Theoretical And Numerical Study Of The Flow Between Rotating Conical Cylinders

The flow between two concentric rotating cylinders is called Taylor-Couette flow, it allows a steady solution of the form u= uφ(r)eφwith p= p(r) in cylindrical coordinates. The behavior of flow between two concentric conical cylinders has been studied by some research workers, both numerically and experimentally. Most results are about the transition to Taylor vortices and turbulence, the existence of steady solution of this problem hasn't received general concern.In this paper we will use the method of "proof by contradiction" to show that there doesn't exist one-dimensional steady solution of the form u= ur(r)er+uφ(r)eφ+uz(r)ez with P= p(r). In the case of small gap and under neglecting the effect of the top and bottom boundaries, the non-existence of this steady solution is verified by numerical simulation. As a result, the average pressure is dependent on z.And then we continue to use the method of "proof by contradiction" to show that there doesn't exist two-dimensional steady solution of the form u= ur(r,z)er+uφ(r,z)eφand u=uφ(r, z)eφ+ uz(r, z)ez with p= p(r,z) for any angular velocities of the inner and outer conical cylinders.The numerical simulation is again used to investigate the existence of a stationary flow state with inner cone rotating and the outer one at rest under the condition of small gap and neglecting the effect of the top and bottom boundaries. It is shown that the total kinetic energy remains approximately constant. This implies that a stationary flow state has been achieved. The stationary solution would be three-dimensional according to the streamlines of the flow, as a consequence, there must exist a three-dimensional steady solution.The last part of this paper is the numerical investigation of the transition to Taylor vortices, as well as the distribution of the extreme values of pressure and velocity under the condition of rigid boundary, where the rotation of the inner cylinder result in fluid motion.