Oldroyd model

The three constant Oldroyd model is a nonlinear constitutive equation of the differential corrotational type, such as the Zaremba-Fromm-Dewitt (ZFD) fluid (Eq. 3.3-11). [For details, see R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Second Edition, Vol. 1, Wiley, New York, 1987, Table 7.3-2.]... 

Solving the previous set of equations, especially with realistic boundary conditions, is a formidable task and a lot of issues are still unanswered. This is not surprising because of the complexity of the equations, and because of their recent derivation, around 1950 for the first nonlinear models, the Oldroyd models. On the other hand, the mathematical theory for the Euler and the Navier-Stokes equations for incompressible Newtonian fluids is still not complete though these equations were derived in 1755 md 1821 respectively ... 

Remark 4.8 The results of Theorem 4.3 depend crucially on the model (the Oldroyd model). It would be interesting to know what happens for one dimensional flows of general differential models with a Newtonian contribution. 

Similar results can be obtained for Couette or Poiseuille flows of several fluids in parallel layers these flows are important in particular in the modelling of coextrusion experiments. Le Meur [50] has studied the existence, uniqueness and nonlinear stability with respect to one dimensional perturbations of such flows. The behaviour of each fluid is governed by an Oldroyd model such as (16)-(17), where the nondimensional numbers Re and We are defined locally in each fluid. On the rigid top or bottom walls, the velocity is given—zero on both walls for Poiseuille flow, and zero or one depending on the wall for Couette flow. The interface conditions on the given interfaces are... 

Following [47] we restrict now the study of stability to Oldroyd models (where di = 0). It is easy to check that the steady Couette flow, solution of the steady equations corresponding to system (16)-(17), is given by... 

The situation for the plane Poiseuille flow for Oldroyd models is not as simple, as shown by the following result. 

Three-constant Oldroyd model for viscoelastic fluids. Phys. Fluids. 5,... 

Kaloni used Oldroyd model, Schtimmer a fourth order fluid model, while Wissler a nonhnear Maxwell model Employing the perturbation method, the authors observed that the inclusion of second-order perturbation terms (which bring in the non-Newtonian effects) predicted velocity profiles with superimposed secondary circulation patterns. 

Thus, the Oldroyd model represents a special case of the general hereditary model [7.2.10] with appropriate choice of parameters. Usually the maximum relaxation time in the spectrum is taken for Ai in equation [7.2.15] and therefore it can be used for quantitative description and estimates of relaxation effects in non-steady flows of polymeric systems. 

Kaloni used the Oldroyd model and Schiimmer a fourth- Kaloni 1965 Schiimmer 1967 ... 

The dynamic test data in the kHz region of ionic emulsions could be described using either model. The emulsion elasticity originates in the interphase deformation. For non-ionic emulsions, only one relaxation time was observed. The data were interpreted in terms of the second Oldroyd model in which the interfacial tension is more important than the viscoelasticity of the interphase. The steady state viscosities of both ionic and non-ionic systems at the volume fraction < 0.2 were found to follow Simha s equation (Eq. (2.8)). 

If, as shown in Table 1, simple postulates are made for the anisotropic tensors (case I is due to Giesekus ), then a variety of constitutive equations can be obtained, including some which are special cases of the Oldroyd model. 

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