Slow flow approximation

The IPM is a simple application of the slow-flow approximation to the pressure equation (2)... 

Though the stress at time t in memory fluids is expected to depend on the history of the deformation, the dependence is stronger for recent deformations than for ancient ones. In other words, these fluids exhibit fading memory (23). The slow flow and small deformation approximations have been used to establish constitutive equations for memory fluids. In the slow flow approximation (23), a sequence of deformation histories is assumed in which each history differs from a reference history in that the time scale is slowed by a... 

Since laminar flow itself occurs at low values of Re( = Dupl/x), the most likely situations are those characterized by low velocity (u) or high viscosity (p,), such as those involving the slow flow of polymers in extrusion reactors, or of blood in certain organs in animals. Even if not a close approximation in some cases, the predictable performance of an LFR may serve as a limiting model for actual performance. 

We will now find the RDT for several models of tubular reactors. We noted previously that the perfect PFTR cannot in fact exist because, if flow in a tube is sufficiently fast for turbulence (Rco > 2100), then turbulent eddies cause considerable axial dispersion, while if flow is slow enough for laminar flow, then the parabolic flow profile causes considerable deviation from plug flow. We stated previously that we would ignore this contradiction, but now we will see how these effects alter the conversion from the plug-flow approximation. 

The basic similarity solution for this ignition problem is derived from the slow flow (6,7J approximation, characterized by (1) flow velocities which are small compared to the speed of sound, and (2) an essentially constant pressure field. The energy and velocity equations may then be written as... 

A simplifying approximation often made in fluid mechanics, where the terms arising due to the inertia of fluid elements is neglected. This is justified if the Reynolds number is small, a situation that arises, for example, in the slow flow of viscous liquids such as when pouring honey over toast. 

Place 0.375 g ytterbium (III) trifluoromethanesulfonate hydrate catalyst (ytterbium triflate) into a 25-mL round-bottom flask. Add 10 mL of 1,2-dichloroethane solvent followed by 0.400 mL of concentrated nitric acid (automatic pipette). Add two boiling stones to the flask. To fhis solution, weigh out and add approximately 6 millimoles of the aromatic substrate. Connect the round-bottom flask to a reflux condenser and clamp it into place on a ring stand. Use a very slow flow of water through the condenser. With a hot plate, heat the mixture to reflux for 1 hour. 

First, consider a simpliflcation in which it is assumed that the contour length of the primitive chain remains at the equilibrium length L under the imposed deformation. This assumes an inextensible primitive chain and is seen as a reasonable approximation for slow flows or long times. Then, the deformation of the primitive chain is given by considering that the segment in the middle of the chain changes position affinely as... 

Beyond the few categories discussed above, it is difficult to work with the ISF in its full form. However, by either assuming slow flows or small deformations, it is possible to obtain approximate expressions in the form of expansions. When the ISF is expanded for slow flows one obtains the order fluids we discussed earlier. When we expand for small deformations, the ISF yields integral relations which account for the fading memory. At the 0th order we again have the hydrostatic pressure, but now at the 1st order we get the constitutive equation for linear viscoelasticity, viz. 

For slow flows in which d,y and its derivatives are assumed to be small, successive approximations of Eq. (3P.2) give... 

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