Expression for the stress tensor

Next we consider how to calculate the stress tensor, say the component aaz oc = x,y, z). To this end we consider a region of volume V in the 

Now Sa consists of two parts, the force 5 which acts through the solvent fluid and the force 5 which acts directly between the beads. The former is written as 

The two 6 functions in eqn (3.124) restrict the bead n to the upper part and the bead m to the lower part. 

Equations (3.121) and (3.133) together may be regarded as a constitutive equation for a given macroscopic velocity gradient K p(t), the distribution function is obtained from eqn (3.121) and the stress is calculated using eqn (3.133). 

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The first ingredient in any theory for the rheology of a complex fluid is the expression for the stress in terms of the microscopic structure variables. We derive an expression for the stress-tensor here from the principle of virtual work. In the case of flexible polymers the total stress arises to a good approximation from the entropy of the chain paths. At equilibrium the polymer paths are random walks - of maximal entropy. A deformation induces preferred orientation of the steps of the walks, which are therefore no longer random - the entropy has decreased and the free energy density/increased. So... 

In this section, we use the Cartesian force of Section VI to derive several equivalent expressions for the stress tensor of a constrained system of pointlike particles in a flow field with a macroscopic velocity gradient Vv. The excess stress of any system of interacting beads (i.e., point centers of hydrodynamic resistance) in a Newtonian solvent, beyond the Newtonian contribution that would be present at the applied deformation rate in the absence of the beads, is given by the Kramers-Kirkwood expression [1,4,18]... 

An alternative class of expressions for the stress tensor, analogous to those obtained in Section VIII for the drift velocity, may be obtained by using Eqs. (2.201) to expand the divergence of in the first hne of Eq. (2.388). This expansion yields an elastic stress... 

The expression for the stress tensor -t in a fluid mixture is the same as that for a pure substance. The tensor components given in Eqs. (7) and (8) may be summarized conveniently in tensor notation ... 

Thus, an equation, which has the sense of a law of conservation of momentum has been obtained. There is an expression for the momentum flux pviVj — Uij under the derivation symbol, which allows one to write down the expression for the stress tensor... 

Furthermore, it is convenient to switch to normal co-ordinates (1.13). We can use the expressions for forces (2.26) to rewrite the expression for the stress tensor in normal co-ordinates... 

To obtain the expression for the stress tensor for the set of Brownian particles suspended in a viscoelastic liquid, we use equation (6.7), in which the elastic and internal viscosity forces are specified in Section 3.2... 

To calculate the dynamic modulus, we turn to the expression for the stress tensor (6.46) and refer to the definition of equilibrium moments in Section 4.1.2, while memory functions are specified by their transforms as... 

However, in this case, the stresses in entangled systems can also be related to the tensor of mean orientation of the segments with a relation similar to equation (7.45), but with other coefficient of proportionality, because we deal with non-equilibrium situation in this case. In this way one can correspond the two expressions for the stress tensor to each other and relate the introduced variables xft and u"k to the tensor of mean orientation of the segments... 

The expressions for the stress tensor together with the equations for the moments considered as additional variables, the continuity equation, and the equation of motion constitute the basis of the dynamics of dilute polymer solutions. This system of equations may be used to investigate the flow of dilute solutions in various experimental situations. Certain simple cases were examined in order to demonstrate applicability of the expressions obtained to dilute solutions, to indicate the range of their applicability, and to specify the expressions for quantities which were introduced previously as phenomenological constants. 

The expression for the stress tensor (6.7) allows us to investigate the non-linear with respect to velocity gradient effects. We use the normal co-ordinates (1.13) to write equation (6.7) in the form... 

When the elastic force and the force of internal viscosity are defined, at N = 1, the expression for the stress tensor directly follows relation (6.7)... 

We note that the second term in (108) is the familiar Kirkwood expression for the stress tensor in terms of the n-particle distribution function. 

