Dissipative stress tensor

The equation of momentum conservation, along with the linear transport law due to Newton, which relates the dissipative stress tensor to the rate of strain tensor = 1 (y. 4, and which introduces two... 

As structured fluids such as liquid crystals are at least partially fluid, we also need to consider the forces and torques produced by friction. The frictional forces are given by a dissipative stress tensor, which is most conveniently derived from the dissipative function (j)F It is a homogeneous positive definite quadratic function of the time derivatives of the strains and rotations (the time derivatives of the torsions can be generally ignored) giving ... 

The Reynolds-averaged approach is widely used for engineering calculations, and typically includes models such as Spalart-Allmaras, k-e and its variants, k-co, and the Reynolds stress model (RSM). The Boussinesq hypothesis, which assumes pt to be an isotropic scalar quantity, is used in the Spalart-Allmaras model, the k-s models, and the k-co models. The advantage of this approach is the relatively low computational cost associated with the computation of the turbulent viscosity, fit. For the Spalart-Allmaras model, one additional transport equation representing turbulent viscosity is solved. In the case of the k-e and k-co models, two additional transport equations for the turbulence kinetic energy, k, and either the turbulence dissipation rate, s, or the specific dissipation rate, co, are solved, and pt is computed as a function of k and either e or co. Alternatively, in the RSM approach, transport equations can be solved for each of the terms in the Reynolds stress tensor. An additional scale-determining equation (usually for s) is also required. This means that seven additional transport equations must be solved in 3D flows. 

Vedula, P., P. K. Yeung, and R. O. Fox (2001). Dynamics of scalar dissipation in isotropic turbulence A numerical and modeling study. Journal of Fluid Mechanics 433, 29-60. Verman, B., B. Geurts, and H. Kuertan (1994). Realizability conditions for the turbulent stress tensor in large-eddy simulations. Journal of Fluid Mechanics 278, 351-362. Vervisch, L. (1991). Prise en compte d effets de cinetique chimique dans lesflammes de diffusion turbulente par Tapproche fonction densite de probabilite. Ph. D. thesis, Universite de Rouen, France. 

The third term on the right side of eq 68 represents viscous dissipation, the heat generated by viscous forces, where r is the stress tensor. This term is also small, and all of the models except those of Mazumder and Cole - neglect it. The fourth term on the right side comes from enthalpy changes due to diffusion. Finally, the last term represents the change in enthalpy due to reaction... 

Hence, the collisional rate of dissipation and the collisional stress tensor are related to the coefficient of restitution by Eq. (5.284) and Eq. (5.277). 

In Eqs. (6) and (7) e represents the internal energy per unit mas, q the heat flux vector due to molecular transport, Sh the volumetric heat production rate, ta, the mass fraction of species i, Ji the mass flux vector of species i due to molecular transport, and 5, the net production rate of species i per unit volume. In many chemical engineering applications the viscous dissipation term (—t Vm) appearing in Eq. (6) can safely be neglected. For closure of the above set of equations, an equation of state for the density p and constitutive equations for the viscous stress tensor r, the heat flux vector q, and the mass flux vector 7, are required. In the absence of detailed knowledge on the true rheology of the fluid, Newtonian behavior is often assumed. Thus, for t the following expression is used ... 

In the most systematic application of this approach, Harlow and co-workers at Los Alamos have derived a transport equation for the full Reynolds stress tensor pu u j. They have coupled this equation with a scalar dissipation transport equation and have utilized with various semi-empirical approximations to evaluate the numerous unknown velocity, velocity-pressure, and velocity-temperature correlations which appear in the formulation. While this treatment is fairly vigorous, extensive compu-... 

Stress required to break the aggregate network. Pa Static yield stress. Pa Tangential components of stress. Pa Stress dissipated due to viscous drag. Pa Stress tensor... 

While the fiber contribution to the steady-state stress tensor at steady-state is modest for shearing flow, its contribution to the stress in extensional flow is large at steady state. In a uniaxial extensional flow, the fibers orient in such a way that the viscous dissipation is maximized. Large values of the extensional viscosity are the result from Batchelor s (1971) theory the uniaxial extensional viscosity is... 

