Eccentric cylinder flows

Kumar, S., and Homsy, G. M., Chaotic advection in creeping flow of viscoelastic fluids between slowly modulated eccentric cylinders. Phys. Fluids 8,1774-1787 (19%). 

Fig. 7.10 Poincare sections of viscous Newtonian flow in alternately turning eccentric cylinders. The inner cylinder turned counterclockwise for a given time, and then the outer cylinder was turned clockwise for 800 periods. There were 11 initial particles. [Reprinted by permission from J. Chaiken, R. Chevray, M. Tabor, and Q. M. Tan, Experimental Study of Lagrangian Turbulence in Stokes Flow, Proc. R. Soc. London A, 408, 165-174 (1986).]...

Grosso, M. Maffettone, P.L. Halin, P. Keunings, R. Legat, V. Flow of nematic polymers in eccentric cylinder geometry influence of... 

As stated, we begin with the special problem of flow between two rotating cylinders whose axes are parallel but offset to produce the eccentric cylinder geometry shown in Fig. 5 1. In the concentric limit, this is the famous Couette flow problem, which was analyzed in Chap. 3. 

There have recently been rapid advances in our understanding of Taylor-Couette flow between a rotating inner cylinder and a concentric stationary outer cylinder. Hiis progress is reviewed and its implications in the case of eccentric cylinders or journal bearings considered. 

The presence of weak vortex motions at Taylor ntmbers less than T have been reported by many investigators and have been described in detail by Jackson, Robati and Mobbs (3), for both concentric and eccentric cylinders. That these motions are due to end effects has been shown by numerical solutions of the Navier-Stokes equations for the flow in a finite length annulus. Solutions have been obtained by Alziary de Roquefort and Grillaud (4) for concentric cylinders with end plates rotating with the inner cylinder, Preston (5) for concentric cylinders with fixed end plates and El-Dujaily (6) for concentric and eccentric cylinders with fixed end plates. 

For eccentric cylinders, no systematic wave frequency measurements have been made, although O Brien and Mobbs (19) have ccnmented on the azimuthal dependence of the flow. Since x/d varies around the azimuth the situation is far more ccmplex than in the concentric cylinder case. 

Much further work is required to extend the present knowledge of superlaminar flow between concentric cyliniders to the eccentric cylinder/journal bearing case. indications are that within the range of Taylor nonbers encountered in practice the flow consists primarily of Taylor vortices with superimposed travelling waves and any possible turbulence is weak. 

EL-DUJAILK, M.J. "End effects on sub-critical and Taylor vortex flow between concentric and eccentric cylinders." KiD. Ttiesis. Department of Mechanical Engineering, university of Leeds, 1983. 

Figure 13.9. Flow of a Bingham material in a planar channel with an eccentric cylinder, Bn 125. (Calculation by J. P. Singh.)...

The ratio of the fluid relaxation time to the timescale for flow If defines a dimensionless group termed the Deborah number, De = /tf. This group has been used in the literature to characterize deviations from Newtonian flow behavior in polymers [6]. Specifically, in flows such as simple steady shear flow where a single flow time tf = y can be defined, it has been observed that for De 1, a Newtonian fluid behavior is observed, whereas for De 1, a non-Newtonian fluid response is observed. However, in flows where multiple timescales can be identified, for example, shear flow between eccentric cylinders, the Deborah number is clearly not unique. In this case, it is generally more useful to discuss the effect of flow on polymer liquids in terms of the relative rates of deformation of material lines and material relaxation. In a steady flow, this effect can be captured by a second dimensionless group termed the Weissenberg number, Wi = k A, where k is a characteristic deformation rate and A is a characteristic fluid relaxation time. For polymer liquids, A is typically taken to be the longest relaxation time Ap, and for steady shear flow, k = y, which leads to Wi = y Ap. 

This result holds in particular for concentric spheres and very long concentric cylinders as here the assumption of isothermal surfaces applies more easily. If, however, body 1 lies eccentric in the enclosure surrounded by body 2, Fig. 5.64, then the two surfaces will generally not be isothermal, as the radiation flow is much higher in the regions where the two surfaces are close to each other than where a large distance exists between them. 

The method of domain perturbations was used for many years before its formal rationalization by D. D. Joseph D. D. Joseph, Parameter and domain dependence of eigenvalues of elliptic partial differential equations, Arch. Ration. Mech. Anal. 24, 325-351 (1967). See also Ref. 3f. The method has been used for analysis of a number of different problems in fluid mechanics A. Beris, R. C. Armstrong and R. A. Brown, Perturbation theory for viscoelastic fluids between eccentric rotating cylinders, J. Non-Newtonian Fluid Mech. 13, 109-48 (1983) R. G. Cox, The deformation of a drop in a general time-dependent fluid flow, J. Fluid Mech. 37, 601-623 (1969) ... 

It is of interest to note that, by judicious definition of the characteristic diameter of nonspherical bodies, good agreement with the equations for spherical solids was obtained. A diameter defined by the total surface area of the body, divided by the perimeter normal to flow, was successfully used for spheres, hemispheres, cubes, prisms, and cylinders (PI), yielding a = 0 b - 0.692 m = 0.514 and n = [Eq. (4)]. Similar results were obtained for spheroids (S14), namely a = 0 6 = 0.74 w = 0.5 and n =. The commonly used equivalent diameter of a sphere of the same volume as the body yields transfer coefficients increasing with eccentricity (SI4). 

The main sources of error in the concentric cylinder type measuring geometry arise from end effects (see above), wall shp, inertia and secondary flows, viscous heating effects and eccentricities due to misahgnment of the geometry [Macosko, 1994],... 

JACKSON, P.A., HDBATI, B. and MOBBS, F.R. "Secondary flows between eccentric rotating cylinders at sub-critical Taylor nunbers."... 

MOBBS, F.R. and OZOGAN, M.S. "Study of sub-critical Taylor votex flow between eccentric rotating cylinders torque measurements and visual observations." International journal of Heat and Fluid Flow, 1984, Vol. 5, no. 4, 251-253. 

CASTLE, P. and MC S, F.R. "I rodynamic stability of the flow between eccentric rotating cylinders." Proc. Instn. Mech Engrs, 1968, 182, 41-52. 

JCNES, C.D. "A study of secondary flow and turbulence between eccentric rotating cylinders." PhD. Thesis Mech. Eng. Dept., university of Leeds. 1973. 

Besides wall slip there can be departure from ideal Couette flow due to the ends of the cylinder, to inertia, and to eccentricity. There also can be errors due to shear heating. These are discussed below, but first we examine normal stress effects. 

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