Convected derivative

Note that convected derivatives of the stress (and rate of strain) tensors appearing in the rheological relationships derived for non-Newtonian fluids will have different forms depending on whether covariant or contravariant components of these tensors are used. For example, the convected time derivatives of covariant and contravariant stress tensors are expressed as... 

We start by considering an arbitrary measurable10 one-point11 scalar function of the random fields U and 0 Q U, 0). Note that, based on this definition, Q is also a random field parameterized by x and t. For each realization of a turbulent flow, Q will be different, and we can define its expected value using the probability distribution for the ensemble of realizations.12 Nevertheless, the expected value of the convected derivative of Q can be expressed in terms of partial derivatives of the one-point joint velocity, composition PDF 13... 

In summary, due to the linear nature of the derivative operator, it is possible to express the expected value of a convected derivative of Q in terms of temporal and spatial derivatives of the one-point joint velocity, composition PDF. Two-point information about the random fields U and

expected value and derivative operators commute, and does not appear in the final expression (i.e., (6.9)). 

The expected value of the convected derivative of Q can also be written in a second independent form starting with15... 

Whereas inviscid flow is a useful reference point for high Reynolds number flows, a different simplification known as the creeping flow approximation applies at very low Re. From Eq. (1-3), the terms on the right-hand side dominate as Re 0, so that the convective derivative may be neglected. In dimensional... 

With the convective derivatives eliminated and the properties constant, the thermal-energy equation is completely decoupled from the system. Moreover the energy equation is a simple, linear, parabolic, partial differential equation. 

The explicit time derivative is zero because the problem is in steady state. Parallel flow v = 0 and w = 0 requires the second two convective derivatives to vanish. The u(du/dz) term vanishes since (du/dz) = 0. The u velocity enters the z face, but since no flow can enter from any other face, there is no way for u to change—it must flow out the opposite z face with the same velocity. Reference to Fig. 4.3 helps visualize this concept. 

The stability of a FD representation deals with the behavior of the truncation error as the calculation proceeds in time or marches in space, typically, transient problems, and problems with convection-convection derivatives. A stable FD scheme will not allow the errors to grow as the solution proceeds in time or space. The issue of stability for transient problems will be analyzed in depth later in this chapter. 

Note that Pg > 2 is critical, because the solution presents a sign change, which means the solution becomes unstable (see Figure 8.17). The root of the problem is explained by the info-travel concept. To generate the difference equation (eqn. (8.66)) we used a central finite difference for the convective derivative, which is incorrect, because the information of the convective term cannot travel in the upstream direction, but rather travels with the velocity ux. This means that to generate the FD equation of a convective term, we only take points that are up-stream from the node under consideration. This concept is usually referred to as up-winding technique. For low Pe the solution is stable because diffusion controls and the information comes from all directions. 

A. Lagrangian Framework. An ideal subgrid model should be constructed on a Lagrangian hydrodynamics framework moving with the macroscopic flow. This requirement reduces purely numerical diffusion to zero so that realistic turbulence and molecular mixing phenomena will not be masked by non-physical numerical smoothing. This requirement also removes the possibility of masking purely local fluctuations by truncation errors from the numerical representation of macroscopic convective derivatives. 

If the domain is deforming, x is a function of time and its convective derivative is ... 

The right-hand side of the equation shows the derivatives of the property inside the control wo m, d pd )tdt, and the property on the boundary of the control volume, / (ppv hdA. Because f ppv hdA belongs in the same equation as d yppd )ldt, it is also loosely called a derivative and, because this particular derivative is conveyed by a velocity, it is called a convective derivative. In other words, the convective derivative is conveyed from outside of the control volume into the inside of the control volume and vice versa. These derivatives describe what an... 

Now, let us determine the expression for h dA. This is the convective derivative and it only applies to the boundary. In the case of the reactor, there are two portions of this boundary the inlet boundary and the outlet boundary. Let the inflow to the reactor be Q this will also be the outflow. Note that Q comes from V h dA and, because the velocity vector and the unit vector are in opposite directions at the inlet, Q will be negative at the inlet. At the outlet, because the two vectors are in the same directions, Q will be positive. The concentration at the outlet will be the same as the concentration inside the tank, which is [A]. Thus, letting the concentration at the inflow be [X ]... 

