneous motion of a fluid with microstructure employs two independent vector fields. The first is the usual velocity v(x, t) and the second is an axial vector w(x, t) which, in the case of polar fluids, represents the angular velocity of the polar fluid particle at position x at time t. In the context of liquid crystals, w is interpreted as being the local angular velocity of the liquid crystal material element, that is, it represents the local angular velocity of the director n. In ordinary continuum theory the only independent field is the velocity v of the fluid because the angular velocity in such theories equals one half of the curl of the velocity. We denote this particular angular velocity by w defined... 

In literature, some researchers regarded that the continuum mechanic ceases to be valid to describe the lubrication behavior when clearance decreases down to such a limit. Reasons cited for the inadequacy of continuum methods applied to the lubrication confined between two solid walls in relative motion are that the problem is so complex that any theoretical approach is doomed to failure, and that the film is so thin, being inherently of molecular scale, that modeling the material as a continuum ceases to be valid. Due to the molecular orientation, the lubricant has an underlying microstructure. They turned to molecular dynamic simulation for help, from which macroscopic flow equations are drawn. This is also validated through molecular dynamic simulation by Hu et al. [6,7] and Mark et al. [8]. To date, experimental research had "got a little too far forward on its skis however, theoretical approaches have not had such rosy prospects as the experimental ones have. Theoretical modeling of the lubrication features associated with TFL is then urgently necessary. 

As noted before, thin film lubrication (TFL) is a transition lubrication state between the elastohydrodynamic lubrication (EHL) and the boundary lubrication (BL). It is widely accepted that in addition to piezo-viscous effect and solid elastic deformation, EHL is featured with viscous fluid films and it is based upon a continuum mechanism. Boundary lubrication, however, featured with adsorption films, is either due to physisorption or chemisorption, and it is based on surface physical/chemical properties [14]. It will be of great importance to bridge the gap between EHL and BL regarding the work mechanism and study methods, by considering TFL as a specihc lubrication state. In TFL modeling, the microstructure of the fluids and the surface effects are two major factors to be taken into consideration. 

From experimental results, the variation of film thickness with rolling velocity is continuous, which validates a continuum mechanism, to some extent in TFL. Because TFL is described as a state in which the film thickness is at the molecular scale of the lubricants, i.e., of nanometre size, common lubricants may exhibit microstructure in thin films. A possible way to use continuum theory is to consider the effect of a spinning molecular confined by the solid-liquid interface. The micropolar theory will account for this behavior. 

Ranka, J. K., Windeler, R. S. and Stentz, A. J. (2000). Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm. Opt. Lett. 25, 25-7. 

Molecular calculations provide approaches to supramolecular structure and to the dynamics of self-assembly by extending atomic-molecular physics. Alternatively, the tools of finite element analysis can be used to approach the simulation of self-assembled film properties. The voxel4 size in finite element analysis needs be small compared to significant variation in structure-property relationships for self-assembled structures, this implies use of voxels of nanometer dimensions. However, the continuum constitutive relationships utilized for macroscopic-system calculations will be difficult to extend at this scale because nanostructure properties are expected to differ from microstructural properties. In addition, in structures with a high density of boundaries (such as thin multilayer films), poorly understood boundary conditions may contribute to inaccuracies. 

J.K. Ranka, R.S. Windeler, and A.J. Stentz Efficient Visible Continuum Generation in Air-Silica Microstructure Optical Fibers with Anomalous Dispersion at 800 nm . In Conference on Lasers and Electro-Optics CLEO, postdeadline paper CD-8, Washington D.C. (1999)... 

In some cases, one is interested in the structures of complex fluids only at the continuum level, and the detailed molecular structure is not important. For example, long polymer molecules, especially block copolymers, can form phases whose microstructure has length scales ranging from nanometers almost up to microns. Computer simulations of such structures at the level of atoms is not feasible. However, composition field equations can be written that account for the dynamics of some slow variable such as 0 (x), the concentration of one species in a binary polymer blend, or of one block of a diblock copolymer. If an expression for the free energy / of the mixture exists, then a Ginzburg-Landau type of equation can sometimes be written for the time evolution of the variable 0 with or without flow. An example of such an equation is (Ohta et al. 1990 Tanaka 1994 Kodama and Doi 1996)... 

Within the FPM, we can extend further the capabilities of the discrete particle method to the mesoscopic regime and show that they are competitive to standard simulation techniques with continuum equations. These methods establish a foundation for cross-scale computations ranging from nanoscales to microns and can provide a framework for studying the interaction of microstructures and large-scale flow, which may be of value in blood flow and other applications in polymeric flows (Banfield et al. 2000 Schwertman et al 1999 Hiemstra and VanReimsdijk 1999). 

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