Once the structure of a complex fluid has been simulated, computed, or derived by analytic theory, one would like to calculate the stress tensor a and compare it to experimental stress measurements. The appropriate expression for the stress tensor depends on the type of complex fluid. However, if the idealized microstructure is built out of many small, point-like elements located at positions x,, i = 1,2,..., N, and on each such point a nonhydrodynamic force F, is exerted by the rest of the microstructure, then a can be obtained from the general Kirkwood (1949) formula (Doi and Edwards 1986) ... 

Although the expression for the stress tensor in the Doi-Edwards model is the same as that of the temporary network model, except for the coefficient [see Eq. (3-13)],... 

The first two terms in this expression for the stress tensor are elastic terms due to Brownian motion and the nematic potential, respectively. The last term is a purely viscous term produced by the drag of solvent as it flows past the rod-like molecules [see Eq. (6-36)]. is a drag coefficient, which for modestly concentrated solutions is predicted to follow the dilnte-solution formula (Doi and Edwards 1986 see Section 6.3.1.4) ... 

To derive a molecular expression for the stress tensor component which is the basic quantity if one wishes to compute pseudo-experimental data F (/i) /R, we start from the thermodynamic expression for the exact differential given in Eq. (1.59), where the strain tensor [Pg.202]

Two expressions may be obtained for the polymer contribution to the stress tensor. The first is obtained by inserting F(c) for rigid dumbbells from Eq. (25.12) into the general expression for the stress tensor in Eq. 

The second expression for the stress tensor is obtained by eliminating the term from the foregoing equation by using Eq. (25.17) ... 

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. 

With these assumptions it can be shown (see, for example, Batchelor 1967) that the expression for the stress tensor in a Newtonian fluid becomes... 

The Flux Expressions. We begin with the relations between the fluxes and gradients, which serve to define the transport properties. For viscosity the earliest definition was that of Newton (I) in 1687 however about a century and a half elapsed before the most general linear expression for the stress tensor of a Newtonian fluid was developed as a result of the researches by Navier (2), Cauchy (3), Poisson (4), de St. Venant (5), and Stokes (6). For the thermal conductivity of a pure, isotropic material, the linear relationship between heat flux and temperature gradient was proposed by Fourier (7) in 1822. For the difiiisivity in a binary mixture at constant temperature and pressure, the linear relationship between mass flux and concentration gradient was suggested by Pick (8) in 1855, by analogy with thermal conduction. Thus by the mid 1800 s the transport properties in simple systems had been defined. 

For two-phase systems Einstein (34) in 1906 was the first to obtain the viscosity for a very dilute suspension of solid spheres the resulting stress expression is Newtonian. However, it has been only within the past decade that nonlinear viscoelastic expressions for the stress tensor in dilute suspensions have been obtained deformable spheres (35), ellipsoids (36), emulsions (37, 38), For a survey of activities in this field, see the summary by Barthes-Biesel and Acrivos (39),... 

As shown in Appendix 16.A, the expression for the stress tensor given in Eq. (16.10) is equivalent to that expressed in terms of connector force and bond vector in Chapters 6 and 7 (see Eq. (6.35)). 

In Appendix A we address some problems that arise in connection with the uniqueness of the expression for the stress tensor. In Appendix B we derive a fairly general stress-diffusion relation for polymer solutions. Finally Appendix C deals with an equation of change for the temperature. 

Sect. 6 the hydrodynamic equation of continuity for each species and a formal expression for the mass flux vector of each species Sect. 7 the hydrodynamic equation of motion for the liquid mixture and a formal expression for the stress tensor Sect. 8 the energy equation for the liquid and a formal expression for the heat flux vector... 

That is, the sum of the Brownian, external, intramolecular, and hydrodynamic forces acting on a bead is zero. The introduction of this assumption makes the solution of the equation for the singlet configuration-space distribution function simpler. It is also useful in developing alternative expressions for the stress tensor (see, e.g., DPL Tables 13.3-1 and 15.2-1). 

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