HALL EFFECTS ON UNSTEADY FLOW DUE TO NON-COAXIALLY ROTATING DISK AND A FLUID AT INFINITY

An investigation is made on the unsteady magnetohydrodynamic (MHD) flow caused by the non-coaxial rotation of a disk and a fluid at infinity being permeated by a transverse magnetic field. The disk is porous and nonconducting and executes oscillations in its own plane. The Laplace transform method is used to obtain the exact solution of the velocity field. The structure of the steady and unsteady flow fields is investigated. It is shown that the ultimate steady-state blowing solution is established in the presence of Hall current also for resonant frequency, which was not possible in the hydrodynamic case. The combined effects of Hall current, rotation, and suction or blowing are examined. The physical significance of mathematical results is given with various limiting cases.     

EFFECTS OF HALL CURRENT ON UNSTEADY FLOW OF A SECOND GRADE FLUID IN A ROTATING SYSTEM

Rotating flow of a second grade conducting fluid on an infinite oscillating plate is investigated when the fluid is permeated by a transverse magnetic field and the Hall effects are taken into account. It is once again found that an asymptotic solution exists in the presence of both suction and blowing at the plate. For fixed magnetic field parameter the boundary layer thickness increases with the increase in Hall parameter. The present analysis is more general than any previous investigations.     

EFFECTS OF HALL CURRENT ON UNSTEADY FLOW OF A SECOND GRADE FLUID IN A ROTATING SYSTEM

Rotating flow of a second grade conducting fluid on an infinite oscillating plate is investigated when the fluid is permeated by a transverse magnetic field and the Hall effects are taken into account. It is once again found that an asymptotic solution exists in the presence of both suction and blowing at the plate. For fixed magnetic field parameter the boundary layer thickness increases with the increase in Hall parameter. The present analysis is more general than any previous investigations.     

ON THE EFFECTIVNESS OF ION SLIP AND UNIFORM SUCTION OR INJECTION ON STEADY MHD FLOW DUE TO A ROTATING DISK WITH HEAT TRANSFER AND OHMIC HEATING

The steady flow and heat transfer of a conducting fluid due to the rotation of an infinite nonconducting porous disk in the presence of an axial uniform steady magnetic field are studied considering ion slip and ohmic heating. A uniform injection or suction is applied through the surface of the disk. The relevant equations are solved numerically using finite differences, and the solution shows that the inclusion of ion slip and the injection or suction through the surface of the disk gives some interesting results. It is found that the influence of the Hall and ion slip parameters on the velocity components is more pronounced in the injection than in the suction case. Also, ohmic heating has a marked effect on the heat transfer rate and it is of interest to see the reversal of the sign of the heat transfer rate for some values of the ion slip and suction parameters.     

ON THE EFFECTIVNESS OF ION SLIP AND UNIFORM SUCTION OR INJECTION ON STEADY MHD FLOW DUE TO A ROTATING DISK WITH HEAT TRANSFER AND OHMIC HEATING

The steady flow and heat transfer of a conducting fluid due to the rotation of an infinite nonconducting porous disk in the presence of an axial uniform steady magnetic field are studied considering ion slip and ohmic heating. A uniform injection or suction is applied through the surface of the disk. The relevant equations are solved numerically using finite differences, and the solution shows that the inclusion of ion slip and the injection or suction through the surface of the disk gives some interesting results. It is found that the influence of the Hall and ion slip parameters on the velocity components is more pronounced in the injection than in the suction case. Also, ohmic heating has a marked effect on the heat transfer rate and it is of interest to see the reversal of the sign of the heat transfer rate for some values of the ion slip and suction parameters.     

Transient three-dimensional heat transfer from rotating spheres with surface blowing

Transient heat transfer and thermal patterns around a rotating spherical particle with surface blowing are studied numerically for Reynolds numbers in the range 10?Re?300 and non-dimensional angular velocities up to Ω=1. This range of Reynolds number includes three distinct wake regimes: steady and axisymmetrical, steady but non-symmetrical, and unsteady with vortex shedding. The Navier-Stokes and energy equations for an incompressible viscous flow are solved numerically by a finite-volume method in a three-dimensional and time-accurate manner. The transient aspects of the thermal wakes associated with the aforementioned wake regimes have been explored. An interesting feature associated with particle rotation and surface blowing is that they can affect the near wake structure in such a way that an unsteady three-dimensional flow with vortex shedding develops at lower Reynolds numbers as compared to flow over a solid sphere in the absence of these effects, and thus, the temperature distributions around the particle are significantly affected. Despite the fact that particle rotation brings about major changes locally, the surface-averaged heat transfer rates are not influenced appreciably even at high rotational speeds; consequently, it is shown that the total heat transfer rates associated with rotating spheres with surface blowing can be calculated from heat transfer correlations developed for flow over evaporating droplets.     

