Tensor extra stress

In the simplest case of Newtonian fluids (linear Stokesian fluids) the extra stress tensor is expressed, using a constant fluid viscosity p, as... 

A frequently used example of Oldroyd-type constitutive equations is the Oldroyd-B model. The Oldroyd-B model can be thought of as a description of the constitutive behaviour of a fluid made by the dissolution of a (UCM) fluid in a Newtonian solvent . Here, the parameter A, called the retardation time is de.fined as A = A (r s/(ri + s), where 7]s is the viscosity of the solvent. Hence the extra stress tensor in the Oldroyd-B model is made up of Maxwell and solvent contributions. The Oldroyd-B constitutive equation is written as... 

The extra-stress tensor t = (Ty) is assumed to satisfy an equation of the form... 

The three-field formulation should reduce to a convenient approximation of the Stokes problem when applied to a Newtonian flow. Hence a second inf-sup condition is necessary to obtain stability. If the approximation (Tv)h of the extra-stress tensor is continuous, this supplementary condition can be satisfied by using a sufficient number of interior nodes in each element. On the contrary if this approximation is discontinuous, this can be done by imposing that the derivatives DUh of the approximated velocity field are in the space of (Tv)h- Various possible choices concerning the satisfaction of the inf-sup condition and the introduction of upwinding have been explored since 1987. In the following we will recall the basic steps (see [10], [24] and [38] for details). 

This difficulty can be overcome by the use of a viscoelastic model limiting the effect of the singularity in the transport equations. In the Modified Upper Convected Maxwell (MUCM) proposed by Apelian et al. (see [1]), the relaxation time X is a function of the trace of the deviatoric part of the extra stress tensor ... 

For the PIT model with a single relaxation time, we adopt the general form for the extra-stress tensor ... 

By the convention used here, a positive pressure p is equivalent to a negative stress. Stresses that exist in addition to a hydrostatic pressure are expressed as the extra stress tensor, a. Thus... 

Leslie recognized from early experiments that the anisotropy of the materials calls for multiple viscosity coefficients corresponding to different orientation of the LC relative to the flow. Combining this idea with the Ericksen theory leads to the Leslie-Ericksen (LE) theory, which comprises two elements one describing the evolution of n(r) in a flow field, and the other prescribing an extra stress tensor due to the evolving (r) field. 

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. 

Yi = pjp is the mass fraction of species i and j, is its diffusirm mass flux, where the subscript /, i = 1,..N, denotes species i. The specific enthalpy is h = e + pip, where e denotes the specific internal energy and p is the pressure q is the mass specific heat flux, a is the extra stress tensor and d/dt denotes the material or substantial derivative. 

Species mass concentration Tensor in heat-flux vector expression Species contribution to extra stress tensor Potential energy for all molecules in liquid Tensor used in heat-flux expression Potential energy for single molecule Potential energy for single molecule in external field... 

Here a.. are components of the extra stress tensor referred to the cartesian axes in Figure 1 and a.. > 0 for tensile stresses. Three material functions can also be defined for shear flow ... 

P is an extra stress tensor and Uj, b, rj, and c are functions of tr as indicated. These formulations are much more arbitrary in character than the approach of Eqs. (46) and (47). 

The choice of constitutive equations depends on the particular problems investigated. If the flow phenomena are dominated by the shear-rate dependent viscosity, it makes sense to use inelastic, or generalized Newtonian fluids, for which the extra stress tensor is proportional to the rate of deformation in the form... 

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