0000000000089903
AUTHOR
Gunter Gerbeth
Breakdown of Burton-Prime-Slichter approach and lateral solute segregation in radially converging flows
A theoretical study is presented of the effect of a radially converging melt flow, which is directed away from the solidification front, on the radial solute segregation in simple solidification models. We show that the classical Burton-Prim-Slichter (BPS) solution describing the effect of a diverging flow on the solute incorporation into the solidifying material breaks down for the flows converging along the solidification front. The breakdown is caused by a divergence of the integral defining the effective boundary layer thickness which is the basic concept of the BPS theory. Although such a divergence can formally be avoided by restricting the axial extension of the melt to a layer of fi…
Stability of an electromagnetically levitated spherical sample in a set of coaxial circular loops
This paper presents a theoretical study of oscillatory and rotational instabilities of a solid spherical body, levitated electromagnetically in axisymmetric coils made of coaxial circular loops. We apply our previous theory to analyze the static and dynamic stability of the sample depending on the ac frequency and the position of the sample in the coils for several simple configurations. We introduce an original analytical approach employing a gauge transformation for the vector potential. First, we calculate the spring constants that define the frequency of small-amplitude oscillations. For static stability, the spring constants must be positive. Dynamic instabilities are characterized by …
Experimental and numerical study of anomalous thermocapillary convection in liquid gallium
Thermocapillary Marangoni convection of liquid gallium was studied experimentally and numerically. A specially designed experimental setup ensured an oxide-free surface of the liquid gallium for a very long time. The convective flow at the free surface was found to be directed opposite to both buoyancy-driven and ordinary thermocapillary convection. The anomalous direction of the thermocapillary flow was explained by the presence of a small amount of a surface-active contaminant—lead adsorbed at the free surface. Two different approaches were used to describe the observed phenomenon. First, the flow was treated as a pure thermocapillary convection with a modified dependence of the surface t…
Influence of growth parameters and melt convection on the solid-liquid interface during RF-floating zone crystal growth of intermetallic compounds
Abstract The influence of growth parameters and melt convection on the solid–liquid interface of the intermetallic compound Ni3Si grown by the RF-floating zone technique was investigated experimentally as well as numerically. Numerical simulations showed that the heat transfer is strongly influenced by the electromagnetically driven and Marangoni convections whereas both the buoyancy and feed rotation have a negligible effect. It was found experimentally that the inductor design, the rod diameter and the length of the molten zone influence the solid–liquid interface shape significantly. The electromagnetically driven convection increases dramatically with increasing zone length due to the r…
Stability of melt flow due to a traveling magnetic field in a closed ampoule
The linear three-dimensional instability of the flow due to a low frequency traveling magnetic field in a regular cylinder is studied numerically for height-to-diameter ratios in the range [0.5:2.5]. The first instability has the form of an azimuthal wave with wave numbers between 1 and 6 depending on the aspect ratio. It is shown that the flow is stable if the Reynolds number is below 290. Even stronger stationary flows may be obtained in a flattened cylinder.
Three-dimensional linear stability analysis of the flow in a liquid spherical droplet driven by an alternating magnetic field
The paper presents a numerical stability analysis of the flow driven by an alternating (AC) magnetic field in an electromagnetically levitated liquid metal droplet. The basic axisymmetric flow is found to become unstable at Reynolds numbers in the order of 100. The critical Reynolds number Rec and the corresponding most unstable azimuthal wave number m are found for several configurations of the magnetic field depending on the skin-depth d. For a uniform external AC magnetic field the azimuthal wave number of the most unstable mode is m=3. An additional steady (DC) magnetic field imposed along the axis of symmetry increases the stability of the flow.