UNSTEADY MAGNETOHYDRODYNAMIC NON-NEWTONIAN FLOW DUE TO NON-COAXIAL ROTATIONS OF DISK AND A FLUID AT INFINITY

An analysis is carried out to study the flow generated in a semi-infinite expanse of an incompressible second-grade fluid bounded by a porous oscillating disk. The flow is due to non-coaxial rotations of a disk and a fluid at infinity. The fluid is electrically conducting in the presence of a uniform transverse magnetic field. The solutions of the developed flow are obtained for the cases when the angular velocity is greater than, smaller than, or equal to the frequency of oscillation. The velocity field is found analytically by a Laplace transform technique. It is found that for uniform suction and blowing at the disk, shear oscillations are confined to the Ekman-Hartmann layer near the disk for all values of the frequencies.     

EXTENDED KALMAN FILTER CONTROLLER: FIRST PRINCIPLES MODELS TO NEURAL NETWORKS

In this study we have obtained an exact solution to the problem of heat and mass transfer in a rotating vertical porous channel taking into account the effects of Hall current. A strong magnetic field of uniform strength is applied along the axis of rotation. The entire system rotates about the axis normal to the plates with a uniform angular velocity. The porous channel is subjected to a constant suction/injection velocity as well as uniform free stream velocity. The nonlinear and coupled governing equations are solved by perturbation technique. The analytical expressions for primary and secondary velocity components, temperature and concentration fields, and shear stresses are obtained. The effects of the magnetic field, rotation of the channel, buoyancy force, Hall current, injection-suction parameter, and the temperature oscillation frequency are described during the course of discussion. The results are presented graphically and discussed.     

EFFECTS OF HALL CURRENT ON MHD FLOW IN A ROTATING CHANNEL PARTIALLY FILLED WITH A POROUS MEDIUM

In a rotating system, magnetohydrodynamic fully developed flow in a parallel-plate channel partially filled with a fluid-saturated porous medium and partially with a clear fluid is considered in the presence of an inclined magnetic field. Hall effects are taken into account and exact solutions of the governing MHD differential equations are obtained in a closed form. The effects of pertinent parameters such as the Hall current parameter (m), rotation parameter (R), permeability parameter (k), viscosities ratio (φ), and the angle of inclination ′θ′ of applied magnetic field on the velocity profiles and induced magnetic field are depicted graphically and discussed.     

THERMOSOLUTAL INSTABILITY OF A COMPRESSIBLE ROTATING WALTERS' (MODEL B′) ELASTICO-VISCOUS FLUID IN THE PRESENCE OF HALL CURRENTS

Effect of Hall currents is considered on Walters' (Model B′) elastico-viscous fluid heated and soluted from below in the presence of a vertical magnetic field. A dispersion relation governing the effects of viscoelasticity, salinity gradient, rotation, magnetic field, and Hall current is derived. For the case of stationary convection, the Walters' (Model B′) fluid behaves like an ordinary Newtonian fluid. The compressibility, stable solute gradient, rotation, and magnetic field postpone the onset of thermosolutal instability, whereas Hall currents are found to hasten the onset of thermosolutal instability in the absence of rotation. In the presence of rotation, Hall currents postpone/hasten the onset of instability depending upon the value of wave numbers as small/large. Again, the dispersion relation is analyzed numerically and the results depicted graphically. The viscoelasticity, solute gradient, and magnetic field (and corresponding Hall currents) introduce oscillatory modes in the system that were nonexistent in their absence. The case of over-stability is discussed and sufficient conditions for nonexistence of over-stability are derived.     

Liquid flow circulating within an unbaffled vessel agitated with an unsteady forward–reverse rotating impeller

Liquid‐phase mixing is a common operation, often performed in vessels using mechanically rotating impellers. To enhance axial mixing the vessels are generally equipped with baffles; however, in industries where cleaning the vessel interior is a major concern, i.e. food and pharmaceuticals, and crystallization, where baffles can disturb particle growth, unbaffled vessels are preferred. One method of agitation in unbaffled vessels is an impeller that periodically changes either the direction or rate of rotation: so‐called unsteady rotation. For use in an enhanced agitation vessel, an agitation technology using an unsteady forward–reverse rotating impeller in an unbaffled vessel was investigated. Such unsteady agitation is expected to enhance mixing. However, knowledge of the liquid flows in such an apparatus remains elusive. Thus an aim of this work was to characterize the circulation flow in such a system. Circulation by a disk turbine impeller with six flat blades was studied through examination of tracer particle trajectories. Images showing flow patterns with the forward–reverse rotating impeller resembled those obtained with a unidirectionally rotating impeller in a baffled vessel. The pattern was characterized by a circulation loop whose pathway exits from the impeller rotational region and returns to that region past the wall and bottom of the vessel. Time‐series particle tracking velocimetry (PTV) images obtained during one forward–reverse rotation of the impeller showed that the flow near the vessel wall reduced the periodic fluctuation downstream and that a flow that was almost independent of time was induced near the vessel bottom. For the flow from the bottom to the impeller, unsteadiness was provided by proximity to the impeller. Based on the intensity distribution of the unsteady flow produced by this type of forward–reverse rotating impeller within the vessel, the unsteady flow was shown to have the potential to reach the region near the vessel wall. Copyright © 2010 Society of Chemical Industry     

