Circular cylinder shear flows

Viscometric flows used for measurements include well known flows, such as flow in a narrow gap concentric cylinder device and between a small angle cone and a flat plate. In both of these cases the flows established in these devices approximate almost exactly simple shearing flow. There are other viscometric flows in which the shear rate is not constant throughout, these include the wide gap concentric cylinder flow and flow in a circular pipe, discussed above. 

Figure 9-16. A qualitative sketch of the streamlines near a nonrotating circular cylinder in a simple 2D shear flow.

To begin our analysis of the cylinder problem, it is necessary to consider briefly the velocity field. The fluid mechanics solution for a circular cylinder rotating with an imposed angular velocity il in a simple shear flow was given originally by Bretherton26 for the creeping-flow limit. In this case, the velocity field can be specified in terms of a stream-function (see Chap. 7) such that... 

Figure 9-18. Streamlines for a freely rotating circular cylinder in simple, linear shear flow (9-306). Contours 0 to 0.75 in increments of 1/16.

A general study of the streamlines for a circular cylinder in simple shear flow can he found in the following papers C. R. Robertson and A. Acrivos, Low Reynolds number shear flow past a rotating circular cylinder, Part I, Momentum Transfer, J. Fluid Mech. 40, 685-704 (1970) R. G. Cox, I. Y. Z. Zia, and S. G. Mason, Particle motions in sheared suspensions, 15. Streamlines around cylinders and spheres, J. Colloid Interface Sci. 27, 7-18 (1968). 

Problem 9-5. Heat Transfer From a Freely Rotating Circular Cylinder in Shear Flow... 

A heated circular cylinder is suspended in a fluid that is undergoing a simple shear flow. Assume that the cylinder does not rotate so that the streamfunction representing flow in the (inner) region near the cylinder is... 

Fixed cylinder. Let us consider a fixed circular cylinder in an arbitrary steady-state linear shear flow of viscous incompressible fluid in the plane normal to the cylinder axis. The velocity field of such a flow remote from the cylinder in the Cartesian coordinates X, X% can be represented in the general case as follows ... 

Figure 2.10. Linear shear flow past a fixed circular cylinder (a) straining flow (Ei = 0 and Cl = 0) (b) simple shear flow (Ei = 0 and Ei = -Cl)...

Figure 2.11. Linear shear flow past a freely rotating circular cylinder in the 7Z -plane (the limit streamlines P = Ps are marked bold) (a) simple shear flow ( STg = 1) (b) general case of plane shear flow (0 < n < 1)...

Freely rotating cylinder. Now let us consider a circular cylinder freely floating in an arbitrary linear shear Stokes flow (Re —i 0). The velocity distribution for such a flow remote from the cylinder is still given by relations (2.7.8). 

In view of the no-slip condition on the surface of the circular cylinder freely floating in a shear flow, this cylinder rotates at the constant angular velocity equal to the rotation flow velocity at infinity. This means that the following boundary conditions for the fluid velocity components must be satisfied on the cylinder surface ... 

Mass transfer between a circular cylinder of radius a and a simple shear flow (G 2 = 1, the other Gkm = 0) was studied in [132]. For the dimensionless total diffusion flux per unit length of the cylinder, the following expression was obtained as Pe 0 ... 

The solution of the corresponding mass exchange problem for a circular cylinder and an arbitrary shear flow was obtained in [353] in the diffusion boundary layer approximation. It was shown that an increase in the absolute value of the angular velocity Cl of the shear flow results in a small decrease in the intensity of mass and heat transfer between the cylinder and the ambient... 

It follows from (4.11.3) that in the region -1 < fl < +1, the mean Sherwood number varies only slightly (the relative increment in the mean Sherwood number as Iflfil varies from 0 to 1 is at most 1.3%). In the special cases of purely straining (CIe = 0) and purely shear (Ifi l = 1) linear Stokes flow past a circular cylinder, formula (4.11.3) turns into those given in [342, 343]. 

Freely rotating cylinder. Now let us consider convective mass transfer to the surface of a circular cylinder freely suspended in an arbitrary linear shear Stokes flow (Re -> 0). In view of the no-slip condition, the cylinder rotates at a constant angular velocity equal to the angular velocity of the flow at infinity. The fluid velocity distribution is described by formulas (2.7.11). The streamline pattern qualitatively differs from that for the case of a fixed cylinder. For 0 0, there are no stagnation points on the surface of the cylinder and there exist two qualitatively different types of flow. For 0 < Ifigl < 1, there are both closed and open streamlines in the flow, the region filled with closed streamlines is adjacent to the surface of the cylinder, and streamlines far from the cylinder are open (Figure 2.11). For Ifl l > 1, all streamlines are open. 

In the mass exchange problem for a circular cylinder freely suspended in linear shear flow, no diffusion boundary layer is formed as Pe - oo near the surface of the cylinder. The concentration distribution is sought in the form of a regular asymptotic expansion (4.8.12) in negative powers of the Peclet number. The mean Sherwood number remains finite as Pe - oo. This is due to the fact that mass and heat transfer to the cylinder is blocked by the region of closed circulation. As a result, mass and heat transfer to the surface is mainly determined by molecular diffusion in the direction orthogonal to the streamlines. In this case, the concentration is constant on each streamline (but is different on different streamlines). 

An experimental verification [405] of the fact that the leading term of the asymptotic expansion of the mean Sherwood number for Pe 1 is independent of the Peclet number for a freely rotating circular cylinder in a simple shear flow ( ff = 1) showed good qualitative and quantitative agreement with the theoretical results [132]. The measured mean Sherwood number was 2.65, which is close to the corresponding asymptotic value (4.11.4). 

A particularly interesting phenomenon connected with transition in the boundary layer occurs with blunt bodies, e.g., spheres or circular cylinders. In the region of adverse pressure gradient (i.e., dP/dx > 0 in Fig. 1.9) the boundary layer separates from the surface. At this location the shear stress goes to zero, and beyond this point there is a reversal of flow in the vicinity of the wall, as shown in Fig. 1.9. In this... 

The boundary of the cavern can be defined as the surface where the local shear stress equals the fluid yield stress. If it is assumed that the predominant flow in the cavern is tangential [and EDA studies suggest that this is a reasonable approximation (Hirata et al., 1994)] and that the cavern shape, fluid yield stress, and impeller power number are known, the cavern size may be determined. A right circular cylinder of height He and diameter Dc centered on the impeller is a good model for the cavern shape, which allows for the effect of different impellers (Elson et al., 1986). Thus,... 

Various arrangements at the bottom of the inner cylinder are available in Figure 3.2 an indentation is provided so that an air gap is formed and shearing in the sample below the inner cylinder is negligible. Another arrangement is to make the bottom of the inner cylinder a cone. When one of the cylinders is rotated, a Couette flow is generated with fluid particles describing circular paths. The only non-zero velocity component is ve and it varies in the r-direction. In order to minimize secondary flow (Taylor vortices) it is preferable that the outer cylinder be rotated however, in most commercial instruments it is the inner cylinder that rotates. In this case, the fluid s velocity is equal to IXR, at the surface of the inner cylinder and falls to zero at the surface of the outer cylinder. The shear stress is uniform over the curved surface of the inner cylinder and over the outer cylinder (to the bottom of the annular gap). 

Two of this class of flows will now be considered. The first is circular flow between closely-fitting, narrow-gap concentric cylinders, and the second is circular flow in a small-angle, cone-and-plate geometry, see figure 1. Also shown is the cone-cylinder or Mooney geometry where the shear rates in the concentric-cylinder and the cone-and-plate parts of the geometry are arranged to be the same. 

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