Stability Couette flow,

The breakup or bursting of liquid droplets suspended in liquids undergoing shear flow has been studied and observed by many researchers beginning with the classic work of G. I. Taylor in the 1930s. For low viscosity drops, two mechanisms of breakup were identified at critical capillary number values. In the first one, the pointed droplet ends release a stream of smaller droplets termed tip streaming whereas, in the second mechanism the drop breaks into two main fragments and one or more satellite droplets. Strictly inviscid droplets such as gas bubbles were found to be stable at all conditions. It must be recalled, however, that gas bubbles are compressible and soluble, and this may play a role in the relief of hydrodynamic instabilities. The relative stability of gas bubbles in shear flow was confirmed experimentally by Canedo et al. (36). They could stretch a bubble all around the cylinder in a Couette flow apparatus without any signs of breakup. Of course, in a real devolatilizer, the flow is not a steady simple shear flow and bubble breakup is more likely to take place. 

Similar results can be obtained for Couette or Poiseuille flows of several fluids in parallel layers these flows are important in particular in the modelling of coextrusion experiments. Le Meur [50] has studied the existence, uniqueness and nonlinear stability with respect to one dimensional perturbations of such flows. The behaviour of each fluid is governed by an Oldroyd model such as (16)-(17), where the nondimensional numbers Re and We are defined locally in each fluid. On the rigid top or bottom walls, the velocity is given—zero on both walls for Poiseuille flow, and zero or one depending on the wall for Couette flow. The interface conditions on the given interfaces are... 

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. 

Following [47] we restrict now the study of stability to Oldroyd models (where di = 0). It is easy to check that the steady Couette flow, solution of the steady equations corresponding to system (16)-(17), is given by... 

Concerning the Liapunov (nonlinear) stability of the Couette flow under one dimensional perturbations, we have for instance the following result [47]. 

We go back to the linear stability of Couette flow of an Oldroyd fluid. 

Nonlinear stability results for viscoelastic fluids are very few. They essentially concern Jeffreys-type fluids. We have already mentioned those of [47] for the one-dimensional stability of Couette flows (see Section 5.3), and for the stability of flows of Jeffreys-type fluids which are small perturbations of the rest state (see Corollary 4.1). 

Renardy and Renardy [66,73] have investigated the stability of plane Couette flows for Maxwell-type models involving the derivative (2). The flow lies between parallel plates at a = 0 euid x = 1, which are moving in the j/-direction with velocities 1, such as in Figure 6. 

Many studies have been devoted to the Taylor-Couette problem (flow between two concentric cylinders with radii R and R2, Ri < R2, of infinite length, and rotating with angular velocities fij and 02 repectively). For instance Zielinska and Demay [74] consider the general Maxwell models with —1 <0 < 1. They show that the axisymmetric steady flow (the Couette flow) does not exist for 2dl values of parameters where the steady state exists moreover all models, except for a very close to —1, predict stabilization of the Couette flow in the spectral sense, for small enough values of the Weissenberg number. (See also [55].)... 

M. Renardy, A rigorous stability proof for plane Couette flow of an uppe-convected Maxwell fluid at zero Reynolds number, Eur. J. Mech. B, 11 (1992) 511-516. 

V.A. Romanov, Stability of plane-parallel Couette flow, Funct. Anal. AppL 7 (1993), 137-146. 

M. Renardy and Y. Renardy, Linear stability of plane Couette flow of an upper-convected Maxwell fluid, J. Non-Newtonian Fluid Mech., 22 (1986) 23-33. 

C.H. Li, Stability of two superposed elastoviscous liquids in plane Couette flow, Phys. Fluids, 12 (1969) 531-538. 

Park, Y. (2006). Development and optimization of novel emulsion liquid membranes stabilized by non-Newtonian conversion in Taylor-Couette flow for extraction of selected organic and metallic contaminants. Ph.D. thesis, Georgia Institute of Technology, Atlanta, GA, USA. 

Next we turn to the stability of Couette flow for parallel rotating cylinders. This is an important flow for various applications, and, though it is a shear flow, the stability is dominated by the centrifugal forces that arise because of centripetal acceleration. This problem is also an important contrast with the first two examples because it is a case in which the flow can actually be stabilized by viscous effects. We first consider the classic case of an inviscid fluid, which leads to the well-known criteria of Rayleigh for the stability of an inviscid fluid. We then analyze the role of viscosity for the case of a narrow gap in which analytic results can be obtained. We show that the flow is stabilized by viscous diffusion effects up to a critical value of the Reynolds number for the problem (here known as the Taylor number). 