Now, let us determine the expression for q. This expression can be readily derived from a material balance using the Reynolds transport theorem. This theorem is derived in any good book on fluid mechanics and will not be derived here. The derivation is, however, discussed in the chapter titled Background Chemistry and Fluid Mechanics. It is important that the reader acquire a good grasp of this theorem as it is very fundamental in understanding the differential form of the material balance equation. This theorem states that the total derivative of a dependent variable is equal to the partial derivative of the variable plus its convective derivative. In terms of the deposition of the material q onto the filter bed, the total derivative is... 

In the previous equation, the total derivative is also called Lagrangian derivative, material derivative, substantive derivative, or comoving derivative. The combination of the partial derivative and the convective derivative is also called the Eulerian derivative. Again, it is very important that this equation be thoroughly understood. It is to be noted that in the enviromnental engineering literature, many authors confuse the difference between the total derivative and the partial derivative. Some authors use the partial derivative instead of the total derivative and vise versa. As shown by the previous equation, there is a big difference between the total derivative and the partial derivative. If this difference is not carefully observed, any equation written that uses one derivative instead of the other is conceptually wrong this... 

The various changes that may be carried out can be either on the convected derivative or in the right term of equation (35) or both these imply the removal of some assmnptions of the initial model. Such a possible modification, that was claimed to give a correct description of the essential phenomena of the nonUnear viscoelastic behaviour of polymer melts, is that proposed by Phan Thien and Tanner [44-46] involving the use of a special convected derivative and special kinetics of the junction. 

The first kind of modification to the UCM model that may be conceivable is that of the convected derivative. This leads one to consider that the motion of the network junctions is no more that of the continuum and thus, the afiine assumption of the Lodge model is removed. Among the various possibilities, Phan Thien and Tanner suggested the use of the (Jordon-Schowalter derivative [47], which is a linear combination of the upper- and lower-convected derivatives, instead of the upper-convected derivative ... 

It requires that the principal stress axes should coincide with the principal strain axes. This rrile has been experimentally checked hy many authors [24, 56] Actually, the use of the Gordon-Schowalter derivative involves the violation of the Lodge - Meissner rule, indeed when a equals 0 or 2, either the upper or the lower convected derivatives implies that the relationship is respected. In the general case, the double value of the slip parameter is a natural way to accommodate this rule. 

In its general form, the Phan Thien Tanner equation includes two different contributions of strain to the loss of network junctions, through the use of a particular convected derivative which materializes some slip of the junctions and through the use of stress-dependent rates of creation and destruction of junctions. The use of the Gordon-Schowalter derivative brings some improvement in shear and a second normal stress is predicted, whereas the... 

One may try to avoid the problem by the use of the upper-convected derivative, which ensures the coincidence of the principal axes of stress and strain. But doing that, it appears that any kinetics based on the stress amplitude is improper, since materials which exhibits thickening behaviour in elongation are, to the contrary, shear-thinning. Consequently no unique dependence can be expected for these two kinematics. The determination of a single set of parameters in various flows in then bound to be a compromise. 

R.G.Larson, Convected derivatives for differential constitutive equations, J. of Non-Newt. Fluid Mech. 2i (1987), 331-342. 

C.J.S.Petrie, Measures of deformation and convected derivatives, J. of Non-Newt.Fluid Mech. 5 (1979), 147-176. 

In [62] Renardy proves the linear stability of Couette flow of an upper-convected Maxwell fluid under the 2issumption of creeping flow. This extends a result of Gorodtsov and Leonov [63], who showed that the eigenvalues have negative real parts (I. e., condition (S3) holds). That result, however, does not allow any claim of stability for non-zero Reynolds number, however small. Also it uses in a crucial way the specific form of the upper-convected derivative in the upper-convected Maxwell model, aind does not generalize so far to other Maxwell-type models. 

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