Disturbance axisymmetric

In laminar flow region, the breakup time tb and length L corresponding to a small axisymmetric disturbance 8() are expressed as [38]... 

Even when great care was taken to ensure that the liquid feed was introduced to the disc in an axisymmetric manner with the minimum disturbance, the smooth inner him always broke down into an array of spiral ripples, as shown in Figures 5 and 6. These spiral structures then broke down further until the wave pattern became utterly chaotic, provided that the disc was big enough. It is known that liquid him how over a surface is intrinsically unstable, and the phenomenon has been studied by several workers (3-7). It appears to be qualitatively equivalent to the breakdown of a smoke plume rising from a lighted cigarette, where a chaotic zone is generated about 20 cm above the source. The behavior can also be observed when a liquid him hows over a stationary surface such as a windowpane or a dam spillway. 

The same processes occur in three-dimensional flows. For a random array of fixed rigid spheres (see Koch and Brady, 1985 [340]), the axisymmetric velocity disturbance in the far wake of a test sphere tends to... 

Figure 7-3. A schematic representation of the proof that a spherical particle cannot undergo lateral migration in either 2-D or axisymmetric Poiseuille flow if the disturbance flow is a creeping flow. In (a) we suppose that the undisturbed flow moves from left to right and the sphere migrates inward with velocity u. Then, in the creeping-flow limit, if direction of the undisturbed flow is reversed, the signs of all velocities including that of the sphere would also have to be reversed, as shown in (b). Because the problems (a) and (b) are identical other than the direction of the flow through the channel or tube, we conclude that = 0.

We assume that the velocity and pressure fields are axisymmetric even in the disturbance fields. Although comparison with experiment shows this assumption to be valid for the linear perturbations that are unstable, it eventually breaks down far enough beyond the initial critical point in terms of the cylinder rotation rates. The corresponding linearized [i.e., 0(e)] equations of motion and continuity in dimensional form are... 

Problem 12-3. Capillary Instability for a Thread in a Second Immiscible Fluid. In this problem, we consider the effect on capillary instability if, instead of being surrounded by air, the thread of liquid is surrounded by a second viscous immiscible fluid that is assumed to be unbounded in the radial direction. Derive a condition from which you could in principle, calculate the growth-rate parameter for an axisymmetric disturbance, a = a(k, Re, 7.) where k is the axial wave number and 7. is the ratio of the external fluid viscosity to the viscosity of the liquid thread. This condition can be simplified if either Re I or the thread is inviscid (though viscous effects still remain in the outer fluid). Evaluate a for several k values in each of these two cases. What is the qualitative effect of the second viscous fluid For example, is the range of unstable k values changed Is the fastest-growing linear mode changed relative to the case of a thread in air ... 

It was shown from linear stability theory that a circular jet will become unstable to axisymmetric disturbances whose wavelength X = 2 irlk is... 

It, however, has to be noticed that this statement is valid provided that the fluctuations of the gas flow do not contribute to droplet coalescence, which can occur when the gas stream reaches a fuUy developed turbulent profile around the liquid jet breakup region. Above the critical We value of 40, Gafidn-Calvo pointed out that the sinuous non axisymmetric disturbances become apparent, coupled to the axisymmetric ones. It is also mentioned that increasing again the We number will lead to a nonlinear growth rate of the sinous disturbances, which will overcome the axissymmetric ones and produce polydisperse drops. ... 

A simple solution for the jet instability is that of an inviscid stationary jet (m = V = 0) subject to an axisymmetric disturbance (no perturbation in 9 direction). For this problem, the linearized equations are ... 

Rayleigh [5] used a periodic perturbation in both axial and angular direction of the following form R = a + cos kx cos mO, here, w = 1 is termed kink mode and m > 2 are termed flute modes. However, he showed that the linear instability analysis provides that the jet is stable for all angular disturbances, and it is only the axial disturbances that may be unstable. This is related to the fact that only the axisymmetric perturbations can reduce the surface energy. Therefore, we only consider the axial disturbances. 