FLOW OF A NON-NEWTONIAN FLUID WITH HEAT TRANSFER ABOVE A ROTATING DISK CONSIDERING THE ION SLIP

The steady flow and heat transfer of a conducting non-Newtonian fluid due to the rotation of an infinite nonconducting disk in the presence of an axial uniform steady magnetic field are studied considering the ion slip. The governing nonlinear equations are solved numerically using finite differences, and the solution shows that the ion slip and the non-Newtonian fluid characteristics give some interesting results.     

FLOW OF A NON-NEWTONIAN FLUID WITH HEAT TRANSFER ABOVE A ROTATING DISK CONSIDERING THE ION SLIP

The steady flow and heat transfer of a conducting non-Newtonian fluid due to the rotation of an infinite nonconducting disk in the presence of an axial uniform steady magnetic field are studied considering the ion slip. The governing nonlinear equations are solved numerically using finite differences, and the solution shows that the ion slip and the non-Newtonian fluid characteristics give some interesting results.     

Numerical Modeling of Particle Dynamics in a Cylindrical Chamber Containing a Rotating Disk

Numerical modeling was performed to study the submicron particle dynamics in a confined flow field containing a rotating disk, temperature gradient, and various inlet gas flow rates. The Lagrangian model was employed to compute particle trajectories under the temperature gradient, disk rotation speed, and inlet gas flow rate effects. The trajectories of particles with diameters of 1 μm, 0.1 μm, and 0.01 μm were examined in this study. When the inlet gas temperature was lower than that of the disk, particle-free zones were created due to upward thermophoretic force for 1 μm and 0.1 μm particles. Disk rotation was found to depress the size of the particle-free zone. Particle deposition onto the disk for 0.01 μm particles was possible because of the Brownian motion effect. A detailed evaluation of the particle-free zone size as a function of the temperature gradient, disk rotation speed, and inlet gas flow rate was performed. When the inlet gas temperature was higher than the disk temperature, particle deposition onto the disk was enhanced due to the downward thermophoretic force for 1 μm and 0.1 μm particles. Disk rotation was found to increase the deposition rate. For 0.01 μm particles, Brownian motion was more important than thermophoretic force in controlling particle behavior. The particle deposition rates as a function of the temperature gradient, disk rotation speed, and inlet gas flow rate were performed.     

EFFECTS OF COMBINED TEMPERATURE- AND DEPTH-DEPENDENT VISCOSITY AND HALL CURRENT ON AN UNSTEADY MHD LAMINAR CONVECTIVE FLOW DUE TO A ROTATING DISK

The present study investigates the effects of mixed temperature- and depth-dependent viscosity and Hall current on an unsteady flow of an incompressible electrically conducting fluid on a rotating disk in the presence of a uniform magnetic field. We assume that the fluid viscosity strongly depends on temperature and depth, which may be directly applicable to the earth's mantle and a uniform mid-ocean ridge basalt reservoir in whole mantle flow. The system of axial symmetric nonlinear partial differential equations governing the unsteady flow and heat transfer is written in cylindrical polar coordinates and reduced to nonlinear ordinary differential equations by introducing suitable similarity parameters. Solutions for the flow and temperature fields are obtained numerically assuming large Prandtl number by using Runge-Kutta and shooting methods. The nature of radial, tangential, and axial velocities and temperature in the presence of a uniform magnetic field is presented for changing various nondimensional parameters at different layers of the medium. The coefficients of skin frictions and the rate of heat transfer are calculated at different parameters. Comparison has been made for steady flow (C = 0) and shows excellent agreement with Sparrow and Gregg (1959 Sparrow , E. M. , and Gregg , J. L. ( 1959 ). Heat transfer from a rotating disk to fluids of any Prandtl number , J. Heat Transfer C , 81 , 249 – 251 . [Google Scholar]), hence encouragement for the use of the present numerical computations.     

