Stress tensor total

Continuum theory has also been applied to analyse tire dynamics of flow of nematics [77, 80, 81 and 82]. The equations provide tire time-dependent velocity, director and pressure fields. These can be detennined from equations for tire fluid acceleration (in tenns of tire total stress tensor split into reversible and viscous parts), tire rate of change of director in tenns of tire velocity gradients and tire molecular field and tire incompressibility condition [20]. 

The pseudopotential density-functional technique is used to calculate total energies, forces on atoms and stress tensors as described in Ref. 13 and implemented in the computer code CASTEP. CASTEP uses a plane-wave basis set to expand wave-functions and a preconditioned conjugate gradient scheme to solve the density-functional theory (DFT) equations iteratively. Brillouin zone integration is carried out via the special points scheme by Monkhorst and Pack. The nonlocal pseudopotentials in Kleynman-Bylander form were optimized in order to achieve the best convergence with respect to the basis set size. 5... 

Note that since i and j each represent any of three possible directions, there are a total of nine possible components of the stress tensor (at any given point in a fluid). However, it can readily be shown that the stress tensor is symmetrical (i.e., the ij components are the same as the ji components), so there are at most six independent stress components. 

Explicit forms for the stress tensors d1 are deduced from the microscopic expressions for the component stress tensors and from the scheme of the total stress devision between the components [164]. Within this model almost all essential features of the viscoelastic phase separation observable experimentally can be reproduced [165] (see Fig. 20) existence of a frozen period after the quench nucleation of the less viscous phase in a droplet pattern the volume shrinking of the more viscous phase transient formation of the bicontinuous network structure phase inversion in the final stage. 

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... 

The stress tensor in the gel is written as p0<5y + fly, where p0 is the solvent pressure outside the gel held at a constant and fly is the stress due to the polymer. Once we have the free energy functional, it can be calculated as follows. We deform the gel infinitesimally as - Xt + 8Xt. Then the resultant change of the total free energy... 

In general, as discussed in Chapter 5, the total stress tensor is defined by... 

The above governing equations are supplemented by initial conditions and boundary conditions on the cluster-macro void interface T js. Denote N the unit normal exterior to Cls. continuity of mass, concentrations, streaming potentials, total flux of the species and the normal component of the stress tensor give (where tpf = (RTtp /F))... 

For an element in equilibrium with no body forces, the equations of equilibrium were obtained by Lame and Clapeyron (1831). Consider the stresses in a cubic element in equilibrium as shown in Fig. 2.3. Denote 7y as a component of the stress tensor T acting on a plane whose normal is in the direction of e and the resulting force is in the direction of ej. In the Cartesian coordinates in Fig. 2.3, the total force on the pair of element surfaces whose normal vectors are in the direction of ex can be given by... 

T Total stress tensor V microslip, defined by Eq. (2.120) Poisson s ratio... 

Equations (5.139) to (5.142) are the basic equations for a gas-solid flow. More detailed information on both the fluid-particle interacting force Fa and the total stresses T and Tp must be specified before these equations can be solved. One approach to formulate the fluid-particle interacting force FA is to decompose the total stress into a component E representing the macroscopic variations in the fluid stress tensor on a scale that is large compared to the particle spacing, and a component e representing the effect of detailed variations of the point stress tensor as the fluid flows around the particle [Anderson and... 

Y Scalar quantity, defined by Eq. (5.176) er Total stress tensor of the fluid phase... 

If the segments are dissolved within a solvent of viscosity, the total stress tensor is... 

Turning back to Eq. 2.5-6, the surface forces Fs can now be expressed in terms of the total stress tensor n as follows ... 

The symbols used follow the recent recommendations of the Society of Rheology SI units are used. We follow the stress tensor convention used by Bird et al., namely, n = P6 + x, where n is the total stress tensor, P is the pressure, and x is that part of the stress tensor that vanishes when no flow occurs both P and x, are positive under compression. 

Applying the Divergence theorem to the volume integral, and adding an osmotic pressure IT to account for variations in ion concentration, yields the total force in terms of a stress tensor consisting of both osmotic and electrostatic components ... 

Even this simple hydrostatic formula clarifies the nature of the surface tension. The concentration variation within the interfacial region leads to a nonuniform stress tensor. The neglect of this nonuniformity gives rise to the conventional description of bulk phases. The iterative subtractive procedure demanded by the convergent expression given by Equation 5 corrects for this oversimplification at the boundary of the phases and yields an asymptotic correction (2) to the free energy of the total system in terms of its geometrical properties. 

In equations (32-33), 36 denotes the limiting surface of the complementary domain Bg = Sq u Sj u S2 (Fig 12). and o are the total stress tensor and vector, respectively, n is the outward unit vector normal to the surface. Sq and S2 are upstream and downstream surfaces limiting the flow domain imder consideration, perpendicular to the z-axis. In the cases under consideration (ducts involving symmetries), it can be shown that equations (32-33) reduce into one scalar equation [55] ... 

This scaling law, Eq. (9-48), implies that all components of the stress tensor are linear in the shear rate. Consider for example, a constant-shear-rate experiment. At steady state, not only is the shear stress predicted to be proportional to the shear rate, but so also is the first normal stress difference N This prediction has been nicely confirmed in recent experiments by Takahashi et al. (1994), who studied mixtures of silicon oil and hydrocarbon-formaldehyde resin. Both these fluids are Newtonian, and have the same viscosity, around 10 Pa s. Figure 9-18 shows that both the shear stress o and the first normal stress difference N = shear rate, so that the shear viscosity rj = aly and the so-called normal viscosity rjn = N /y are constants. The first normal stress difference in this mixture must be attributed entirely to the presence of interfaces, since the individual liquids in the mixture have no measurable normal stresses. A portion of the shear stress also comes from the interfacial stress. Figure 9-19 shows that the shear and normal viscosities are both maximized at a component ratio of roughly 50 50. At this component ratio, the interfacial term accounts for roughly half the total shear stress. 

The total stress tensor in the Leslie-Ericksen theory is the sum of the viscous stress of Eq. (10-10), an isotropic pressure, and the Frank distortional stress, given by... 

From eqn (6.30) it is clear that the virial of the electronic forces, which is the electronic potential energy, is totally determined by the stress tensor a and hence by the one-electron density matrix. The atomic statement of the virial theorem provides the basis for the definition of the energy of an atom in a molecule, as is discussed in the sections following Section 6.2.2. 

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