Hyperbolic flow

Fig. 37. Strain rate distribution along the centerline in a 2-dimensional hyperbolic flow (the flow geometry is shown as an insert). The solid curve, redrawn according to ref. 131, corresponds to a viscoelastic fluid (the spike at x = 2 is a calculation artefact) the dotted curve is calculated with POLYFLOW for a Newtonian liquid...

Rumscheidt, F. D., and Mason, S. G., Particle motions in sheared suspensions. XII. Deformation and burst of fluid drops in shear and hyperbolic flow. J. ColloidScL 16,238-261 (1961). Rwei, S. P., Manas-Zloczower, I., and Feke, D. L., Observation of carbon black agglomerate dispersion in simple shear flows. Polym. Eng. ScL 30, 701-706 (1990). 

Theoretical predictions relating to the orientation and deformation of fluid particles in shear and hyperbolic flow fields are restricted to low Reynolds numbers and small deformations (B7, C8, T3, TIO). The fluid particle may be considered initially spherical with radius ciq. If the surrounding fluid is initially at rest, but at time t = 0, the fluid is impulsively given a constant velocity gradient G, the particle undergoes damped shape oscillations, finally deforming into an ellipsoid (C8, TIO) with axes in the ratio where... 

For the limiting cases, N 0 and k oo, Eqs. (12-47) and (12-49) reduce to equations derived by Taylor (T2, T3) for fluid particles in steady-state shear or hyperbolic flows, i.e.,... 

F. D. Rumscheidt and S. G. Mason, Particle Motion in Sheared Suspensions. XII. Deformation and Burst of Fluid Drops in Shear and Hyperbolic Flow, J. Colloid Sci., 16, 238-261 (1961). 

A similar analysis can be performed for three-dimensional lattices subjected to the same flow. The corresponding maximum concentration curve in three dimensions is shown in Fig. 3 as a function of the flow parameter X. This curve displays a discontinuous dependence on X in the neighborhood of X = 0, revealing a very special feature of simple shear flow. The saw-tooth property characterizing hyperbolic flows (X > 0) is derived from the best estimates... 

Finally, the self-reproducibility in time of the lattice configuration (for two-dimensional flows) must be addressed. In the elliptic streamline region (A < 0), the lattice necessarily replicates itself periodically in time owing to closure of the streamlines. For hyperbolic flows (A > 0), the lattice is not generally reproduced however, in connection with research on spatially periodic models of foams (Aubert et al., 1986 Kraynik, 1988), Kraynik and Hansen (1986, 1987) found a finite set of reproducible hexagonal lattices for the extensional flow case A = 1. It is not clear how this unique discovery can be extended, if at all. 

FIGURE 11.7 Two types of laminar flow, and the effect on deformation and breakup of drops at increasing velocity gradient ( ). The flow is two-dimensional, i.e., it does not vary in the z-dircction. More precisely, the flow type in (b) is plane hyperbolic flow. ... 

Note The part after the second equal sign in Eq. [11.6] only applies in simple shear flow. In plane hyperbolic flow, 2t]c should be inserted in the denominator see below. 

The shear stress acting on the drop is given by r c times the velocity gradient u (d)/d. The flow type is probably close to plane hyperbolic flow, and Figure 11.8 shows that in this case Wecr will not strongly depend on the viscosity... 

The first observation of shear-induced increase of the LCST was reported for PS/PVME by Mazich and Carr [1983]. The authors concluded that shear stress can enhance miscibility by 2-7°C. Larger effects, AT < 12°C, were reported for the same system in hyperbolic flow [Katsaros et al., 1986]. In a planar extensional flow at 8 = 0.012 - 26 s the phase separated PS/PVME was homogenized at temperatures 3 to 6°C above... 

Figure 9.5. Drop deformability in a planar hyperbolic flow as a function of total strain. The dependence was calculated from Cox equation, Eq 9.11.

The above relations are valid for Newtonian systems undergoing small. Unear deformations, smaller than that, which would lead to a breakup. As Figure 9.5 indicates, in planar hyperbolic flow about 10 units of strain are required to get into an equihbrium. In simple shear, t = 25 of the reduced time scale is required [Elemans, 1989]. 

Hydroperoxide Hypalon Hyperbolic flow Hyperbolic interfaces Hyperbolic point... 

Hinze also discussed various well-defined flow forms and the types of droplet deformation associated with them. The flow patterns described are parallel flow, plane hyperbolic flow, rotating flow, axisymmetric hyperbolic flow, Couette flow, and irregular flow (turbulent). 

Rumscheidt, E D., and S. G. Mason. 1961. Particle motions in sheared suspensions XII. Deformation and burst of fluid drops in shear and hyperbolic flow. Journal of Colloid Science 16 238-261. 

Fig. 9.6 Drop deformability in a planar hyperbolic flow at three values of X and k. The dependencies were computed from the Cox equation (Eq. 9.11)...

The value of relaxation parameter, w, ranges from 0 to 2, being classified as a process of over-relaxation for 1 < w < 2, connmonly used for parabolic and hyperbolic flow problems, or as a sub-relaxation, for 0 < w < 1, which is often used in elliptic or oscillatory flows. The proper choice for the value of w, often w 2/3 for the case of elliptical oscillatory flows and w 4/3 for the parabolic/hyperbolic case, can reduce the computational cost needed to solve the system of discretized equations in one order of magnitude. 

J. E. Orlanski, A simple boundary condition for unbounded hyperbolic flows, J. Comput. Phy. 21, 251-269 (1976). 

The form taken by / (//a// c) depends on the nature of the flow. In laminar flow, the contributions of rotational and elongational components can be expressed via the value of a parameter ot, which ranges between 0 and 1. For simple shear a = 0, and for pure elongational (hyperbolic) flow a = 1 [43]. With low a values, Wecrit (now equivalent to the critical capillary number Caoit) passes through a minimum as increases. After the rninimum is reached, relatively small increases in... 

Figure 2.4 Critical Weber number for break-up of drops in various types of flow. Singledrop experiments in two-dimensional simple shear (a — 0), hyperbolic flow (a = l) and intermediate types/-as well as a theoretical result for axisymmetrical extensional flow (ASE)." The hatched area rrfers to apparent We values obtained in a colloid mill" ...

The definition of the homoclinic and heteroclinic points needs first the introduction of hyperbolic and elliptic points. A two-dimensional flow always consists of hyperbolic and/or elliptic points (Fig. 6.26). At the hyperbolic point the fluid moves toward it in one direction and away from it in another direction. At an elliptic point the fluid moves in closed pathlines. A periodic point is defined as the point at which a particle in a periodic flow returns after a number of periods. The number of periods defines also the order of the periodic point, as periodic point of period 1, 2, and so on. Note that the periodic elliptic points should be avoided should we want enhanced mixing. A point where the outflow of one hyperbolic point intersects the inflow of another hyperbolic flow is called transverse heteroclinic point. When the inflow and outflow refer to the same hyperbolic point, the point is called transverse homoclinic point. 

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