Calculation of laminar flow of a viscous fluid between rotating disks

The results are given of a numerical investigation of the laminar flow of a viscous incompressible fluid with heat transfer from the periphery to the center between two rotating disks. The system is a simplified model of one of the elements of the cooling circuit of a gas turbine. The complete Navier—Stokes equations in the vortlcity—flow function variables were solved by an explicit conservative scheme with appoximation of the convective terms of divergence type by directed differences. The calculations were made in a wide range of variation of the dimensionless determining parameters of the problem. The results agree well with the known experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 76–81, January–February, 1982.We thank V. M. Kapinos for discussion and helpful comments.     

Magnetohydrodynamic laminar source flow between two parallel disks

The steady axisymmetrical laminar source flow of an incompressible conducting fluid between two circular parallel disks in the presence of a transverse magnetic field is analytically investigated. A solution is obtained by expanding the velocity and the pressure distribution in terms of a power series of 1/r. Velocity, induced magnetic field, pressure and shear stress distributions are determined and compared with the case of the hydrodynamic solution. Pressure is found to be a function of both r and z in the general case and the flow is not parallel. At high magnetic fields, the velocity distribution degenerates to a uniform core surrounded by a boundary layer near the disks.Nomenclature C _f skin friction coefficient - H ₀ impressed magnetic field - H _r induced magnetic field in the radial direction, H _r /H ₀ - M Hartmann number, H ₀ t(/)^1/2 - P dimensionless static pressure, P*t ⁴/Q - P* static pressure - P ₀ reference dimensionless pressure - Q source discharge - R outer radius of disks - Rm magnetic Reynolds number, Q/t - Re Reynolds number, Q/t - 2t channel width - u dimensionless radial component of the velocity, u*t ²/Q - u* radial component of the velocity - w dimensionless axial component of the velocity, w*t ²/Q - w* axial component of the velocity - z, r dimensionless axial and radial directions, z*/t and r*/t, respectively - z*, r* axial and radial direction, respectively - magnetic permeability - coefficient of kinematic viscosity - density - electrical conductivity - ² LaPlacian operator in axisymmetrical cylindrical coordinates     

Magnetohydrodynamic laminar flow between porous disks

The steady two-dimensional laminar flow of an incompressible conducting fluid between two parallel circular disks in the presence of a transverse magnetic field is investigated. A solution is obtained by perturbing the creeping flow solution and it is valid only for small suction or injection Reynolds numbers. Expressions for velocity, induced magnetic field, pressure, and shear stress distribution are determined and are compared with the creeping flow and hydrodynamic solutions. It is found that the overall effect of the magnetic field on the flow is the same as that in the Hartmann flow.Nomenclature stream function - 2h channel width - z, r axial and radial coordinates - radius of the disk - U _r radial component of velocity - U _r average velocity in the radial direction, U _r d - U _z axial component of velocity - U ₀ injection or suction velocity - dimensionless axial coordinate, z/h - f() function defined in (8) - density - coefficient of kinematic viscosity - electrical conductivity - magnetic permeability - H ₀ impressed magnetic field - h _r induced magnetic field, H _r /H ₀ - M Hartmann number, H ₀ h(/)^1/2 - R Reynolds number, U ₀ h/ - R _m magnetic Reynolds number, U ₀ r - A constant defined in (15) - K constant defined in (27) - C ₂ constant defined in (26) - p pressure - C _p pressure coefficient - C _f skin friction coefficient     

Hydromagnetic flow between two porous disks rotating about non-coincident axes

Hydromagnetic flow between two porous disks rotating with same angular velocity Ω about two noncoincident axes has been studied in the presence of a uniform transverse magnetic field. An exact solution of the governing equations has been obtained in a closed form. It is found that the primary velocity f/Ωl increases and the secondary velocity g/Ωl decreases with increase in either Reynolds number Re or the Hartman number M. It is also found that the torque at the disk η= 0 increases with increase in either M^2 or K^2. On the other hand there is no torque at the disk η= 1 for large M^2 and K^2. The heat transfer characteristic has also been studied on taking viscous and Joule dissipation into account. It is seen that the temperature increases with increase in either M^2 or K^2. It is found that the rate of heat transfer at the disk η= 0 increases with increase in either M or K. On the other hand the rate of heat transfer at the disk η= 1 increases with increase in K but decreases with increase in M.     

Flow between two unsteadily rotating disks

Summary This is a theoretical investigation of the unsteady laminar flow of a viscous incompressible fluid between two infinite parallel disks, which are rotating with angular velocities varying with time. The solution is obtained in the form of a series expansion about the quasi-steady state. The deviation of the actual instantaneous state of the flow from the quasi-steady state is determined.     