LIQUID FLOW IN IMPELLER REGION OF AN UNBAFFLED AGITATED VESSEL WITH UNSTEADILY FORWARD-REVERSE ROTATING IMPELLER

For an unbaffled agitated vessel with an unsteadily forward-reverse rotating impeller whose rotation proceeds with repeated acceleration, deceleration, and stop-reverse processes, the liquid flow in the impeller region was studied based on photographs showing path lines of tracer particles. An image series taken during one cycle of the forward-reverse rotation was analyzed to characterize the internal stream inside the impeller rotational region and the discharge stream outside its region when a disk turbine impeller with six flat blades was rotated unsteadily. Because of the unsteady flow generated inside the impeller rotational region, the velocity vector of outflow from its region fluctuated periodically with the change of the impeller rotation rate. The circumferential velocity was almost in phase with the impeller rotation rate, oscillating periodically. The radial velocity exhibited larger values in the process for the impeller from decelerating to stopping and reversal. The radial flow, whose velocity decreased downstream outside the impeller rotational region, was clarified to be transformed into upward and downward axial flows that are almost uniform in the circumferential direction throughout the region near the vessel wall.     

UNSTEADY FLOW DUE TO NON-COAXIAL ROTATION OF A POROUS DISK AND A FLUID AT INFINITY THROUGH POROUS MEDIUM

An exact solution of the unsteady hydrodynamic flow in a porous medium due to non-coaxial rotations of a porous disk and a fluid at infinity has been obtained. An analytical solution of the problem is obtained at small and large times by the Laplace transform method. It is found that the primary velocity decreases while the secondary velocity increases with the increase in porosity parameter σ. It is also found that the porosity takes more time to reach the steady state than that in the case of no porosity.     

LIQUID FLOW IN IMPELLER REGION OF AN UNBAFFLED AGITATED VESSEL WITH UNSTEADILY FORWARD-REVERSE ROTATING IMPELLER

For an unbaffled agitated vessel with an unsteadily forward-reverse rotating impeller whose rotation proceeds with repeated acceleration, deceleration, and stop-reverse processes, the liquid flow in the impeller region was studied based on photographs showing path lines of tracer particles. An image series taken during one cycle of the forward-reverse rotation was analyzed to characterize the internal stream inside the impeller rotational region and the discharge stream outside its region when a disk turbine impeller with six flat blades was rotated unsteadily. Because of the unsteady flow generated inside the impeller rotational region, the velocity vector of outflow from its region fluctuated periodically with the change of the impeller rotation rate. The circumferential velocity was almost in phase with the impeller rotation rate, oscillating periodically. The radial velocity exhibited larger values in the process for the impeller from decelerating to stopping and reversal. The radial flow, whose velocity decreased downstream outside the impeller rotational region, was clarified to be transformed into upward and downward axial flows that are almost uniform in the circumferential direction throughout the region near the vessel wall.     

Radiation-absorption,chemical reaction,Hall and ion slip impacts on magnetohydrodynamic free convective flow over semi-infinite moving absorbent surface

The investigation of radiation-absorption,chemical reaction,Hall and ion-slip impacts on unsteady MHD free convective laminar flow of an incompressible viscous,electrically conducting and heat generation/absorbing fluid enclosed with a semi-infinite porous plate within a rotating frame has been premeditated.The plate is assumed to be moving with a constant velocity in the direction of fluid movement.A uniform transverse magnetic field is applied at right angles to the porous surface,which is absorbing the fluid with a suction velocity changing with time.The non-dimensional governing equations for present inves-tigation are solved analytically making use of two term harmonic and non-harmonic functions.The graphical results of velocity,temperature and concentration distributions on the analytical solutions are displayed and discussed with reference to pertinent parameters.It is found that the velocity profiles decreased with an increasing in Hartmann number,rotation parameter,the Schmidt number,heat source parameter,while it increased due to an increase in permeability parameter,radiation-absorption param-eter,Hall and ion slip parameters.However,the temperature profile is an increasing function of radiation-absorption parameter,whereas an increase in chemical reaction parameter,the Schmidt num-ber Sc or frequency of oscillations decrease the temperature profile on cooling.Also,it is found that the concentration profile is decreased with an escalating in the Schmidt number or the chemical reaction parameter.     

非中心入射流旋转圆盘水跃现象的实验与模拟

针对非中心入射流旋转圆盘,设计了利用高速摄像系统记录液膜流动的旋转圆盘装置,对圆盘表面的流体流动现象进行了研究。利用VOF（Volume of Fluid）多相流模型,建立了非中心入射流旋转圆盘进口区域气液两相流模型,并与实验测量结果进行对比,模拟值与实验值的平均相对误差为9%,证明了CFD模型的准确性。随后通过CFD方法分别研究了进口流量和圆盘旋转速度对非中心入射流旋转圆盘进口区域水跃现象的影响,结果表明在进口旋转方向前方,进口流量越大,圆盘旋转速度越小,水跃半径越大,水跃现象越明显;在进口旋转方向后方,进口流量越大,水跃半径越大,而随着圆盘旋转速度的增加,水跃半径先增大后减小。对进口旋转方向后方的水跃半径与进口流量和旋转速度的关系进行拟合,得到相关经验关联式,拟合值与模拟值的误差范围在15%以内。