Two-layer film flow over a rotating disk

Unsteady two-layer liquid film flow on a horizontal rotating disk is analyzed using asymptotic method for small values of Reynolds number. This analysis of non-linear evolution equation elucidates how a two-layer film of uniform thickness thins when the disk is set in uniform rotation. It is observed that the final film thickness attains an asymptotic value at large time. It is also established that viscous force dominates over centrifugal force and upper layer thins faster than lower layer at large time.     

The flow due to a rough rotating disk

Von Kármáns problem of a rotating disk in an infinite viscous fluid is extended to the case where the disk surface admits partial slip. The nonlinear similarity equations are integrated accurately for the full range of slip coefficients. The effects of slip are discussed. An existence proof is also given.     

Hydromagnetic forced flow against a porous rotating disk

This paper deals with the steady forced flow of a viscous, incompressible and electrically conducting fluid against a porous rotating disk when a uniform magnetic field acts perpendicular to the disk surface. For small suction the equations of motion are integrated numerically by Kármán-Pohlhausen method, but for large suction a series solution in the inverse powers of the suction parameter is obtained. The effects of disk porosity and magnetic field on the various flow parameters are discussed in detail.     

The flow due to a rough rotating disk

Von Kármáns problem of a rotating disk in an infinite viscous fluid is extended to the case where the disk surface admits partial slip. The nonlinear similarity equations are integrated accurately for the full range of slip coefficients. The effects of slip are discussed. An existence proof is also given.     

Unsteady viscous flow over a rotating stretchable disk with deceleration

In this work, the laminar unsteady flow over a stretchable rotating disk with deceleration is investigated. The three dimensional Navier–Stokes (NS) equations are reduced into a similarity ordinary differential equation group, which is solved numerically using a shooting method. Mathematically, two solution branches are found for the similarity equations. The lower solution branch may not be physically feasible due to a negative velocity in the circumferential direction. For the physically feasible solution branch, namely the upper solution branch, the fluid behavior is greatly influenced by the disk stretching parameter and the unsteadiness parameter. With disk stretching, the disk can be friction free in both the radial and the circumferential directions, depending on the values of the controlling parameters. The results provide an exact solution to the whole unsteady NS equations with new nonlinear phenomena and multiple solution branches.     

On the numerical solution of the flow between a rotating and a stationary disk

The flow between two co-axial, infinite disks, one rotating with constant angular velocity and one stationary is treated in this paper. The problem is reduced to that of finding the solution of a two-point boundary value for a sixth order nonlinear ordinary differential equation and three boundary conditions at each of a finite interval. The numerical solutions are obtained by using a fourth order Runge-Kutta integration scheme in modification due to Gill and in conjunction with a modified shooting method to correct the initial guesses at one boundary. The numerical calculations for different Reynolds numbers are carried out. The results obtained by this method are compared with available results. The comparison shows excellent agreement.     

On the steady flow between a rotating and a stationary disk with a uniform suction at the stationary disk

Summary The steady axisymmetric flow of a viscous incompressible fluid between two coaxial disks, one rotating and the other stationary, with uniform suction at the stationary disk is discussed. Similarity solutions of Navier-Stokes equations are obtained by the method of regular perturbation for small suction Reynolds number and numerically for an arbitrary suction Reynolds number by a series expansion method and also by integrating directly by a Newton-Raphson technique. It is found that for a fixed rotational Reynolds number and varying suction Reynolds number there is an overall increase in the velocity components as the suction increases.

---

Zusammenfassung Es wird die stationäre axisymmetrische Strömung zwischen zwei koaxialen Platten untersucht, von denen die eine rotiert und die andere stationär ist, wobei an der stationären Platte eine gleichförmige Absaugung stattfindet. Ähnlichkeitslösungen der Navier-Stokesschen Gleichungen werden erhalten, durch eine reguläre Störungsrechnung bei kleinen Absaug-Reynoldszahlen, während bei beliebigen Werten dieses Parameters numerische Resultate sowohl durch eine Reihenentwicklung wie auch durch direkte Integration mit Hilfe der Newton-Raphson-Technik gefunden werden. Bei fester Rotations-Reynoldszahl und variabler Absaugung findet man eine allgemeine Erhöhung der Geschwindigkeits-komponenten mit der Absaugung.

     

Large eddy simulations of the turbulent flow between a rotating and a stationary disk

Large eddy simulations of the flow between a rotating and a stationary disk have been performed using a dynamic and a mixed dynamic subgrid-scale model. The simulations were compared to direct numerical simulation results. The mixed dynamic model gave better overall predictions than the dynamic model. Modifications of the near-wall structures caused by the mean flow three-dimensionality were also investigated. Conditional averages near strong stress-producing events led to the same conclusions regarding these modifications as studies of the flow generated by direct numerical simulation, namely a distinct asymmetry of the vortices producing sweeps and ejections.     

Large eddy simulations of the turbulent flow between a rotating and a stationary disk

Large eddy simulations of the flow between a rotating and a stationary disk have been performed using a dynamic and a mixed dynamic subgrid-scale model. The simulations were compared to direct numerical simulation results. The mixed dynamic model gave better overall predictions than the dynamic model. Modifications of the near-wall structures caused by the mean flow three-dimensionality were also investigated. Conditional averages near strong stress-producing events led to the same conclusions regarding these modifications as studies of the flow generated by direct numerical simulation, namely a distinct asymmetry of the vortices producing sweeps and ejections.     