For Taylor numbers exceeding Tc, the flow develops a secondary flow pattern in which ur and uz are both nonozero. A sketch of the stability criteria given by (3-86) is shown in Fig. 3 8. The reader who is interested in a detailed description of the stability analysis that leads to the criterion (3-86) is encouraged to consult Chap. 12 or one of the standard textbooks on hydrodynamic stability theory (see Chandrashekhar [1992] for a particularly lucid discussion of the instability of Couette flows).12... 

Figure 3-8. Stability diagram for Couette flow with 0 < . 2 / -2 i < 1, plotted as Taylor number T versus ih/ The flow is stable for T < Tc and unstable for T > Tc.

In this section, we study the stability of Couette flow between a pair of concentric rotating cylinders. The steady flow was considered earlier in Section C of Chap. 3. 

In the case of Couette flow, the base flow whose stability we wish to study, was shown in Chap. 3, Section C, to be... 

To analyze the linear stability of a Couette flow, we begin with the Navier Stokes and continuity equations in a cylindrical coordinate system. The frill equations in dimensional form can be found in Appendix A. We wish to consider the fate of an arbitrary infinitesimal disturbance to the base flow and pressure distributions (12 114) and (12 116). Hence we consider a linear perturbation of the form... 

If 22 = 0, for example, the Rayleigh criterion indicates that Couette flow is unstable for any 12]. On the other hand, the approximate stability criterion (12-158) for a viscous fluid says that the system will remain stable for... 

In the case of non-Newtonian polymer solutions (and narrow gaps) the stability limit increases. In situations where the outer cylinder is rotating, stable Couette flow may be maintained rmtil the onset of turbulence at a Reynolds nmnber. Re, of ca. 50 000 where Re = pQR2(R2 — R )/p- [Van Wazer et al, 1963],... 

Whether the string phase corresponds to a real situation or is an artifact of the simulations due to the use of an inaccurate expression for the secondary flow in regimes where the hydrodynamic stability of planar Couette flow is lost is difficult to ascertain at the present time [205]. Experiments on colloidal suspensions have not provided a clear answer, though at moderate strain rates structural behavior similar to that found in the simulations is observed [222]. 

The critical Taylor number T for the onset of Taylor vortices can be predicted by examining the stability of snail amplitude disturbances when superimposed on the basic Couette flow. The use of this linear stability analysis for concentric cylinders has been extensively reviewed by Chandrasekhar (1) and Stuart (2). All such analyses assune that the cylinders are infinitely long. In addition to T they predict an initial Taylor vortex celf axial length, . ... 

Hinvi, LA., Monwanou, A.V, Orou, J.B.C., 2013a. Linear stability analysis of hydromagnetic couette flow with small injection/suction through the modified Orr-Sommerfeld equation. arXiv 1308.5530 [physics.flu-dyn]. 

In SSF or Couette flow, it is not possible to break a drop if the viscosity ratio is greater than about 3. The deformed drop shape is stabilized by internal circulation. It has been concluded that extension is more effective than shear at breaking drops. With respect to practical flows, it is important not to interpret this statement too literally since in practical applications, a single steady shear gradient rarely exists. Bear in mind that the physical definitions of shear and extension depend on the environment seen by the drop along its trajectory, while the mathematical definitions are related to the choice of coordinate system. 

RE. Cladis and S. Torza, Stability of Nematic Liquid Crystals in Couette Flow, Phys. Rev. Lett, 35, 1283-1286 (1975). 

Linear stability theory results match quite well with controlled laboratory experiment for thermal and centrifugal instabilities. But, instabilities dictated by shear force do not match so well, e.g. linear stability theory applied to plane Poiseuille flow gives a critical Reynolds number of 5772, while experimentally such flows have been observed to become turbulent even at Re = 1000- as shown in Davies and White (1928). Couette and pipe flows are also found to be linearly stable for all Reynolds numbers, the former was found to suffer transition in a computational exercise at Re = 350 (Lundbladh Johansson, 1991) and the latter found to be unstable in experiments for Re > 1950. Interestingly, according to Trefethen et al. (1993) the other example for which linear analysis fails include to a lesser degree, Blasius boundary layer flow. This is the flow which many cite as the success story of linear stability theory. 

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