Transient rotational fluid motion

Statement of the problem. Two problems on the transient rotational motion of an incompressible viscous fluid around a cylindrical surface which suddenly begins to rotate are also considered in [449]. In this case only the velocity component V p is nonzero. This variable satisfies the equation 

Fluid motion inside a hollow rotating cylinder. The first problem corresponds to the motion of a fluid inside the cylinder (in the region 1Z a). In this case the solution has the form 

Here Jj(A) is the Bessel functions of the first kind and order one. 

Fluid motion outside a rotating cylinder. The second problem corresponds to the rotation of a cylinder in an infinite fluid. In this case Eq. (1.9.29) must be considered in the region 1Z a. The boundary condition (1.9.31) is supplemented by the condition that the velocity decays at infinity, 

It should be noted that the spectrum of eigenvalues of this problem is continuous, 0 P oo. 

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Vibrational spectroscopy in rigid matrices has been used extensively for studying species that would be transient in fluid media [45]. Quenching of rotational motion at low temperature typically gives lines much sharper than those observed in fluids. Studies of organic mechanisms by this technique have tended to focus on using spectra to identify the chemical structure of trapped intermediates or products. 

Fluid motion between rotating cylinders. The problem about the transient motion of an initially stagnant fluid in the gap between two coaxial cylinders of radii b and a (b < a) that suddenly begin to rotate at angular velocities uib and u>a is considered in [40]. The only nonzero velocity component Vv satisfies Eq. (1.9.29) with the initial condition (1.9.30) and the following boundary conditions ... 

To create the mixer geometry, a cylindrical mesh is generated for the tank. Two other, completely independent meshes are defined for the blades. The three meshes are then combined into one. As the blades rotate, the transient flow pattern in the tank can be calculated and illustrated by the dispersion of tracer particles, as shown in the figure. As the total number of rotations increases, the tracer becomes more uniformly distributed. After six rotations, the dispersion of the tracer particles in the horizontal plane is satisfactory. Note, however, that the particles have moved little in the vertical direction. This is becanse the anchor impellers in use impart little or no axial momentum to the fluid. Twisted blades, which also impose an axial motion on the flow, might perform better to distribute the tracer throughout the vessel. The mesh superposition technique is well suited to study such systems. For other examples of flow in planetary mixers, see Tanguy et al. (1999) and Zhon et al. (2000). 

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