Velocity measurements in the liquid metal flow driven by a two-phase inductor
We present the results of velocity measurements obtained by ultrasonic Doppler velocimetry and local potential probes in the flow of GaInSn eutectic melt driven by a two-phase inductor in a cylindrical container. This type of flow is expected in a recent modification to the floating zone technique for the growth of small-diameter single intermetallic compound crystals. We show that the flow structure can be changed from the typical two toroidal vortices to a single vortex by increasing the phase shift between the currents in the two coils from 0 to 90 degrees. The latter configuration is thought to be favourable for the growth of single crystals. The flow is also computed numerically and a …
Energy oscillations and a possible route to chaos in a modified Riga dynamo
Starting from the present version of the Riga dynamo experiment with its rotating magnetic eigenfield dominated by a single frequency we ask for those modifications of this set-up that would allow for a non-trivial magnetic field behaviour in the saturation regime. Assuming an increased ratio of azimuthal to axial flow velocity, we obtain energy oscillations with a frequency below the eigenfrequency of the magnetic field. These new oscillations are identified as magneto-inertial waves that result from a slight imbalance of Lorentz and inertial forces. Increasing the azimuthal velocity further, or increasing the total magnetic Reynolds number, we find transitions to a chaotic behaviour of th…
Contactless generation of cavitation in high temperature liquid metals and its impact on particle dispersion in solidified iron and steel samples
Abstract A recently developed method for the contactless magnetic generation of cavitation is demonstrated for high-melting-point metals. The approach is based on the floating-zone technique, which is truly contactless and crucible-free as it uses electromagnetic forces. Using this method, ultra-high-temperature ceramic particles, such as TiN, TiB2 and TiC, are admixed in liquid iron and 316L steel. The dispersion and particle refinement caused by cavitation treatment during melting and solidification are investigated. Magnetic fields up to 8 T that correspond to pressure oscillation amplitude of 0.83 MPa are used. The signal emitted by the collapsing bubbles is captured and visualized for …
Stability analysis of an electromagnetically levitated sphere
We present a combined numerical and analytical approach to analyze the static and dynamic stabilities of an electromagnetically levitated spherical body depending on the ac frequency and the configuration of a three-dimensional (3D) coil made of thin winding which is modeled by linear current filaments. First, we calculate numerically the magnetic vector potential in grid points on the surface of the sphere and then use Legendre and fast Fourier transforms to find the expansion of the magnetic field in terms of spherical harmonics. Second, we employ a previously developed gauge transformation to solve analytically the 3D electromagnetic problem in terms of the numerically obtained expansion…
Magnetic field controlled FZ single crystal growth of intermetallic compounds
Abstract Intermetallic rare-earth-transition-metal compounds with their coexistence of magnetic ordering and superconductivity are still of great scientific interest. The crystal growth of bulk single crystals is very often unsuccessful due to an unfavorable solid–liquid interface geometry enclosing concave fringes. The aim of the work is the contactless control of heat and material transport during floating-zone single crystal growth of intermetallic compounds. This control is provided by a tailored design of the electromagnetic field and the resulting electromagnetically driven convection. Numerical simulations for the determination of the electromagnetic field configuration induced by th…
History and results of the Riga dynamo experiments
On 11 November 1999, a self-exciting magnetic eigenfield was detected for the first time in the Riga liquid sodium dynamo experiment. We report on the long history leading to this event, and on the subsequent experimental campaigns which provided a wealth of data on the kinematic and the saturated regime of this dynamo. The present state of the theoretical understanding of both regimes is delineated, and some comparisons with other laboratory dynamo experiments are made.