Retracted: Unsteady flow and heat transfer on a rotating disk in a porous medium

This article has been retracted. See retraction notice DOI: 10.1002/mma.850 . An unsteady flow and heat transfer in a porous medium of a viscous incompressible fluid over a rotating disk in an otherwise ambient fluid are studied. The unsteadiness in the flow field is caused by the angular velocity of the disk which varies with time. The new self‐similar solution of the Navier–Stokes and energy equations is obtained numerically. The solution obtained here is not only the solution of the Navier–Stokes equations, but also of the boundary layer equations. Also, for a simple scaling factor, it represents the solution of the flow and heat transfer in the forward stagnation‐point region of a rotating sphere or over a rotating cone. The asymptotic behaviour of the solution for a large porosity or for a large independent variable is also examined. The surface shear stresses in the radial and tangential directions and the surface heat transfer increase as the acceleration parameter increases. Also, the surface shear stress in the radial direction and the surface heat transfer decrease with increasing porosity, but the surface shear stress in the tangential direction increases. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical simulation of flow of a micropolar fluid between a porous disk and a non-porous disk

Two dimensional steady, laminar and incompressible motion of a micropolar fluid between an impermeable disk and a permeable disk is considered to investigate the influence of the Reynolds number and the micropolar structure on the flow characteristics. The main flow stream is superimposed by constant injection velocity at the porous disk. An extension of Von Karman’s similarity transformations is applied to reduce governing partial differential equations (PDEs) to a set of non-linear coupled ordinary differential equations (ODEs) in dimensionless form. An algorithm based on finite difference method is employed to solve these ODEs and Richardson’s extrapolation is used to obtain higher order accuracy. The numerical results reflect the expected physical behavior of the flow phenomenon under consideration. The study indicates that the magnitude of shear stress increases strictly and indefinitely at the impermeable disk while it decreases steadily at the permeable disk, by increasing the injection velocity. Moreover, the micropolar fluids reduce the skin friction as compared to the Newtonian fluids. The magnitude of microrotation increases with increasing the magnitude of R and the micropolar parameters. The present results are in excellent comparison with the available literature results.     

Numerical modelling of the hydrodynamic field coupled to the transport of chemical species through the finite-element method

We describe the main features of a numerical simulator currently under development for studying water physico-chemical properties during the flooding of hydroelectric plants reservoirs. The work is sponsored by the brazilian Electric Energy National Agency ANEEL and conducted in a joint work of brazilian universities rearchers, with Furnas Centrais Elétricas S. A., the leading brazilian power utility company. An overview of the simulator requirements and of the involved partners is given. The software modules and engineering solutions are briefly discussed, including the finite element based transport module. Some results, future steps and partnerships are presented and discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the linearization of the equations of motion of a rotating disk

In this paper, we present a consistent approach to reduce the fully nonlinear equations of a rotating disk to the classical linear equation derived by Lamb and Southwell and the nonlinear equations derived by Nowinski. The approach recognizes the fact that the out-of-plane deflection and the in-plane deflections are of different orders of magnitude. By using the ratio between the plate thickness and the outer radius as a measurement and carefully examining the reasonable magnitudes of all the variables involved, the fully nonlinear equations can be non-dimensionalized with all the terms being sorted according to their orders of magnitude. It is found that the classical linear equation derived by Lamb and Southwell can be recovered if all the terms of the lowest order of magnitude in the fully nonlinear equations are retained. If all the terms of the lowest two orders of magnitude are retained, Nowinski’s equations can then be recovered. Furthermore, the terms arising from in-plane deformation and rotary inertia are of the highest order and can be ignored in most of the applications.     

A note on magneto-hydrodynamics of a finite rotating disk

Zusammenfassung Es wird gezeigt, dass in elektrisch leitender Flüssigkeit bei zylindrischer Symmetrie die Radialkomponente des Magnetfeldes klein ist im Vergleich zur Axialkomponente, sobald die axiale Ausdehnung des Gefässes klein ist im Vergleich zu seinem Durchmesser.     

On the existence and uniqueness of solutions for certain boundary value problems governing the species distribution in a rotating binary mixture over an infinite rotating disk

The questions of existence and uniqueness of solutions for certain boundary value problems arising in connection with the barodiffusion of species, in an incompressible, isothermal, rotating binary mixture over an infinite, impermeable, rotating disk are considered. For arbitrary values of the Schmidt number, solutions are shown to exist by a constructive proof and their uniqueness is also established, when the disk rotates faster than the mixture far from it.     

Effect of Hall current on the unsteady MHD flow due to a rotating disk with uniform suction or injection

The unsteady flow of a viscous conducting fluid due to the rotation of an infinite, non-conducting, porous disk in the presence of an axial uniform steady magnetic field is studied without neglecting the Hall effect. The fluid is acted upon by a uniform injection or suction through the disk. The relevant equations are solved numerically with a special technique to resolve the conflict between the initial and boundary conditions. The solution shows that the inclusion of the injection or suction through the surface of the disk in addition to the Hall current gives some interesting results.