Fig. 18.13 Spike behavior at the tip of an electrified column, (a) Cross-sectional view of a 2D jet [3] showing the effect of initial deformation and (b) side view of an axisymmetric jet [70] showing the effect of charging level. The jet is disturbed at ka= 1.3 and bla = 10 (Courtesy of Elsevier)...

A gradually-varied Hoek-Brown disturbance factor for analyzing an axisymmetrical cavern... 

ABSTRACT In this study the disturbance factor in the general Hoek-Brown (HB) criterion is considered to be a gradually-attenuated variable from the excavation surface to the deep surrounding rocks. The elasto-plastic analytical solution is formulated for an axisymmetrical cavern model in which there exist a supported pressure at the wall of tunnel and a far-field pressure at infinity. The presented analytical model can well reflect the disturbance of the HB rock mass triggered by drilling and blasting excavation. 

However, the T)-value is often fixed to be a constant in the existing analysis for the underground engineering projects in the HB rock mass (Chen Tonon 2011, Fraldi Guarracino 2010, Li et al. 2009, Park Kim 2006, Shen et al. 2010, Zhong et al. 2009, Zhou Li 2011). In this study D is treated as a variable. To formulate the elasto-plastic analysis solution for an axisymmetrical cavern, a linear function is chosen to quantitatively describe D. Compared with the elastic perfectly-plastic and elastic-brittle-plastic results, the present analysis can objectively reflect the excavation disturbance of the surrounding rocks. 

In absence of support disturbances, equilibrium becomes possible in the axisymmetric field of Fig. 1 where ions and electrons drift in antiparallel directions along

charge separation, also in presence of a poloidal electric field. 

Wave motion at a surface or interface can result from both gravitational and capillary forces. Lx)rd Rayleigh studied the break-up of a jet of liquid into a gas [20]. Capillary forces make liquid jets unstable when their length, L, exceeds their circumference. Axisymmetric disturbances of the surface grow in amplitude until the jet is pinched off (Figure 13). Rayleigh s theory correctly predicted the drop size and the dependence of L on surface tension. Subsequent work [21] for liquid/gas and liquid/liquid jets has shown that the breakup is dependent upon ... 

SHARIF, N.S. "An experimental study of the fluid dynamics of axisymmetric and non-axisymmetric disturbances in Taylor-Conette flow." rtiD. Thesis. Mech Eng. Dept., university of Leeds, 1986. 

Fig. 22.2 Laminar operating rotary atomizer during spray experiments with viscous PVP solution. The flow rate is 201/h. In both cases, laminar conditions of the open channel flow are present. The reshaping from the open channel flow into the cylindrical thread is visible close to the LamRot edge. In (a), 3000 rpm are applied and the threads break up by axisymmetric disturbances. Higher rotary speed of 4000 rpm (b) leads to a more irregular break up obviously influenced by the ambient gas [14]...

The authors then perform a linear stability analysis solving the dynamics resulting from small perturbations of the jet radius or charge density, or of the electrie field. These take expressions of the form, r=ro +where m indicates the growth rate of the instability, k is the disturbance wavenumber along the z direction, and the perturbation amplitude, Ar, is much smaller than yq. In this way, three instability modes are identified (for an introduction to the main axisymmetric and non-axisymmetric instabilities observed in electrospinning, see Section 2.1.3) ... 

S is the membrane area. Snapshots of vesicle and RBC shapes in flow are shown in Fig. 32 for a reduced volume of V = 0.59, where the vesicle shape at rest is a discocyte. For sufficiently small flow velocities, the discocyte shape is retained. However, the discocyte is found not in a coaxial orientation instead the shortest eigenvalue of the gyration tensor is oriented perpendicular to the cybnder axis [187]. Since two opposite sides of the rim of the discocyte are closer to the wall where the flow velocity is small, the rotational symmetry is slightly disturbed and the top view looks somewhat triangular, see Fig. 32a. With increasing flow velocity, a shape transition to an axisymmetric shape occurs. In the case of fluid vesicles this is a... 

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