Controlling melt convection—an innovation potential for concerted microstructure evolution of Nd-Fe-B alloys
The solidification process and the resulting microstructure of Nd-Fe-B alloys in consideration of melt convection has been investigated experimentally with a specially designed forced crucible rotation technique. Samples were subjected to well-defined forced rotation and vibration, respectively, during induction heating and solidification. A concerted microstructure evolution is possible by enhancement or suppression of the melt convection. As a result, the microstructure pattern, mainly the volume fraction of the soft magnetic a-Fe phase, vary strongly with the strength of the internal flow motion. A distinct reduction of the a-Fe volume fraction in samples with strong melt rotation was ob…
Magnetohydrodynamic experiments on cosmic magnetic fields
It is widely known that cosmic magnetic fields, i.e. the fields of planets, stars, and galaxies, are produced by the hydromagnetic dynamo effect in moving electrically conducting fluids. It is less well known that cosmic magnetic fields play also an active role in cosmic structure formation by enabling outward transport of angular momentum in accretion disks via the magnetorotational instability (MRI). Considerable theoretical and computational progress has been made in understanding both processes. In addition to this, the last ten years have seen tremendous efforts in studying both effects in liquid metal experiments. In 1999, magnetic field self-excitation was observed in the large scale…
Physical modelling of Czochralski crystal growth in horizontal magnetic field
Abstract This study addresses experimentally the heat transfer, the temperature azimuthal non-uniformity and the onset of oscillations in a low temperature physical model of a medium-sized Czochralski crystal growth process with a strong horizontal magnetic field (HMF). It is observed that under certain conditions the integral heat flux may decrease with increasing magnetic field strength at the same time as the flow velocity increases. The azimuthal non-uniformity of the temperature field in the melt near the crystal model rim is only little influenced by its rotation rate outside of a narrow range where the centrifugal force balances the buoyant one. The flow oscillation onset has been ob…
Casting technology for ODS steels – dispersion of nanoparticles in liquid metals
Dispersion of particles to produce metal matrix nanocomposites (MMNC) can be achieved by means of ultrasonic vibration of the melt using ultrasound transducers. However, a direct transfer of this method to produce steel composites is not feasible because of the much higher working temperature. Therefore, an inductive technology for contactless treatment by acoustic cavitation was developed. This report describes the samples produced to assess the feasibility of the proposed method for nano-particle separation in steel. Stainless steel samples with inclusions of TiB2, TiO2, Y2O3, CeO2, Al2O3 and TiN have been created and analyzed. Additional experiments have been performed using light metals…
Instability of the melt flow in VGF growth with a traveling magnetic field
The linear instability of a thermally stratified melt flow in the VGF configuration driven by a traveling magnetic field (TMF) is considered numerically and experimentally. The dependency of the instability threshold on the governing parameters is found for several cuts through the parameter space covering a wide range of possible applications. In a first approximation the linear instability occurs when the dimensionless TMF forcing parameter reaches the magnitude of the Grashof number. This is particularly true in a medium-sized crucible where the first instability is axisymmetric and sub-critical. As the Grashof number increases the flow develops self-similar boundary layers and the insta…
Contactless magnetic excitation of acoustic cavitation in liquid metals
A steady axial magnetic field is applied to a liquid metal zone heated by induction currents. The resulting alternating Lorentz force causes pressure oscillations that being strong enough lead to cavitation in the molten metal. Amplitude of the pressure oscillations is proportional to the product of the induced currents and the steady axial magnetic field induction. We follow an approach where the acoustic pressure is maximized by the induction currents. The onset of cavitation is identified by the occurrence of sub-harmonics of the drive frequency in sound recorded at the surface of the experimental cell. It is demonstrated that cavitation in a liquid metal may be excited by a superimposed…
Analytical solution for the diffusion of a capacitor discharge generated magnetic field pulse in a conductor
Powerful forces arise when a pulse of a magnetic field in the order of a few tesla diffuses into a conductor. Such pulses are used in electromagnetic forming, impact welding of dissimilar materials and grain refinement of solidifying alloys. Strong magnetic field pulses are generated by the discharge current of a capacitor bank. We consider analytically the penetration of such pulse into a conducting half-space. Besides the exact solution we obtain two simple self-similar approximate solutions for two sequential stages of the initial transient. Furthermore, a general solution is provided for the external field given as a power series of time. Each term of this solution represents a self-sim…
Tailoring the microstructure and mechanical properties of Ti–Al alloy using a novel electromagnetic stirring method
The effect of melt convection during the solidification of Ti45Al55 alloys was investigated in terms of microstructure evolution and the resulting mechanical properties. The samples were subjected to conventional induction melting as well as enhanced melt stirring by an external magnetic field using a specially designed floating zone arrangement. The stirred samples showed a significant improvement of plastic deformability. A strong change in the morphology from dendritic to spherical and an increased properitectic phase fraction were observed after stirring.