Numerical investigations of flow of a viscoelastic fluid between rotating coaxial disks

Summary A numerical investigation is made of the problem of flow of a viscoelastic fluid of the Rivlin-Ericksen type between rotating coaxial disks. The finite-difference analogues of the governing nonlinear ordinary differential equations are written and the resulting equations are solved using point successive overrelaxation method (SOR) under the appropriate boundary conditions. Two cases of interest are treated, namely, when one of the disk is at rest while the other rotates with a constant angular velocity and when both the disks rotate with constant angular velocities but in the opposite directions. Typical examples of Reynolds numberR in the range 10 R 1000 are described for various values of the non-Newtonian parameters and results are compared with those for a classical viscous fluid.

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Zusammenfassung Die Strömung einer viskoelastischen Flüssigkeit vom Rivlin-Ericksen-Typ zwischen zwei rotierenden koaxialen Kreisscheiben wird numerisch untersucht. Die das Problem beschreibenden nichtlinearen Differentialgleichungen werden in analogen Gleichungen für finite Differenzen umgeschrieben und unter angepaßten Randbedingungen mit einem punktweisen Über-Relaxations-Verfahren (SOR) gelöst. Es werden zwei interessierende Fälle behandelt, zuerst der Fall, bei dem eine Platte ruht und die andere mit konstanter Winkelgeschwindigkeit rotiert, als zweiter derjenige, bei dem beide Platten mit gleichen konstanten Winkelgeschwindigkeiten, jedoch entgegengesetztem Drehsinn rotieren. Typische Beispiele für Reynoldszahlen zwischen 10 und 1000 werden für verschiedene Werte der nicht-newtonschen Parameter beschrieben und die Ergebnisse mit denen für newtonsche Flüssigkeiten verglichen.

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With 9 figures     

Theoretical and experimental investigation of the flow of a viscoelastic fluid in the gap between two rotating disks

A theory of the nonlinear viscoelastic behavior of polymer fluids has been constructed in [1]. The theory was used in [2] to investigate the motion of a nonlinear viscoelastic medium under steady and unsteady deformation rates in simple shear flow, and a comparison was made with experiment. The experiments in [2], which were performed on a cone-plate Weissenberg rheogoniometer, indicate that this arrangement is unsuitable for measurements of normal stresses under unsteady conditions in fluids with a fairly high viscosity. Below, we will show the suitability of using a disk-disk Weissenberg rheogoniometer to measure normal stresses in this case for unsteady conditions (transition to steady flow and stress relaxation). In this regard, a theoretical study of the flow of a viscoelastic fluid in the gap between rotating disks is needed. Note that in this case new information will be obtained from a comparison with simple uniform shear flow, since in the flow of a polymer between two disks all three normal stress components contribute to the axial force, while in the gap between a cone and a plate only the first normal stress difference contributes to the normal force.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 25–30, March–April, 1976.     

Squeeze flow of a second-order fluid between two parallel disks or two spheres

The normal viscous force of squeeze flow between two arbitrary rigid spheres with an interstitial second-order fluid was studied for modeling wet granular materials using the discrete element method. Based on the Reynolds‘ lubrication theory, the small parameter method was introduced to approximately analyze velocity field and stress distribution between the two disks. Then a similar procedure was carried out for analyzing the normal interaction between two nearly touching, arbitrary rigid spheres to obtain the pressure distribution and the resulting squeeze force. It has been proved that the solutions can be reduced to the case of a Newtonian fluid when the non-Newtonian terms are neelected.     

MHD flow and heat transfer of micropolar fluid between two porous disks

A numerical study is carried out for the axisymmetric steady laminar incompressible flow of an electrically conducting micropolar fluid between two infinite parallel porous disks with the constant uniform injection through the surface of the disks. The fluid is subjected to an external transverse magnetic field. The governing nonlinear equations of motion are transformed into a dimensionless form through von Karman’s similarity transformation. An algorithm based on a finite difference scheme is used to solve the reduced coupled ordinary differential equations under associated boundary conditions. The effects of the Reynolds number, the magnetic parameter, the micropolar parameter, and the Prandtl number on the flow velocity and temperature distributions are discussed. The results agree well with those of the previously published work for special cases. The investigation predicts that the heat transfer rate at the surfaces of the disks increases with the increases in the Reynolds number, the magnetic parameter, and the Prandtl number. The shear stresses decrease with the increase in the injection while increase with the increase in the applied magnetic field. The shear stress factor is lower for micropolar fluids than for Newtonian fluids, which may be beneficial in the flow and thermal control in the polymeric processing.     

Stability of viscous flow between rotating and stationary disks

The linear problem of the stability of viscous flow between rotating and stationary parallel disks is solved in the locally homogeneous formulation using the method of normal modes. The main flow is assumed to be selfsimilar with respect to the radial coordinate. The system of sixth-order equations, derived for the amplitude functions of the disturbances, is integrated by a finite difference method. The stability characteristics with respect to disturbances of four types are calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 79–87, November–December, 1991.     

