Circular Couette Flow - A One-Dimensional Analog to Unidirectional Flows

In the present book, we focus our attention on the solution of the Navier-Stokes equations for laminar flows, frequently without any attempt to analyze the stability (or experimental realizability) of the resulting solutions. In using these solutions, it is therefore quite apparent that we must always reserve judgment as to the range of parameter values where they will exist in practice. We have already noted that experimental observation shows that Poiseuille flow exists for Reynolds numbers only less than a critical value. A general introduction to hydrodynamic stability theory is given in Chap. 12. It should be noted, however, that the stability of Poiseiulle flow is a very difficult problem, and only a short introductory section that is relevant to this problem is provided.7 

CIRCULAR COUETTE FLOW - A ONE-DIMENSIONAL ANALOG TO UNIDIRECTIONAL FLOWS 

We have analyzed several simple examples of steady, unidirectional flows in the preceding section. Returning to the case of flow between two infinite plane walls, the case of simple shear flow is of special significance because it is one of the flows used by rheologists to measure the viscosity and generally characterize the flow behavior of viscous and viscoelastic liquids.8 The main reason is that the form of the flow between two plane boundaries, one of which is moving and the other stationary is the same, namely 

Whatever the nature of the constitutive equation, provided the fluid is homogeneous, this implies that the corresponding component of the rate-of-strain tensor, 

One immediate comment before discussing the simplification and solution of these equations is that they appear much more complex than the same equations for a Cartesian coordinate system. To cite just one example, we note that the r component of V2u on the right-hand side of (3-56) not only contains second derivatives with respect tor, 0, and z, but a number of additional terms. These arise because the cylindrical coordinate system is curvilinear, and indeed are often called curvature terms because they arise as a consequence of the curvature of the coordinate lines. The same is true for the u Vu terms in which, for example, (uVu)r/u-V n again because of curvature terms. 

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C. Circular Couette Flow - a One-Dimensional Analog to Unidirectional Flows... 

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