A The Eccentric Cylinder Problem

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. 

To obtain governing equations and boundary conditions, we adopt the circular cylindrical coordinate system that is shown, along with a top view of the eccentric cylinder system, in Fig. 5-1. In this system, the origin of the coordinate system is chosen to be coincident with the central axis of the inner cylinder, which is assumed to have a radius a and an angular velocity 2 in the direction shown. The radius of the outer cylinder is assumed to be a( 1 + e), and its center is offset along the 0 = 0 axis relative to the center of the inner cylinder by an amount ask. In the journal-bearing problem, the outer cylinder does not rotate ( 2 = 0). The surface of the inner cylinder is thus 

Note that we use a prime to signify dimensional variables. 

The parameter k determines the degree of eccentricity. If k = 1, the two cylinders touch at 0 = 7r whereas they are concentric if k = 0. The range of allowable values for k is thus 

The gap width between the cylinders varies as a function of 9, being a maximum at 0 = 0 and a minimum at 9 = jt. The general expression for the dimensional gap width is 

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It is perhaps worthwhile to dwell briefly on the use of the journal-bearing configuration in practical lubrication applications. In such circumstances, the rotating inner cylinder is normally allowed to float to seek a position in which the hydrodynamic force precisely balances the load. We have analyzed the eccentric cylinder problem based on the picture in Fig. 5-1, in which the line of centers between the two cylinders is in the horizontal direction. In that configuration, there is a net vertical force but no horizontal force. Moreover, examination of (5-48) shows that the magnitude of the vertical force for a given pair of cylinders (so that a and e are fixed) is determined by 9 and A. The A dependence is contained in the factor... 

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) ... 

Enclosure problems (Fig. 4.1c) arise when a solid surface completely envelops a cavity containing a fluid and, possibly, interior solids. This section is concerned with heat transfer by natural convection within such enclosures. Problems without interior solids include the heat transfer between the various surfaces of a rectangular cavity or a cylindrical cavity. These problems, along with problems with interior solids including heat transfer between concentric or eccentric cylinders and spheres and enclosures with partitions, are discussed in the following sections. Property values (including P) in this section are to be taken at Tm = (Th+ TC)I2. 

The bearing runout problems discussed above highlight the importance of mechanical construction in rheometers. Clearly a small departure from 90° between the axis of a small angle cone and the surface of its plate will cause the shear rate to be higher on one side than on the other. The same is true for eccentricity in the axes of concentric cylinders (eq. 5.3.43) but with typical gap sizes (> 0.5 mm), alignment is less critical than with the cone and plate. 

Now, the general problem of (5—5)—(5—10) is highly nonlinear and, for an arbitrary occentric cylinder geometry, it can only be solved numerically - i.e., for arbitrary e and X in the range 0 < X < 1.3 However, for Re = 0, an exact analytic solution can be obtained by a coordinate transformation. In addition, for Re / 0, there are two limiting cases for which we can use asymptotic methods to obtain approximate analytic solutions. These are slight eccentricity... 

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