Showing papers on "Hartmann number published in 1993"
TL;DR: In this paper, the authors studied the convection of an electrically conducting liquid in a horizontal cylinder (horizontal Bridgman configuration) with a uniform steady vertical magnetic field externally imposed.
Abstract: We study the convection of an electrically conducting liquid in a horizontal cylinder (horizontal Bridgman configuration). A uniform steady vertical magnetic field is externally imposed. The thermal and magnetic Prandtl numbers are assumed equal to zero. The thermal field is obtained assuming pure conduction and certain conditions at the boundaries, so that the heat flux is axial and uniform. The influence of the cylinder's cross-sectional shape is examined. In the high Hartmann number limit (Ha [Gt ] 1), an analytical solution is found for the fully established flow. With electrically insulating walls, the magnetically damped convective velocity varies as Ha-2 when the cross-section has a horizontal plane of symmetry, while it varies as Ha-1 for non-symmetrical shapes. When the walls are perfectly conducting, the damped velocity always varies as Ha-2.
80 citations
TL;DR: In this paper, a numerical study for magnetohydrodynamic free convection of an electrically conducting fluid in a two-dimensional rectangular enclosure in which two side walls are maintained at uniform heat flux condition.
Abstract: A numerical study is presented for magnetohydrodynamic free convection of an electrically conducting fluid in a two-dimensional rectangular enclosure in which two side walls are maintained at uniform heat flux condition. The horizontal top and bottom walls are thermally insulated. A finite difference scheme comprising of modified ADI (Alternating Direction Implicit) method and SOR (Successive-Over-Relaxation) method is used to solve the governing equations. Computations are carried out over a wide range of Grashof number, Gr and Hartmann number, Ha for an enclosure of aspect ratio 1 and 2. The influences of these parameters on the flow pattern and the associated heat transfer characteristics are discussed. Numerical results show that with the application of an external magnetic field, the temperature and velocity fields are significantly modified. When the Grashof number is low and Hartmann number is high, the central streamlines are elongated and the isotherms are almost parallel representing a conduction state. For sufficiently large magnetic field strength the convection is suppressed for all values of Gr. The average Nusselt number decreases with an increase of Hartmann number and hence a magnetic field can be used as an effective mechanism to control the convection in an enclosure.
59 citations
TL;DR: In this paper, the role of viscosity and its reflection in the Hartmann number in determining the existence and nonlinear evolution of instabilities was investigated for resistive equilibria in periodic cylinders.
Abstract: Magnetohydrodynamic activity near the threshold of instability is studied numerically for resistive equilibria in periodic cylinders. Attention focuses on the role of the viscosity (and its reflection in the Hartmann number) in determining the existence and nonlinear evolution of instabilities. Three identical axisymmetric resistive equilibria without flow are investigated, with three viscosities. The highest viscosity leads to stability of the axisymmetric state. The intermediate value leads to a helically deformed equilibrium with flow, with an (m,n)=(2,1) deformation. The lowest viscosity leads to a 'mixed' helically deformed, final, approximately steady state with flow, with (m,n)=(2,1) and (3,2) deformations and their harmonics, plus other, adjacent, modes near m=2n. It is concluded that it is likely that the numerical value of the viscosity, and possibly the form of the viscous stress tensor, must never be ignored in discussions of near-threshold incompressible MHD activity in driven, dissipative MHD plasmas.
30 citations
TL;DR: In this article, the temperature distribution of viscous incompressible fluid flow between two parallel walls has been analyzed in the presence of a uniform magnetic field applied transversely to the flow and when rotated at an angular velocity about an axis perpendicular to the walls, taking Hall currents into account.
28 citations
01 Jun 1993
TL;DR: Theory of the L -mode confinement in toroidal plasmas is developed in this paper, where the anomalous transport process is caused by self-sustained turbulence in the toroidal plasma.
Abstract: Theory of the L -mode confinement in toroidal plasmas is developed. The Prandtl number, the ratio between the ion viscosity and the thermal conductivity is obtained for the anomalous transport process which is caused by the self-sustained turbulence in the toroidal plasma. It is found that the Prandtl number is of order unity both for the ballooning mode turbulence in tokamaks and for the interchange mode turbulence in helical system. The influence on the anomalous transport and fluctuation level is evaluated. Hartmann number and magnetic Prandtl number are also discussed.
27 citations
TL;DR: Theory of the L -mode confinement in toroidal plasmas is developed in this paper, where the anomalous transport process is caused by self-sustained turbulence in the toroidal plasma.
Abstract: Theory of the L -mode confinement in toroidal plasmas is developed. The Prandtl number, the ratio between the ion viscosity and the thermal conductivity is obtained for the anomalous transport process which is caused by the self-sustained turbulence in the toroidal plasma. It is found that the Prandtl number is of order unity both for the ballooning mode turbulence in tokamaks and for the interchange mode turbulence in helical system. The influence on the anomalous transport and fluctuation level is evaluated. Hartmann number and magnetic Prandtl number are also discussed.
26 citations
TL;DR: In this paper, a numerical technique for searching for the stability boundary for a resistive, straight-cylinder, magnetohydrodynamic equilibrium with spatially-dependent resistivity was developed.
Abstract: A numerical technique is developed for searching for the stability boundary for a resistive, straight-cylinder, magnetohydrodynamic equilibrium with spatially-dependent resistivity. For fixed aspect ratio, the boundary is a curve in the plane whose axes are Hartmann number and pinch ratio (or reciprocal of the safety factor at the wall). The technique is spectral and utilizes orthonormal eigenfunctions of the curl. Nonlinear behavior above the stability boundary is computed for a particular profile, using a nonlinear version of the code.
