Unsteady hydromagnetic flow in a rotating channel with oscillating pressure gradient

Summary The unsteady hydromagnetic flow of a viscous incompressible fluid in the presence of a uniform transverse magnetic field in a rotating parallel plate channel with oscillating pressure gradient is investigated. An exact solution of the governing equations for the fully developed flow is obtained in closed form. The solution in dimensionless form contains three parameters: the Hartman NumberM ²,K ², the reciprocal of the Ekman Number, and the frequency of oscillation. The effects of these parameters on the flow field are studied. For large values ofK ² and there arise thin double-decker boundary layers near the plates of the channel and thin boundary layers for largeM ².

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Instationäre, hydromagnetische Strömung in einem rotierenden Kanal mit oszillierendem Druckgefälle

Zusammenfassung Die instationäre, hydromagnetische Strömung einer viskosen, inkompressiblen Flüssigkeit unter Einwirkung eines homogenen Magnetfelds quer zur Strömungsrichtung in einem durch rotierende, parallele Platten gebildeten Kanal mit oszillierendem Druckgradienten wird untersucht. Es wird eine exakte Lösung der beschreibenden Gleichungen für die voll entwickelte Strömung in geschlossener Form erhalten. In dimensionsloser Darstellung hängt die Lösung von drei Parametern ab: Der HartmanzahlM ²,K ², dem Reziprokwert der Ekmanzahl und , der Frequenz der Schwingung. Die Auswirkungen dieser Parameter auf das Strömungsfeld werden untersucht. Für große Werte vonK ² und entstehen dünne, doppelt gepackte Grenzschichten an den Platten des Kanals und dünne Grenzschichten für großeM ².

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With 8 Figures     

Hydromagnetic couette flow of an Oldroyd-B fluid in a rotating system

The motion of electrically conducting, Oldroyd-B and incompressible fluid between two infinitely extended non-conducting parallel plates under a uniform transverse magnetic field, fixed relative to the fluid has been considered. The lower plate is at rest and the upper plate is oscillating in its own plane. The governing partial differential equation of this problem, subject to boundary conditions are solved analytically. The expressions for the steady and unsteady velocity fields for the conducting Oldroyd-B fluid are obtained. The graphs are plotted for different values of dimensionless parameters of the problem and the analysis of the results showed that the flow field is appreciably influenced by the applied magnetic field, the rotation and the material parameters of the fluid.     

Numerical study of plane couette flow in a rotating framework

Summary A numerical study of laminar plane Couette flow subjected to a steady spanwise rotation is conducted. The full nonlinear Navier-Stokes equations in a steadily rotating framework are solved by a finite difference method for a long, large-aspect-ratio rectangular channel where the outer wall moves at a constant velocity. In this manner, nonlinear and wall end effects which are present in any real laboratory Couette flow experiment are taken into account. The computations demonstrate the existence of a roll instability, at intermediate rotation rates, when the Reynolds number exceeds a critical value of 42. The associated secondary flow in the form of longitudinal rolls is shown to have a severe distortional effect on the primary axial velocity in the interior of the channel which is not linear like its counterpart in an inertial framework. Comparisons are made with previously conducted linear stability analyses as well as with other analogous numerical and experimental studies.With 15 Figures     

Unsteady flow from a source in a rotating fluid

Summary A source, which is situated on the axis of a rotating fluid, commences to expel fluid with constant rate at the time t=0. We describe how the geostrophic forces lead to the formation of a narrow column along the axis, before the eventual development of the viscous Stewartson column along the axis, and how the final steady state is achieved. An understanding of the role of the non-linear inertial forces in the neighbourhood of the source is given. The results are also extended by considering the effect of placing the source between two infinite discs situated perpendicular to the axis of rotation.     

Numerical solution of the stability of hydromagnetic couette flow with hall currents

In the present problem, we have studied numerically the effects of Hall currents and the magnetic Prandtl number $\text{P}{}_{\mathrm{m}}$ on the stability of hydromagnetie Couette flow. In obtaining a numerical solution of the problem by Runge-Kutta-Merson method (here after will be called as RKM method), we have used the approximate solution found by modified Galerkin method as the starting solution. The accuracy of the method is tested by comparing the numerical solution with the exact solution of Chandrasekhar [1]. It is found that even though Hall currents are destabilizing for small and large $\text{P}{}_{\mathrm{m}}$, they tend to stabilize for certain intermediate values.     

Unsteady MHD rotating flow over a rotating sphere near the equator

The unsteady rotating flow of a laminar incompressible viscous electrically conducting fluid over a rotating sphere in the vicinity of the equator has been studied. The fluid and the body rotate either in the same direction or in opposite directions. The effects of surface suction and magnetic field have been included in the analysis. There is an initial steady state that is perturbed by a sudden change in the rotational velocity of the sphere, and this causes unsteadiness in the flow field. The nonlinear coupled parabolic partial differential equations governing the boundary-layer flow have been solved numerically by using an implicit finite-difference scheme. For large suction or magnetic field, analytical solutions have also been obtained. The magnitude of the radial, meridional and rotational velocity components is found to be higher when the fluid and the body rotate in opposite directions than when they rotate in the same direction. The surface shear stresses in the meridional and rotational directions change sign when the ratio of the angular velocities of the sphere and the fluid ₀. The final (new) steady state is reached rather quickly which implies that the spin-up time is small. The magnetic field and surface suction reduce the meridional shear stress, but increase the surface shear stress in the rotational direction.     

