Circular Couette flow

Example 4.4 Thermomechanical coupling in a circular Couette flow For a circular Couette flow (Figure 4.5), the entropy production rate for an incompressible Newtonian fluid held between two coaxial cylinders is... 

The circular Couette flow between concentric cylinders is in the 0-direction only, and satisfies vr = vz = 0, ve = Vfl(r), and T = T(r). The inner cylinder is stationary while the outer cylinder rotates with an angular velocity w. Assuming a steady and laminar flow without end effects, the velocity distribution is... 

Figure 4.5. The circular Couette flow. Reprinted with permission from Elsevier, Int. J. Heat Mass Transfer, 43 (2000) 4205.

Figure 4.6. The Bejan number Be for the circular Couette flow for 7 = 300 K, r0 = 0.02 m, and r = 0.019 m. Reprinted with permission...

Kataoka, K. Taylor vortices and instabilities in circular Couette flows, in N.P. Cheremisinoff (Ed.), Encyclopedia of Eluid Mechanics , vol. 1. Gulf Publishing, Houston (1986), p. 236. 

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

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

On the other hand, because the circular Couette flow is generally adopted as a convenient substitute for a simple shear flow, it maybe tempting to analyze the experimental data as though we exactly had simple shear between two plane boundaries. This would mean dividing rrg r=a with an estimated velocity gradient given by the velocity difference of the two walls divided by the gap width as would be exactly correct for a linear shear flow. The latter is simply... 

Although the full Navier Stokes equations are nonlinear, we have studied a number of problems in Chap. 3 in which the flow was either unidirectional so that the nonlinear terms u Vu were identically equal to zero or else appeared only in an equation for the crossstream pressure gradient, which was decoupled from the primary linear flow equation, as in the ID analog of circular Couette flow. This class of flow problems is unusual in the sense that exact solutions could be obtained by use of standard methods of analysis for linear PDEs. In virtually all circumstances besides the special class of flows described in Chap. 3, we must utilize the original, nonlinear Navier Stokes equations. In such cases, the analytic methods of the preceding chapter do not apply because they rely explicitly on the so-called superposition principle, according to which a sum of solutions of linear equations is still a solution. In fact, no generally applicable analytic method exists for the exact solution of nonlinear PDEs. 

Example 4.4 Thermomechanical coupling in a Circular Couette flow... 

The circular Couette flow between concentric cylinders is in the 0 direction only, and satisfles... 

GORMAN, M., REITH, L.A., and SWINNEY, H.L. "Modulation patterns, multiple frequencies and other lAvencroena in circular Couette flow" in Nonlinear Dynamics" edit by HELLEMAN, R. Annals of the New York Acadan of Sci. 1980. 

R.J."Re-onergent order of chaotic circular Couette flow." Phys. Rev. Lett. 1979, 42(5), 301-463. 

As an example of a simation in which it is important to use an algorithm which conserves angular momentum, consider a drop of a highly viscous fluid inside a lower-viscosity fluid in circular Couette flow. In order to avoid the complications of phase-separating two-component fluids, the high viscosity fluid is confined to a radius r < Ri by an impenetrable boundary with reflecting boundary conditions (i.e., the momentum parallel to the boundary is conserved in collisions). No-slip boundary conditions between the inner and outer fluids are guaranteed because collision cells reach across the boundary. When a torque is applied to the outer circular wall (with no-slip, bounce-back boundary conditions) of radius R2 > Ri, a solid-body rotation of both fluids is expected. The results of simulations with both MPC-AT-a... 

In the case of the phase-separated fluids in circular Couette flow, this implies that if both fluids rotate at the same angular velocity, the inner and outer stresses do not coincide. Thus, the angular velocity of the inner fluid is smaller than the outer one, with vg r) = Q. r for r [Pg.41]

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