Dynamical responses of non-isothermal flow between two independently rotating disks with time-dependent wall conditions

The present work develops a theoretical model of rotational convection and uses it to investigate the dynamical responses of the flow and heat transfer between two disks rotating at different rates under the influences of time-dependent disturbances. The unsteady non-isothermal flow model is formulated by extending a recently developed steady-state similarity model of axi-symmetric rotational convection. In the new model all the rotation-induced buoyancy forces are considered. Using one disk as reference, effects of the time-dependent changes in wall temperature or rotating rate of the other disk on the flow and heat transfer are explored. Various rotational modes with asymptotic or fluctuating change in boundary condition of temperature or disk rotation are studied. The present time-dependent model for this non-isothermal rotating flow is numerically solved by a finite-difference method. By using the present results, the complicated flow and heat transfer mechanisms with thermal-flow coupling in the class of time-dependent rotational convection are manifested.     

An analysis on entrance region effect of the laminar radial flow between two parallel disks

In this paper, B. B. Golubef method^[1] is used for calculating the radial diffuse flow between two parallel disks for the first step. The momentum integral equation together with the energy integral equation is derived from the boundary layer momentum equation, and the expression of secondary approximation explicit function in which the channel length of entrance region varies with the boundary layer thickness can be obtained by using Picard iteration[2] in the solution of the energy integral equation. Therefore, this has made it possible to analyze directly and analytically the coefficients of the entrance region effect. In particular, when the outer diameter of disk is smaller than the entrance region length, the advantage of this method can be prominently manifest.Only because the energy integral equation is employed, the terms in the pressure loss coefficient can be independently derived theoretically. The computable value of the pressure loss coefficient presented in this paper is nearer to the testing value than that in ref. [3] when the entrance correction Reynolds number Re<100. Therefore the results in this paper within Re<100 are both reliable and simple.     

Oscillations of a viscous fluid in a thin layer between two coaxial disks rotating together

A study is made of the problem of the motion of an incompressible viscous fluid in the space between two coaxial disks rotating together with constant angular velocity under the assumption that the pressure changes in time in accordance with a harmonic law. The problem is solved using the equations of unsteady motion of an incompressible viscous fluid in a thin layer. It is shown that the velocity field in this case is a superposition on a steady field of damped oscillations with cyclic frequency equal to twice the angular velocity of the disks and forced oscillations with cyclic frequency equal to the cyclic frequency of the oscillations of the pressure field. It is shown that the amplitude of the forced oscillations of the velocity field depends strongly on the ratio of the cyclic frequency of the oscillations of the pressure field to the angular velocity of the disks. It is shown that there is a certain value of the ratio at which the amplitude of the forced oscillations has a maximal value (resonance). It is shown that even for very small amplitudes of the pressure oscillations the amplitude of the oscillations of the relative velocity at resonance may reach values comparable with the mean velocity of the main flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 166–169, January–February, 1984.     

Laminar thermal convection in rotating cavity between two disks

In solving several technical problems it is necessary to know what takes place in a closed rotating axisymmetric cavity filled with a nonuniformly heated viscous fluid. Such cavities are encountered, for example, in the rotors of steam and gas turbines. The thermal convection in these cavities is studied for a definite temperature condition of the rotors: in [1, 2] some qualitative considerations are presented, and quantitative estimates are given for thermal convection in cavities of turbine rotors; in [3,4] there is presented a very approximate calculation using the method of integral relations of the heat transfer coefficients in the case of a narrow cavity between two rotating disks which have different temperatures. We note that the thermal convection effect in a rotating cavity may be utilized in various technical devices, for example, in equipment for separating isotopes, etc. [5], A solution is presented for the problem of laminar thermal convection in a narrow cavity between two disks which are rotating with the same velocity and which have different temperatures which are constant along the radius. In the case of the narrow cavity we can neglect the influence of the cylindrical cavity rim on the flow in primary portion of the cavity (see [6]); therefore it is sufficient to solve the self-similar problem for two infinite disks.In conclusion I would like to thank A. Z. Serazetdinov and V. L. Karaseva for carrying out the computer calculations.     

Axisymmetric flows of an incompressible fluid between movable rotating disks

Within the Karman family of exact solutions of the Navier-Stokes equations, some non-selfsimilar solutions are considered to the problem of unsteady incompressible flow between two rotating disks one of which moves along a common rotation axis. Three classes of the flow regimes are studied: (i) a flow between the non-rotating disks, (ii) a flow between the disks rotating with identical angular velocities, and (iii) a flow between the disks rotating with opposite velocities. Examples of exact rotationally symmetric solutions for the inviscid-fluid equations, satisfying the no-slip conditions, are given.