Continuum theory viscosity coefficients

This theory takes into account the micro-rotational effects due to rotation of molecules. This becomes important with polymers or polymeric suspensions. The physical model assigns a substructure to each continuum particle. Each material volume element contains microvolume elements which can translate, rotate, and deform independently of the motion of the microvolime. In the simplest case, these fluids are characterised by 22 viscosity coefficients and the problem is formulated in terms of a system of 19 equations with 19 unknowns. The equations for a 2-D case were solved numerically and compared to experimental results. It is concluded that the model based on the micropolar fluid theory gives a better fit than the Navier - Stokes equations. However, it seems that the difference is small. 

A number of important ideas concerning the N, phase have been discussed theoretically - molecular statistical and phenomenological theories, " " continuum theories, " topological theories of de-fects, - " etc. For example, Saupe and Kini " who used different theoretical approaches, have both concluded that the incompressible orthorhombic nematic has 12 curvature elastic constants (excluding three which contribute only to the surface torque) and 12 viscosity coefficients. 

The laws of Pick and Stokes are good examples on transport processes in a continuum. Adolf Pick (1829-1901), who derived the law of diffusion in 1855, was in facta physiologist from Kassel in Germany. The Irishman George Stokes (1819-1903) was a pure mathematician. For us it is important to find out how the diffusion constant (D) and the viscosity coefficient (q) are related to entities in thermodynamics and kinetic gas theory. 

Continuum theory has also been applied to analyse the dynamics of flow of nematics. The equations provide the time-dependent velocity, director and pressure fields. These can be determined from equations for the fluid acceleration, the rate of change of director orientation in terms of the velocity gradients and the molecular field, and the incompressibility condition. Further details can be found in de Gennes and Frost (1993). Various combinations of elements of the viscosity tensor of a nematic define the so-called Leslie coefficients. 

Theoretical treatments of liquid crystals such as nematics have proved a great challenge since the early models by Onsager and the influential theory of Maier and Saupe [34] mentioned before. Many people have worked on the problems involved and on the development of the continuum theory, the statistical mechanical approaches of the mean field theory and the role of repulsive, as well as attractive forces. The contributions of many theoreticians, physical scientists, and mathematicians over the years has been great - notably of de Gennes (for example, the Landau-de Gennes theory of phase transitions), McMillan (the nematic-smectic A transition), Leslie (viscosity coefficients, flow, and elasticity). Cotter (hard rod models), Luckhurst (extensions of the Maier-Saupe theory and the role of flexibility in real molecules), and Chandrasekhar, Madhusudana, and Shashidhar (pre-transitional effects and near-neighbor correlations), to mention but some. The devel-... 

The hydrodynamic continuum theory of nematic liquid crystals was developed by Leslie [1,2] and Ericksen [3, 4] in the late 1960s. The basic equations of this theory are presented in Vol. 1, Chap. VII, Sec. 8. Since then, a great number of methods for the determination of viscosity coefficients have been developed. Unfortunately, the reliability of the results has often suffered from systematic errors leading to large differences between results. However, due to a better understanding of flow phenomena in nematic liquid crystals, most of the errors of earlier investigations can be avoided today. 

Of course, eventually, particles will reappear, through the equation = Po + RTln the existence of R depends on the existence of particles. But a theory of stress-driven deformation of a continuum does not require particles, even with stress-driven self-diffusion coefficients for viscosity and self-diffusion are the only things required. 

In our simulations published earlier [20,71,84], we map the DPD interactions onto the macroscopic parameters of fluid, for example, viscosity, compressibility and diffusion coefficient, by using continuum limit equations obtained from kinetic theory. For example, for a given density Pk and sound speed cp- in fcth DPD fluid, we computed the scaling factor 11 from the following continuum equations [81]... 

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