23 citations
TL;DR: In particular, the Hartmann number as mentioned in this paper is the crucial number describing the onset of MHD activity in voltage-driven dissipative equilibria that model such confinement devices as tokamaks.
Abstract: Despite its limitations, magnetohydrodynamic theory remains the best possibility for a predictive framework for the large-scale dynamics of a magnetized plasma. The mathematical structure is very similar to that for Navier-Stokes fluids, and indeed contains fluid dynamics as a special case. Fluid dynamics would not have gotten very far without understanding the crucial role played by dimensionless numbers (such as the Reynolds number) in classifying its regimes of different kinds of behavior. The situation seems to be much the same in magnetohydrodynamics. In particular, the Hartmann number, familiar in the theory of MHD power generation, seems to be the crucial number describing the onset of MHD activity in voltage-driven dissipative equilibria that model such confinement devices as tokamaks. Stability thresholds are calculable and the supercritical behavior above those thresholds may be quantitatively compared with the numerical computations of Shan et al. (1991, 1993).
20 citations
TL;DR: In this paper, the steady flow of an electrically conducting, incompressible liquid in a duct with a constant rectangular cross section, with thin, electrically-conducting walls and with a non-uniform transverse magnetic field which is parallel to two duct walls is treated.
16 citations
TL;DR: In this paper, a Czochralski flow of liquid metal in a static crucible under either an axisymmetric vertical magnetic field or a horizontal uniform magnetic field was studied.
13 citations
TL;DR: In this article, an analytical solution for the isothermal melt motion and electric current density driven by the differential rotation of the crystal and crucible about their common vertical axis is presented for two simple field configurations, with only one class leading to the presence of the large meridional flow driven by differential rotation.
Abstract: The use of magnetic fields during the growth of semiconductor crystals from the melt in a Czochralski (CZ) crystal puller has shown promise in controlling the heat and mass transport to the growth interface. The magnetic field suppresses turbulence and thermal convection in the melt in which large thermal gradients are present, thus improving the quality of the crystal. In this paper, analytical solutions are presented for the isothermal melt motion and electric current density driven by the differential rotation of the crystal and crucible about their common vertical axis. There is an applied, non-uniform, axisymmetric magnetic field with only radial and axial components which are independent of the azimuthal coordinate. The melt motion with a uniform axial magnetic field represents a singular limit of the flow considered here: as the radial magnetic field component goes to zero, the radial and axial (meridional) velocity components decrease in magnitude by a factor of M-1, where M is the large Hartmann number. The uniform axial field is a singular limit because the centrifugal acceleration due to the azimuthal velocity is exactly perpendicular to the magnetic field. Since the radial isothermal motion near the growth interface controls the radial distributions of dopants and impurities in the crystals, a non-uniform axisymmetric magnetic field is better than the uniform axial field. In addition, the axisymmetric field avoids the detrimental deviations from axisymmetric heat and mass transport associated with a uniform transverse (horizontal) magnetic field.Two classes of shaped fields are considered, with only one class leading to the presence of the large meridional flow driven by differential rotation. The small electrical conductivity of the crystal plays an important role in determining the behaviour of the melt's angular velocity, which is constant along each magnetic field line. Results for two simple field configurations are presented in order to illustrate the effect of the field configuration on the nature of the meridional circulation and the potential for flow tailoring with the shaped field.
TL;DR: In this paper, a theory of self-sustained turbulence for resistive plasma in toroidal devices is presented. But this theory assumes that the current diffusivity prevails upon resistivity, and the turbulence nature changes so as to recover the L mode transport.
Abstract: A theory of the self‐sustained turbulence is developed for resistive plasma in toroidal devices. Pseudoclassical confinement is obtained in the low‐temperature limit. As temperature increases, the current diffusivity prevails upon resistivity, and the turbulence nature changes so as to recover the L‐mode transport. Comparison with experimental observation on this transition is made. The Hartmann number is also given.
TL;DR: In this article, Laminar liquid metal flow fields between two parallel insulator walls under a traveling sine wave magnetic field are numerically solved for a small magnetic Reynolds number.
Abstract: Laminar liquid metal flow fields between two parallel insulator walls under a traveling sine wave magnetic field are numerically solved for a small magnetic Reynolds number. When both average Hartmann number Ha and cubic root of interaction parameter N1/3 are much greater than dimensionless wavelength Λ, a set of turnaround lines of magnetic flux, which exist every half‐wavelength, cause a set of recirculating flows. Consequently a nozzlelike converging and diverging flow is formed between them. The throat width of the nozzlelike flow is estimated to be the greater of N−1/4Λ3/4 or Ha−1/2Λ1/2 compared with the channel height.
TL;DR: In this article, a numerical experiment was carried out to simulate the magnetohydrodynamic flow of a power law fluid through a flat duct, and several techniques were invoked in an attempt to obtain the results for various combinations of values of the power law index of the fluid and the Hartmann number.
TL;DR: In this article, a numerical investigation of the two-dimensional magnetohydrodynamic flow (MHD flow) in a rectangular duct and an error analysis of the traditional calculation of the solution are presented.
Abstract: Publisher Summary This chapter discusses a numerical investigation of the two-dimensional magnetohydrodynamic flow (MHD flow) in a rectangular duct and presents an error analysis of the traditional calculation of the solution. Arbitrary values of the flow parameters—the Hartmann number and the wall conduction ratio—are admitted. The singular perturbation problem is solved and analyzed by means of interval arithmetic and verified enclosure methods (E-Methods), supported by the programming languages PASCAL-SC and FORTRAN-SC. The error analysis of the traditional calculation as applied to this MHD flow shows that there is lack of reliability of published numerical results of this physical problem, and this even for Hartmann numbers M