Determining the Tangential Velocity profile of a Rotating Fluid in a Static Container using Navier–Stokes equations

The shape of the liquid surface for a fluid present in a uniformly rotating cylinder is generally determined by making a Tangential velocity gradient along the radius of the rotating cylindrical container. A very similar principle can be applied if the direction of the produced velocity gradient is reversed, for which the source of rotation will be present at the central axis of the cylindrical vessel in which the liquid is present. Now if the described system is completely closed, the angular velocity will decrease as a function of time. But when the surface of the rotating fluid is kept free, then the Tangential velocity profile would be similar to that of the Taylor-Couette Flow, with a modification that; due to formation of a curvature at the surface, the Navier-Stokes law is to be modified. Now the final equation may not seem to have a proper general solution, but can be approximated to certain solvable expressions for specific cases of angular velocity.

Tangential flow development for laminar axial flow in an annulus with a rotating inner cylinder

The numerical finite-difference procedure of Gosman et al. (1969) is used to predict the growth of the tangential velocity profile and boundary-layer displacement thickness across an isothermal laminar axial flow through a concentric annulus when the inner cylinder is rotated at speeds which are insufficient to generate Taylor vortices. Solutions are obtained for fully developed and for developing axial flow over the ranges 0.05 < R 1 /R 2 < 0.98, 0.0002 < l < 1.0 and 100 < Re < 1700. The axial velocity profile is predicted to be insensitive to core rotation and, if varied, to influence only marginally the development of the tangential velocity profile; this is such that its dimensionless displacement thickness is related to dimensionless axial distance by a power law except near full development and at very low Reynolds number. Predictions at high Re accord extremely well with measurements. Astill’s (1964) stability criterion for the onset of vortices in tangential developing flow is accordingly presented afresh in terms of system parameters readily available to the designer.

The developing tangential velocity profile for axial flow in an annulus with a rotating inner cylinder

A method is described of predicting the growth of a tangential velocity profile in fully developed laminar axial flow through a concentric annulus when the inner surface is rotated at speeds which are insufficient to generate Taylor vortices. The treatment, which is based on simplification and subsequent solution of the Navier-Stokes equations, as Fourier-Bessel series, appears preferable to momentum-integral techniques through greater simplicity of expression and in requiring fewer assumptions about the developing tangential profile. The validity of the predictions is best at high axial Reynolds number.