Perturbed motion of a viscous fluid layer on the surface of a rotating cylinder

We investigate the plane flow of a viscous fluid layer plane flow on the surface of a cylinder rotating with a constant angular velocity with account for inertia forces and the acceleration of gravity.Lomonosov Moscow State University, Moscow; Bauman Moscow State Technical University; Belarusian State University, Minsk. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 66, No. 6, pp. 689–694, June, 1994.     

Velocity regimes of motion of a viscous liquid in a horizontal rotating cylinder

The influence of viscosity on the stability of motion of a liquid tube in the cavity of a stationarily rotating horizontal cylinder is considered. An analytical condition of the stability of motion is obtained. The similarity criteria of steady motion of the liquid are selected. On the basis of experimental data diagrams of the transition boundaries for turbular and nontubular regimes of liquid motion are constructed.Rovno State Pedagogical Institute. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 64, No. 5, pp. 558–565, May, 1993.     

Viscous-fluid flow in a thin layer on the side surface of a rotating braking-upper-end cylinder

An analytical solution is obtained to the hydrodynamic problem on circulating viscous-fluid flow in a thin layer on the side surface of a rotating braking-end cylinder. Calculated data are in satisfactory agreement with the experimental findings and with the numerical integration of the equations of motion.Moscow Engineering Physics Institute. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 62, No. 6, pp. 814–840, June, 1992.     

Shapes of annular layers of fluid on the surface of a rotating cylinder

A qualitative and quantitative study is made of the equilibrium forms of plane and axisymmetric fluid layers.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 55, No. 3, pp. 422–431, September, 1988.     

The shape of a liquid surface in a uniformly rotating cylinder in the presence of surface tension

The free-surface shape of a liquid in a uniformly rotating cylinder in the presence of surface tension is determined before and after the onset of dewetting at the bottom of the cylinder. Two scenarios of liquid withdrawal from the bottom are considered, with and without deposition of thin film behind the liquid. The governing non-linear differential equations for the axisymmetric liquid shapes are solved numerically by an iterative procedure similar to that used to determine the equilibrium shape of a liquid drop deposited on a solid substrate. The numerical results presented are for cylinders with comparable radii to the capillary length of liquid in the gravitational or reduced gravitational fields. The capillary effects are particularly pronounced for hydrophobic surfaces, which oppose the rotation-induced lifting of the liquid and intensify dewetting at the bottom surface of the cylinder. The free-surface shape is then analyzed under zero gravity conditions. A closed-form solution is obtained in the rotation range before the onset of dewetting, while an iterative scheme is applied to determine the liquid shape after the onset of dewetting. A variety of shapes, corresponding to different contact angles and speeds of rotation, are calculated and discussed.     

Form of a liquid layer of constant mass on the surface of a rotating cylinder

Comparison with experimental data reveals satisfactory agreement of results of a theoretical study of the equilibrium form and hydrodynamic stability of a liquid layer on the surface of a rotating horizontal cylinder.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 59, No. 1, pp. 80–84, July, 1990.     

Source-sink flow in a rotating cylinder

Summary This paper describes the axisymmetric source-sink flow in a rapidly rotating cylinder. Relative fluid motion is induced by the presence of a sink in the bottom corner and a ring source located somewhere in the fluid, at some distance from the solid boundaries. In order to neglect nonlinear effects the volumetric flow rates are assumed to be small, i.e. O(E ^1/2), with E the Ekman number of the flow. The transport from the source to the sink is carried by Ekman layers at the end caps, and a Stewartson layer at the sidewall. At the ring source a free Stewartson layer arises, in which the injected fluid is transported towards the Ekman layers. This Stewartson layer consists of layers of thicknesses E ^1/4 and E ^1/3, which both contribute to the vertical O(E ^1/2) transport. The ring source is enveloped by a ring-shaped region of cross-sectional dimensions O(E ^1/2 × E ^1/2), in which the injected fluid is rearranged before erupting into the E ^1/3 layer. As E ^1/2 E ^1/3, this region appears as an isolated singularity in the E ^1/3 layer; in fact it consists of a combination of an upward and a downward directed source, the strengths of which can be determined by transport arguments. The paper presents an analysis of the E ^1/3-layer structure on the basis of a linear theory; it also describes how the analysis can be extended to the situation in which fluid is injected through an array of sources at different heights.     

