Oldroyd derivative

Shephard, D. S., Maschmeyer, T., Sankar, G., Thomas, J. M., Ozkaya, D., Johnson, B. F. G., Raja, R., Oldroyd, R. D. and Bell, R. G. Preparation, characterization and performance of encapsulated copper-ruthenium bimetallic catalysts derived from molecular cluster carbonyl precursors, Chem. Eur. J., 1998, 4, 1214-1224. 

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 ... 

Oldroyd (1953, 1955) derived expressions for the linear viscoelasticity of suspensions of one Newtonian fluid in another. By using an effective-medium approach, he was able to relax the requirement of diluteness. For an ordinary interface whose interfacial tension r remains constant during the deformation, Oldroyd s result gives the following for the complex modulus G = G + iG" ... 

Morcos SK, Oldroyd S, Haylor J Effect of radiographic contrast media on endothelium derived nitric oxide-dependent renal vasodilatation. Br J Radiol 70 154-9,1997... 

We note in passing that the first three terms in the last equation are closely related to the convective derivative of a contravariant tensor [see, for example, Oldroyd (59), Lodge ((46), Eq. 12.58), Fredrickson (30)],... 

Nearly a decade later, Oldroyd [1953, 1955] proposed a constitutive model similar to that of Frohlich and Sack, vahd at small deformations. The model considered low concentration of monodispersed drops of one Newtonian hquid in another. The interfacial tension and the viscoelastic properties of the interfacial film were incorporated by means of convected derivatives. The model provided the following relation for the complex modulus ... 

Parameter P governs the eontribution of the Maxwell element to effective viscosity, T, (Newtonian viscosity of the solution). Equation [7.2.26] is similar to the Oldroyd-type equation [7.2.15] with the only difference that in the former the upper convective derivative is used to account for nonlinear effects instead of partial derivative, d/dt. 

A and are phenomenological coefficients characteristic of the fluid, not the turbulence. The time derivative in Eq. (3) is the upper convected derivative of Oldroyd (7). In what follows,.A.and.. are assumed to be independent of the... 

Historically, Einstein was the first to derive an expression for the viscosity of hard spheres in a dilute suspension [4]. Later, Oldroyd [5] considered the case where the spheres are themselves liquid. Taylor extended the study to a system where the suspending medium, as well as the dispersed spheres, are Newtonian liquids [6,7]. It was observed that when the radius of the drop is great enough or the rate of distortion is high, the drop breaks up. Taylor derived the following two equations ... 

The above equation is generalized to three dimensions by replacing the stress component T by the stress matrix and the strain component 7 by the strain matrix. This procedure works well as long as one confines oneself to small strains. For large strains, the time derivative of the stress requires special treatment to ensure that the principle of material objectivity(78) is not violated. This principle requires that the response of a material not depend on the position or motion of the observer. It turns out that one can construct several different time derivatives all of which satisfy this requirement and also reduce to the ordinary time derivative for infinitesimal strains. By experience over many years, it has been found (see Chapter 3 of Reference 79) that the Oldroyd contravariant derivative also called the codeformational derivative or the upper convected derivative, gives the most realistic results. This derivative can be written in Cartesian coordinates as(79)... 

The VOF-code has been extended to simulate viscoelastic flows. In the current investigation, the Oldroyd-B model is employed which can be derived either from the dilute solution theory with bead-spring dumbbell model or from the stress-... 

The above observed phenomenon can be explained by reviewing the derivation of the Oldroyd-B constitutive model. As shown in Fig. 1.27, the polymer in the fluid is modeled as a dumbbell. The stiffness k of the Hookean spring which coimects the two beads in the dumbbell is related to the fluid relaxation time by... 

Using the Oldroyd determination of derivatives In the law (1.1) and keeping nonlinear terms In the balance equations of mass and momentum, one can get the system of dynamic equations for the medium with Internal oscillators. 

Here, b/br is called the convected derivative due to Oldroyd (1950), and it is the fixed coordinate equivalent of the material derivative of a second-order tensor referred to in convected coordinates. The physical interpretation of the right-hand side of Eq. (2.104) may be given as follows. The first two terms represent the derivative of tensor a j with time, with the fixed coordinate held constant (i.e., Da /Dr), which may be considered as the time rate of change as seen by an observer in a fixed coordinate system. The third and fourth terms represent the stretching and rotational motions of a material element referred to in a fixed coordinate system. This is because the velocity gradient dv fdx (or the velocity gradient tensor L defined by Eq. (2.59)) may be considered as a sum of the rate of pure stretching and the material derivative of the finite rotation. For this reason, the convected derivative is sometimes referred to as the codeformational derivative (Bird et al. 1987). 

However, Eq. (3.3) is valid only for extremely small strain rates because the spring and dashpot mechanical model is based on the premise that the Hookean material is subjected to an infinitesimally small displacement gradient. In order to overcome this limitation, Oldroyd (1950) proposed a generalization of Eq. (3.3) by replacing the partial derivative d/dt with the convected derivative b/bt (see Chapter 2), yielding... 

In this section definitions are given of the main kinematic tensors (see ref. 5, chapter 9) that are needed for the continuum mechanics description of viscoelastic materials as well as for the presentation of kinetic theory results. Some of these tensors are defined naturally in terms of the velocity field of the material, whereas others are defined easily in terms of the displacement functions that describe the motion of fluid particles. Inasmuch as the velocity field and the displacement functions are themselves interrelated, it is possible to interrelate the two groups of kinematic tensors. Here the emphasis is on working definitions of the kinematic tensors and not on their derivation from the motion of a convected coordinate system, which is a standard starting point for the discussion of continuum mechanics an important basic reference for the kinematics and dynamics of continuous media is a paper by Oldroyd. ... 

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