Linear and non-linear stability of a thermally stratified magnetically driven rotating flow in a cylinder
The stability of a thermally stratified liquid metal flow is considered numerically. The flow is driven by the rotating magnetic field in a cylinder heated from above and cooled from below. The stable thermal stratification turns out to destabilise the flow. This is explained by the fact that a stable stratification suppresses the secondary meridional flow, thus indirectly enhancing the primary rotation. The instability in the form of Taylor-Görtler rolls is consequently promoted. It is known from earlier studies that these rolls can be only excited by finite disturbances in the isothermal flow. A sufficiently strong thermal stratification transforms this non-linear bypass instability into …
Rayleigh–Bénard instability of Czochralski configuration in a transverse magnetic field
Abstract The linear instability of a rotating conducting liquid cylinder heated from below in a horizontal magnetic field is considered numerically. A condition for the magnetic suppression of the bulk rotation is obtained. If the bulk is rotation dominated then the linear instability is slightly delayed by the field and sets in as a rotating wave. If the bulk is dominated by the magnetic field then the instability has the form of field aligned convection rolls. Outside thin boundary layers the instability then becomes increasingly similar to the onset in a plain channel. The results are discussed in light of previous silicon growth experiments and existing knowledge from related problems.
Colloquium: Laboratory experiments on hydromagnetic dynamos
Cosmic magnetic fields, including the fields of planets, stars, and galaxies, are believed to be caused by dynamo action in moving electrically conducting fluids. While the theory and numerics of hydromagnetic dynamos have flourished during recent decades, an experimental validation of the effect was missing until recently. We sketch the long history towards a working laboratory dynamo. We report on the first successful experiments at the sodium facilities in Riga and Karlsruhe, and on other experiments which are carried out or planned at various places in the world.
Nanoparticle dispersion in liquid metals by electromagnetically induced acoustic cavitation
Abstract Aim of this study is to investigate experimentally the effect of magnetically induced cavitation applied for the purpose of nanoparticle dispersion in liquid metals. The oscillating magnetic force due to the azimuthal induction currents and the axial magnetic field excites power ultrasound in the sample. If the fields are sufficiently high then it is possible to achieve the acoustic cavitation threshold in liquid metals. Cavitation bubble collapses are known to create microscale jets with a potential to break nanoparticle agglomerates and disperse them. The samples are solidified under the contactless ultrasonic treatment and later analyzed by electron microscopy and energy-dispers…
Matched asymptotic solution for the solute boundary layer in a converging axisymmetric stagnation point flow
Abstract A novel boundary-layer solution is obtained by the method of matched asymptotic expansions for the solute distribution at a solidification front represented by a disk of finite radius R 0 immersed in an axisymmetric converging stagnation point flow. The detailed analysis reveals a complex internal structure of the boundary layer consisting of eight subregions. The development of the boundary layer starts from the rim region where the concentration, according to the obtained similarity solution, varies with the radius r along the solidification front as ∼ln 1/3 ( R 0 / r ). At intermediate radii, where the corresponding concentration is found to vary as ∼ln( R 0 / r ), the boundary …
Magnetically Induced Cavitation for the Dispersion of Particles in Liquid Metals
A contactless excitation of cavitation is possible by superposition of induction heating with a static axial magnetic field. This creates an alternating electromagnetic body force in a liquid metal which in turn produces pressure oscillations. Using this method, the onset of cavitation has been clearly observed in various liquid metals (tin, zinc, aluminum, steel SAE 304) at pressure oscillations in the range of 28…50 kPa. The present study aims to extend the previous work by producing steel metal matrix composites (MMC) and assessing the feasibility of the proposed method for particle dispersion in steel. Stainless steel (SAE 316L) samples with different ceramic inclusions, e.g. TiN, Al2O3…