On compressible flow in a rotating cylinder

Summary Axisymmetric steady flow of a perfect gas in a rotating cylinder is studied by applying a linearised analysis to a small perturbation about isothermal rigid body rotation. Motivated by present day gas centrifuges, special attention is focussed on the effect of a length-to-radius ratio which increases from unit magnitude to infinity and on the effect of a strong radial density gradient associated with the isothermal rigid body rotation. The Ekman number E _*based on the small radial density scale and the density at the cylinder wall is taken to be small. It appears that the flow outside Ekman boundary layers at the end caps consists of three types. These correspond to 1 L _* E _* ^{% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS% qaaSqaaiaaigdaaeaacaaIYaaaaaaa!386D!\[ - \tfrac{1}{2}\]} E _* ^{% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS% qaaSqaaiaaigdaaeaacaaIYaaaaaaa!386D!\[ - \tfrac{1}{2}\]} L _*, E _* ^–1 andE _* ^–1 L _* where L _*is the ratio of the cylinder-length to the radial density scale. For 1 L _* E _* ^{% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS% qaaSqaaiaaigdaaeaacaaIYaaaaaaa!386D!\[ - \tfrac{1}{2}\]} an inviscid flow in a region of limited thickness near the cylinder wall is found. Due to the strong decrease of the density, radial diffusion is not confined to Stewartson boundary layers at the wall (typical for incompressible flow) but extends in the core. This finds expression in two layers in the centre of the cylinder, parallel to the rotation axis, having a structure similar to both Stewartson layers and adjusting the inviscid flow near the wall to a flow dominated by radial diffusion near the rotation axis. For L _* E _* ^{% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS% qaaSqaaiaaigdaaeaacaaIYaaaaaaa!386D!\[ - \tfrac{1}{2}\]} and L _* E _* ^–1 both Stewartson layers become successively of the same thickness as the density scale. At the same time the corresponding layers in the core go to the wall and join. As a result, for L _* E _* ^–1 radial diffusive processes are significant in the entire cylinder, a situation also known from studies of flows in semi-infinite gas centrifuges.     

Temperature distribution in a rotating hollow cylinder

The temperature distribution in a hollow cylinder rotating with a given angular velocity is found for steady-state boundary conditions of the first kind.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 37, No. 4, pp. 705–711, October, 1979.     

Flow of a non-Newtonian liquid in the gap between a rotating cylinder and a permeable surface with rotor granulation

The isothermal process of rotor granulation of a material having the properties of an anomalously viscous liquid is analyzed hydrodynamically.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 54, No. 3, pp. 415–422, March, 1988.     

Nonlinear stability analysis of thin Newtonian film flowing down on the inner surface of a rotating vertical cylinder

Summary The paper presents both of the linear and nonlinear stability theories for characterization of Newtonian film flows down on the inner surface of a rotating in.nite vertical cylinder. After showing the insufficiency of the linear model in characterizing certain flow behaviors, a generalized nonlinear kinematic model is then derived to represent the physical system. The model is solved by the long wave perturbation method in a two-step procedure. In the first step, the normal mode method is used to characterize the linear behaviors. The amplitude growth rates and the threshold conditions are characterized subsequently and summarized as the by-products of the linear solutions. In the second step, an elaborated nonlinear film flow model is solved by using the method of multiple scales to characterize flow behaviors at various states of sub-critical stability, sub-critical instability, supercritical stability, and supercritical explosion. The modeling results indicate that by increasing the rotation speed, X, and the radius of cylinder, R, the film flow will make the flow system more stable.     

