Axial flow deformations

Time evolution of the molecular orientation under axial flow deformation can be computed from (11) using the initial condition Aj(t = 0) = 1. The analytical and numerical solutions calculated at fixed elongation rates are discussed in the earlier papers [12,13]. The chain elongation coefficients Aj(f) satisfy the condition of affine molecular deformation, = X j(t) h j)o,... 

When two gas streams collide, the initial velocity profiles characteristic of a free flow deform in the vicinity of the impingement plane and additional components of velocity (radial, axial, or circumferential, depending on the impinging streams configuration) appear as a result of this deformation (Figures 21.28 and 21.29). 

If a fluid can be considered incompressible, the first principal invariant of the rate of deformation tensor will be zero, Ij = 0. The third principal invariant vanishes in many simple flow situations, like axial flow in pipe, tangential flow between concentric cylinders, etc. In more general terms, the third invariant is zero in rectilinear... 

Most rheological measurements measure quantities associated with simple shear shear viscosity, primary and secondary normal stress differences. There are several test geometries and deformation modes, e.g. parallel-plate simple shear, torsion between parallel plates, torsion between a cone and a plate, rotation between two coaxial cylinders (Couette flow), and axial flow through a capillary (Poiseuille flow). The viscosity can be obtained by simultaneous measurement of the angular velocity of the plate (cylinder, cone) and the torque. The measurements can be carried out at different shear rates under steady-state conditions. A transient experiment is another option from which both y q and ]° can be obtained from creep data (constant stress) or stress relaxation experiment which is often measured after cessation of the steady-state flow (Fig. 6.10). 

Fig. 4.4-2. Causes of Taylor dispersion. In Taylor dispersion, fast diffusion unexpectedly produces little dispersion, and vice versa. The reasons for this are shown here. The initial solute pulse (a) is deformed by flow (b). In fast-flowing regions, diffusion occurs outward, and in the slow flow near the wall, diffusion occurs inward. Thus diffusion in the radial direction inhibits dispersion caused by axial flow (c).

The elongation viscosity defined by Equation (1.19) represents a uni-axial extension. Elongational flows based on biaxial extensions can also be considered. In an equi-biaxial extension the rate of deformation tensor is defined as... 

Finally we will consider the two flow fields that yield extensional deformation. The first is axial elongational in which the fluid is contained between two planar surfaces in relative motion along their planar normals. With the axial extension direction taken as z, one has nzz = e = —2nxx = —2xyy and... 

Fig. 11.9 Types of linear continuous-flow reactors (LCFRs). (a) Continuous plug flow reactor (CPFR) resembling a batch reactor (BR) with the axial distance z being equivalent to time spent in a BR. (b) A tabular flow reactor (TFR) with (tq) miscible thin disk of reactive component deformed and distributed (somewhat) by the shear field over the volume, and (b2) immiscible thin disk is deformed and stretched and broken up into droplets in a region of sufficiently high shear stresses, (c) SSE reactor with (cj) showing laminar distributive mixing of a miscible reactive component initially placed at z = 0 as a thin slab, stretched into a flat coiled strip at z L, and (c2) showing dispersive mixing of an immiscible reactive component initially placed at z — 0 as a thin slab, stretched and broken up into droplets at z — L.

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