The possible development of gradients in the components of the interfacial stress tensor due to flow of an adjacent fluid implies that the momentum flux caused by the the flow of liquid at one side of the interface does not have to be completely transported across the interface to the second fluid but may (partly or completely) be compensated in the interface. The extent to which this is possible depends on the rheological properties of the interface. For small shear stresses the interface may behave elastically or viscoelastically. For an elastic interfacial layer the structure remains coherent the layer will only deform, while for a viscoelastic one it may or may not start to flow. The latter case has been observed for elastic networks (e.g. for proteins) that remciln intact, but inside the meshes of which liquid can flow leading to energy dissipation. At large stresses the structure may yield or fracture (collapse), leading to an increased flow. 

The PGT model represents an extension of the PT models in that the gas turbulence is taken into account by including the Re3moIds stress tensor in the momentum equation for the gas phase. The turbulence model used for the gas phase is similar to the standard single phase k-e turbulence model presented in sect 1.3.5, although additional generation and dissipation terms may be added to consider the presence of particles. In the PGT model the drift velocity is neglected. 

The two expansion coefficients ijs and rjv are called the shear and bulk viscosities, respectively. The shear viscosity is all that is required to describe our gedanken experiment. The bulk viscosity describes the viscous or dissipative part of the response to a compression. This linear constitutive relation is called the Newtonian stress tensor. A fluid correctly described by this form is called a Newtonian fluid.16... 

More recenfly, a complete set of governing relationships was derived from the requirements of the compatibility of dynamics and thermodynamics [Grmela and Ait-ICadi, 1994, 1998 Grmela et al, 1998, 2001]. The authors developed a set of equations governing the time evolution of the functions Q and q. (see Eqs 7.95), as well as the extra stress tensor expressed in their terms. The rheological and morphological behavior was expressed as controlled by two potentials thermodynamic and dissipative. Under specific conditions for these potentials, Lee and Park formalism can be recovered. 

The mechanisms governing deformation and breakup of drops in Newtonian liquid systems are relatively well understood. However, within the range of compounding and processing conditions the molten polymers are viscoelastic liquids. In these systems the shape of a droplet is determined not only by the dissipative (viscous) forces, but also by the pressure distribution around the droplet that originates from the elastic part of the stress tensor. Therefore, the characteristics of drop deformation and breakup in viscoelastic systems may be quite different from those in Newtonian ones. Some of the pertinent papers on the topic are listed in Table 9.3. 

In a dynamic experiment, a small-amplitude oscillatory shear is imposed to a molten polymer confined in the rheometer. The shear stress response of the polymeric system can be expressed as in Equation 22.14. In this equation, G and G" are dynamic moduli related to the elastic storage energy and dissipated energy of the system, respectively. For a viscoelastic fluid, two independent normal stress differences, namely, first and second normal stress differences can be defined. These quantities are calculated in terms of the differences of the components of the stress tensor, as indicated in Equation 22.15a and 22.15b, and can be obtained, for instance, from the radial pressure distribution in a cone-and-plate rheometer [5]. Some other experiments used in the determination of the normal stress differences can be found elsewhere [9, 22] ... 

The last term in Eq. 19 is the non-dissipative part of the stress tensor, representing the hydrodynamic interactions. The parameter C is estimated as jly- flaMr], where the interfacial tension y and aM is the diffusivity. For a fluid with high viscosity, one has C -C 1, which implies that the hydrodynamic interactions can be neglected. The parameter IT (nondimensional pressure) mathematically acts like a Lagrange multiplier, which generates the incompressibility condition V V = 0. The boundary conditions on p and / are as follows (where n represents a normal direction to the boundary surface) ... 

The dissipative (or viscous or deviatoric) part of the stress tensor (II ) is a function of the rheological fluid properties and depends on the local shear rates. For Newtonian fluids, which are isotropic, purely viscous, and without rheological memory , this dependency is linear (Cauchy-Poisson-law) ... 

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