Weakly nonlinear stability analysis of thin viscoelastic film flowing down on the outer surface of a rotating vertical cylinder

The paper presents both of the linear and nonlinear stability theories for characterization of viscoelastic film flows down on the outer surface of a rotating infinite vertical cylinder. After showing the insufficiency of the linear model in characterizing certain flow behaviors, a generalized nonlinear kinematic model is then derived to represent the physical system. The model is solved by the long wave perturbation method in a two-step procedure. In the first step, the normal mode method is used to characterize the linear behaviors. The amplitude growth rates and the threshold conditions are characterized subsequently and summarized as the by-products of the linear solutions. In the second step, an elaborated nonlinear film flow model is solved by using the method of multiple scales to characterize flow behaviors at various states of sub-critical stability, sub-critical instability, supercritical stability, and supercritical explosion. The modeling results indicate that by increasing the rotation speed, Ω, and decreasing the radius of cylinder, R, the film flow will generally make the flow system less stable. In this study, the interaction of the rotation and the radius of cylinder are taken into account. Generally, Reynolds number is divided into three regions, which are Re<3, 3<Re<8 and 8<Re, for discussion corresponding to the pre-selected Rossby number (Ro=0.1) and viscoelastic parameter (k=0.05).     

On the slow motion of a cylinder in a micropolar fluid

Slow steady 2-dimensional motion of an incompressible micropolar fluid in the unbounded region exterior to a cylinder of arbitrary cross section is considered. The possibility of a solution in the strict sense of the Stokes' approximation is examined. It is shown that the near and far boundary conditions can be satisfied simultaneously only when the drag on the cylinder is zero and the velocity and microrotation vectors satisfy an integral condition.     

Experimental investigation of axial dispersion in a horizontal rotating cylinder

This work reports experimental measurements of the dispersion of particles during rotation in a horizontal cylinder. The axial dispersion of a pulse of approximately monodisperse black glass ballotini into a bed of clear glass ballotini of the same size is analysed. This is done using a sectioning technique, where the concentration is determined throughout the cylinder for a given rotation time and speed. The concentration profile is fitted to an appropriate solution of Fick’s second law to determine the dispersion coefficient. The dispersion coefficient is compared for various drum rotation rates and glass ballotini sizes. The cylinder was filled to 35 % by volume and rotated at a range of speeds between 5 and 20 rpm. The particle sizes vary from 1.14 to 3.15 mm. The dispersion coefficient was found to be dependent on both particle size and rotation speed. As the rotation speed, \(\omega \), was increased the dispersion coefficient increased proportionally to \(\omega ^{0.8}\). As the particle diameter, \(d_p\), was increased the dispersion coefficient increased proportionally to \(d_p^{1.84}\). These results are compared with previous experimental and simulation data, in particular the simulations of Third et al. (Powder Technol 203:510, 2010). Strong agreement was found between the simulations of Third et al. and the experimental results.

     

Theoretical and experimetal analyses of the maximum-suppotable fluid load on a rotating cylinder

Thin-film theory is used to derive an implicit criterion for the existence of a viscous, free-surface flow on the outer surface of a rotating circular cylinder. We estimate both the maximum fluid load supportable at a given Stokes number, and the maximum Stokes number permitting the support of a prescribed fluid load. The results of our theory are in excellent agreement with those obtained from an independent, numerical solution of the full Stokes equations; the comparison reveals that our thin-film theory works well beyond its expected range of validity.Explicit existence criteria, in the form of small-parameter expansions, are presented for the extreme cases of small fluxes and large Stokes numbers. We derive an expression for the minimum angular velocity which will support a given load; our results show qualitative agreement with those of basic experiments.     

Laminar flow of a liquid in a rotating cylinder in a gravitational field

The laminar flow of a viscous liquid in a vertical rotating cylinder is studied.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 54, No. 2, pp. 198–202, February, 1988.