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.     

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.     

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.     

Plane couette flow of two immiscible incompressible fluids with uniform suction at the stationary plate

The contact layer in the plane couette flow of two immiscible fluids with uniform suction at the stationary plate has been investigated. The effect of suction is to supply an adverse pressure gradient to the fluid which causes back flow near the stationary plate. The streamlines are drawn to show the effect of uniform suction at the stationary plate on the contact layer.     

On the steady flow of a Newtonian fluid between two parallel disk

Summary We show that the steady flow of a viscous fluid of moderate Reynolds number between two parallel disks has an exact solution which takes the form of a power series. Employing this exact solution the two-point boundary value problem for this class of flow is reduced to a nonlinear algebraic system which is then solved by a standard optimization method. Results are given for two particular cases, the continuous squeezing flow and the coaxial-disk flow.

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Zusammenfassung Wir zeigen, daß die ständige Strömung einer viskosen Flüssigkeit von mäßiger Reynoldszahl zwischen zwei parallelen Scheiben eine exakte Lösung hat, die die Form einer Potenzreihe annimmt. Wenn wir diese exakte Lösung anwenden, wird das Zwei-Punkte-Grenzwertproblem für diese Art von Strömung reduziert auf ein nichtlineares algebraisches System, das dann durch eine Optimierungsmethode gelöst werden kann. Angegeben werden Resultate für zwei Einzelfälle: die kontinuierliche Quetschströmung und die Strömung für Koaxialscheiben.

     

Flow of a viscous incompressible fluid between two parallel plates,one in uniform motion and the other at rest with uniform suction at the stationary plate

In the present paper an exact solution of the Navier-Stoke’s equations reduced to third and second order non-linear differential equations with appropriate boundary conditions is obtained. The longitudinal and transverse velocity profiles for λ=0·1, 1 and R=100 are drawn. It is noted that for large values of\(\bar x\), an adverse pressure gradient is developed which causes a back flow. There is increase in pressure even for very small fluid suction along the stationary plate. The skin friction and the flow coefficient decrease with reference to the suction velocity and the distance along the stationary plate. For λ=0, the results transform to the known results for plane couette flow without suction.     

On stationary two‐dimensional flows around a fast rotating disk

We study the two‐dimensional stationary Navier–Stokes equations describing flows around a rotating disk. The existence of unique solutions is established for any rotating speed, and qualitative effects of a large rotation are described precisely by exhibiting a boundary layer structure and an axisymmetrization of the flow.     

Hydromagnetic flow over a plane wall with uniform suction

Zusammenfassung Diese Abhandlung behandelt die Strömung einer inkompressiblen, zähen und elektrisch leitenden Flüssigkeit über eine ebene Wand mit homogener Absaugung und senkrechtem Magnetfeld. Eine exakte Lösung der hydromagnetischen Gleichungen kann erhalten werden, wenn die magnetische Druckzahl kleiner als eins ist.Aus den Rechnungen findet man, dass sowohl die Reibungsschubspannung als auch die Wärmeübertragung an der Wand unabhängig sind von der hydromagnetischen Wirkung, aber dass die Grenzschichtdicke zunimmt mit der Zunahme des hydromagnetischen Einflusses.Es sei bemerkt, dass keine stationäre Lösung erhalten wird für den Fall des Ausblasens.     

Transient flow in a rotating porous annulus due to suction at the walls

A highly porous material occupies the annular region between two coaxial infinitely long cylinders. A viscous incompressible fluid fills this porous medium and is initially in a state of rigid rotation together with the medium. The flow has been disturbed by imposing suction/injection at the outer/inner cylindrical boundaries respectively. The Brinkman's law has been used to represent the fluid motion. The exact solution for the resulting unsteady flow is obtained by Laplace transformation technique. The transient evolution of the boundary layers and the response of steady boundary layers to the resistance of the medium are discussed.     

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.     

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.     

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.     

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.     

Simultaneous effects of partial slip and thermal-diffusion and diffusion-thermo on steady MHD convective flow due to a rotating disk

The purpose of present research is to derive analytical expressions for the solution of steady MHD convective and slip flow due to a rotating disk. Viscous dissipation and Ohmic heating are taken into account. The nonlinear partial differential equations for MHD laminar flow of the homogeneous fluid are reduced to a system of five coupled ordinary differential equations by using similarity transformation. The derived solution is expressed in series of exponentially-decaying functions using homotopy analysis method (HAM). The convergence of the obtained series solutions is examined. Finally some figures are sketched to show the accuracy of the applied method and assessment of various slip parameter γ, magnetic field parameter M, Eckert Ec, Schmidt Sc and Soret Sr numbers on the profiles of the dimensionless velocity, temperature and concentration distributions. Validity of the obtained results are verified by the numerical results.     

Magnetohydrodynamic shear flow along a flat plate with uniform suction or blowing

An analysis is made of the steady shear flow of an incompressible viscous electrically conducting fluid past an electrically insulating porous flat plate in the presence of an applied uniform transverse magnetic field. It is shown that steady shear flow exists for suction at the plate only when the square of the suction parameter S is less than the magnetic parameter Q. In this case the velocity at a given point increases with increase in either the magnetic field or suction velocity. The shear stress at the plate increases with increase in either S or the free-stream shear-rate parameter σ₁ or Q. The analysis further reveals that solution exists for steady shear flow past a porous flat plate subject to blowing only when the square of the blowing parameter S₁ is less than Q. It is found that the induced magnetic field at a given location decreases with increase in Q. Further the wall shear stress decreases with increase in S₁. No steady shear flow is possible for blowing at the plate when S₁² > Q. Received: June 16, 2004; revised: October 24, 2004     

A numerical study of magnetohydrodynamic flow between a rotating and a stationary porous coaxial discs

The magnetohydrodynamic flow of an electrically conducting viscous fluid between a rotating and a stationary disc has been studied numerically by the Newton-Raphson method and the method of continuation. The results have been calculated for a wide range of parameters involved, thus indicating the efficiency of the methods. The induced electric field has also been calculated for various values of suction Reynolds number.