Current-induced instabilities of hydromagnetic flow in a small gap between rotating conducting cylinders

Summary.  In considering the stability of an electrically conducting fluid between rotating perfectly conducting cylinders with a current-induced pressure gradient acting in the azimuthal direction and with an applied axial magnetic field, the assumption of small-gap approximation is made and the governing equations with respect to both axisymmetric and non-axisymmetric disturbances are solved by a direct numerical procedure. A parametric study covering wide ranges of Q, the Hartmann number which represents the strength of the axial magnetic field, and β, a parameter characterizing the ratio of current-induced and rotation velocities, is conducted for the situation where the outer cylinder is stationary and the inner cylinder is rotating. It is found that the stability characteristics are thoroughly different from those of the case of weakly conducting walls. The variation of the onset mode is shown in the (β, Q)-plane, and the transition of the corresponding neutral curves is discussed in detail. Results for the critical Taylor number and wavenumber pertaining to the critical disturbances are presented. The critical values of radial current density required for the onset of instability are also determined. Received May 8, 2002; revised November 11, 2002 Published online: May 8, 2003 The financial support for this work from National Science Council of Taiwan, ROC, through Grant No. NSC 89-2212-E-132-006 is gratefully acknowledged.     

Unsteady nonsimilar compressible boundary layer flow over a rotating sphere

Summary The problem of determining the development of the nonsimilar viscous compressible boundary layer flow with time over a rotating sphere is considered when both free stream velocity and rotation vary arbitrarily with time. Variations of skin friction, heat transfer and displacement thickness with mass injection, rotation, wall enthalpy and time are discussed.     

Effect of wall conductances on hydromagnetic flow and heat transfer in a rotating channel

Summary Wall conductance effects on the hydromagnetic flow and heat transfer between two parallel plates in a rotating frame of reference has been studied when the liquid is permeated by a transverse magnetic field. An exact solution of the governing equations has been obtained. It is found that the velocity, current density and the temperature depend only on the sum of the wall conductances ₁ + ₂ = but magnetic field depends on the individual values of ₁ and ₂, where ₁ and ₂ are respectively the wall conductance ratios of the upper and lower walls.

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Einfluß der Wandleitfähigkeit auf die hydromagnetische Strömung und den Wärmeübergang in einem rotierenden Kanal

Zusammenfassung In der Arbeit wird der Einfluß der Leitfähigkeit der Wände auf die MHD-Strömung sowie auf den Wärmeübergang untersucht für den Fall, daß sich das zähe Medium zwischen zwei rotierenden Platten befindet und ein Magnetfeld parallel zur Rotationsachse angelegt wird. Es wird eine exakte Lösung des Problems gegeben. Es zeigt sich, daß die Geschwindigkeit, die Stromdichte und die Temperatur nur von der Summe der Leitfähigkeiten der Wände + ₂ = abhängen, dagegen hängt die magnetische Feldstärke von den individuellen Werten ₁ und ₂ ab, wobei ₁ und ₂ die Leitfähigkeiten der oberen bzw. unteren Wand bezeichnen.

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With 5 Figures     

Taylor stability of a thermoviscoelastic fluid in couette flow

Taylor stability in Couette flow of a thermoviscoelastic fluid between two rotating coaxial circular cylinders is investigated employing the constitutive theory of Eringen and Koh. The stability problem is reduced to a characteristic value problem which is solved to obtain stability criteria in terms of critical Values of Taylor numbers. The results are compared with those of the classical viscous fluids and viscoinelastic fluids and the stabilizing effects of the thermoviscoelastic nature of the fluid are discussed.     

侧壁式压水室离心泵小流量非稳态旋转失速特性

用数值计算方法研究具有特殊结构的侧壁式压水室离心泵,分析小流量工况时模型泵的非稳态旋转失速特性,用快速傅里叶变换(FFT)获得压力脉动信号的频谱特征。结果表明,小流量工况时模型泵的扬程曲线呈驼峰状,压水室不同位置处压力分布不均;受叶轮旋转产生的非稳态作用影响,叶轮不同叶片流道内流动结构差异较大。不同流量下,叶轮内部分离涡结构诱发的激励频率各异,0.4ФN工况时模型泵压力脉动频谱图出现0.5fR及高次谐波频率,压力脉动最大幅值出现于4fR频率处;0.2ФN流量时非定常流动结构会诱发0.18fR及高次谐波频率;0.05ФN流量时压力脉动频谱图同时出现0.1fR、0.28fR两种激励频率。旋转失速现象出现时,频谱图中叶频处压力脉动幅值不再起主导作用。     

Hall effects on unsteady couette flow

Hall effects on Couette flow of an electrically conducting fluid between two parallel plates for both the cases of impulsive as well as uniformly accelerated motion of one of the plates is discussed. Expressions for the shear stress components are obtained in terms of two non-dimensional parameters, the Hartmann number and the Hall parameter. It is found that for the case of impulsive as well as accelerated motion of one of the plates, the magnitude of the shear stress component due to the primary flow decrease whereas that due to the secondary flow increase with increase in Hall parameter.     

Heat transfer in generalized couette flow of a nonlinear viscoplastic fluid

The steady-state heat-transfer problem is solved for dissipative pressure flow of a nonlinear viscoplastic fluid between two parallel isothermal plates, one of which is moving at a constant velocity while the other is stationary.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 34, No. 6, pp. 1070–1080, June, 1978.