Diffusion stress tensor

Each stress tensor can then always be expressed as the sum of a mean pressure tensor, a viscous stress tensor, and a viscous diffusion-stress tensor thus, ... 

Note that 7Zu = 0 due to the continuity equation. Thus, the pressure-rate-of-strain tensor s role in a turbulent flow is to redistribute turbulent kinetic energy among the various components of the Reynolds stress tensor. The pressure-diffusion term T is defined... 

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... 

Problems of forced convection diffusion in non-Newtonian flow have to this author s knowledge not yet been attacked. The equations needed for solving such problems are given in this article. The equation of motion in terms of the stress tensor [Eq. (25)] can be used to describe non-Newtonian flow provided that a suitable form for the stress tensor is used examples of two non-Newtonian stress tensors are given in Eqs. (28a) and (28b). 

In the case K > fi, the usual diffusion determines the kinetics for any gel shapes. Here the deviation of the stress tensor is nearly equal to — K(V u)8ij since the shear stress is small, so that V u should be held at a constant at the boundary from the zero osmotic pressure condition. Because -u obeys the diffusion equation (4.18), the problem is trivially reduced to that of heat conduction under a constant boundary temperature. The slowest relaxation rate fi0 is hence n2D/R2 for spheres with radius R, 6D/R2 for cylinders with radius R (see the sentences below Eq. (6.49)), and n2D/L2 for disks with thickness L. However, in the case K < [i, the process is more intriguing, where the macroscopic critical mode slows down as exp(- Q0t) with Q0 oc K. 

Surface tractions or contact forces produce a stress field in the fluid element characterized by a stress tensor T. Its negative is interpreted as the diffusive flux of momentum, and x x (—T) is the diffusive flux of angular momentum or torque distribution. If stresses and torques are presumed to be in local equilibrium, the tensor T is easily shown to be symmetric. 

The state of mechanical equilibrium is characterized by vanishing acceleration dy/dt = 0. Usually, mechanical equilibrium is established faster than thermodynamic equilibrium, for example, in the initial state when diffusion is considered. In the case of diffusion in a closed system, the acceleration is very small, and the corresponding pressure gradient is negligible the viscous part of the stress tensor also vanishes t = 0. The momentum balance, Eq. (3.97), is limited to the momentum conservation equation... 

It is composed of the stress tensor t — 2p S — 1/3 SijTr S)) (momentum equations), the energy flux u-r+< (energy equation) and the diffusive flux Jk... 

The stress tensor for a semidilute solution of rods is given by Eq. (6-36), the formula for dilute solutions. However, if in a thought experiment one holds the shear rate fixed at a low value while increasing the concentration of rods from dilute to semidilute, the Brownian contribution to the stress will greatly increase, since the rotary diffusivity decreases according to (6-44). The viscous stress contribution, however, only increases in proportion to u. Thus, as Doi and Edwards (1986) argued, the ratio of viscous to Brownian stresses decreases as as the concentration increases in the semidilute regime. Hence, in the semidilute... 

Figure 11.23—Comparison of theo-retical and experimental first and second normal stress differences N and N2. The theoretical results (a) were calculated from the Smoluchowski equation (11-3) using the Onsager potential with U = 10.67, the minimum value for a fully nematic state, y/ >r is the dimensionless shear rate (or Deborah number), where Dr is the rotary diffusivity of a hypothetical isotropic fluid at the same concentration. Only the molecular-elastic contribution to the stress tensor was considered. The experimental results (b) are for 12.5% (by weight) PBLG (molecular weight = 238,000) in w-cresol. (Reprinted with permission from Magda et al., Macromolecules 24 4460. Copyright 1991, American Chemical Society.)...

It is valid for each continuum independent of the individual material properties and is therefore one of the fundamental equations in fluid mechanics and subsequently also in heat and mass transfer. The movement of a particular substance can only be described by introducing a so-called constitutive equation which links the stress tensor with the movement of a substance. Generally speaking, constitutive equations relate stresses, heat fluxes and diffusion velocities to macroscopic variables such as density, velocity and temperature. These equations also depend on the properties of the substances under consideration. For example, Fourier s law of heat conduction is invoked to relate the heat flux to the temperature gradient using the thermal conductivity. An understanding of the strain tensor is useful for the derivation of the consitutive law for the shear stress. This strain tensor is introduced in the next section. 

The starting point of a molecular constitutive theory is a simple mechanical model for the molecule that captures its most salient traits. Thus, flexible polymer molecules have been represented by elastic dumbbells and bead-spring chains, and rigid polymers by rigid dumbbells and rigid rods. For its simplicity, the evolution of the model molecule is easily described by a convection-diffusion equation. Then a Fokker-Planck equation is written for the probability distribution function of an ensemble of these molecules. Finally, the macroscopic stress tensor is derived in terms of the distribution function. This kinetic theory framework was pioneered by Kirkwood (see, for example, Ref. ). 

The structure of the expression for totai is that of a bilinear form it consists of a sum of products of two factors. One of these factors in each term is a flow quantity (heat flux q, mass diffusion flux jc, momentum flux expressed by the viscous stress tensor o, and chemical reaction rate rr)- The other factor in each term is related to a gradient of an intensive state variable (gradients of temperature, chemical potential and velocity) and may contain the external force gc or a difference of thermodynamic state variables, viz. the chemical affinity A. These quantities which multiply the fluxes in the expression for the entropy production are called thermodynamic forces or affinities. 

For each balance law, the values of -0, J and 4> defines the transported quantity, the diffusion flux and the source term, respectively, v denotes the velocity vector, T the total stress tensor, gc the net external body force per unit of mass, e the internal energy per unit of mass, q the heat flux, s the entropy per unit mass, h the enthalpy per unit mass, u>s the mass fraction of species s, and T the temperature. 

In order to answer this question one has to find out what modifications are necessary in (a) the diffusion equation for the distribution function, and (b) the expression for the stress tensor. Kirkwood and coworkers (39,40,67) and Kotaka (42)w studied this problem for multibead dumbbells including complete hydrodynamic interaction. If one neglects the hydrodynamic interaction entirely, then from the articles cited above one concludes that all the results for rigid dumbbells can be taken over for the multibead dumbbells by replacing X — (,I / 2kT by XN — XN(N + l)/6(iV — 1) everywhere. For the case of complete hydro-dynamic interaction no such simple replacement is possible. 

The lack of correlation between the fluctuating stress tensor and the fluctuating heat flux in the third expression is an example of the Curie principle for the fluctuations. These equations for fluctuating hydrodynamics are arrived at by a procedure very similar to that exhibited in the preceding section for diffusion. A crucial ingredient is the equation for entropy production in a fluid... 

The momentum equation, as represented by the Navier-Stokes equation, is not restricted to a single-component fluid but is valid for a multicomponent solution or mixture so long as the external body force is such that each species is acted upon by the same external force (per unit mass), as in the case with gravity. In the following section we consider external forces associated with an applied external field, which differ for different species. The reason for there being no distinction between the various contributions to the stress tensor associated with diffusive transport is that the phenomenological relation for the stress is unaltered by the presence of concentration gradients. This is seen from the fact that the stress tensor must be related to the spatial variations in fluid... 

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