Results for “solidification” — Page 57

NASA NTRS1998

Experiments on Nucleation in Different Flow Regimes

The vast majority of metallic engineering materials are solidified from the liquid phase. Understanding the solidification process is essential to control microstructure, which in turn, determines the properties of materials. The genesis of solidification is nucleation, where the first stable solid forms from the liquid phase. Nucleation kinetics determine the degree of undercooling and phase selection. As such, it is important to understand nucleation phenomena in order to control solidification or glass formation in metals and alloys.

Bayuzick, Robert J.↗

NASA NTRS1999

Use of Microgravity to Control the Microstructure of Eutectics

This grant began in June of 1996. Its long term goal is to be able to control the microstructure of directionally solidified eutectic alloys, through an improved understanding of the influence of convection. The primary objective of the projects in the present grant is to test hypotheses for the reported influence of microgravity on the microstructure of eutectics. The prior experimental results on the influence of microgravity on the microstructure of eutectics have been contradictory. With lamellar eutectics, microgravity had a negligible effect on the microstructure. Microgravity experiments with fibrous eutectics sometimes showed a finer microstructure and sometimes a coarser microstructure. Most research has been done on the MnBi/Bi rod-like eutectic. Larson and Pirich obtained a two-fold finer microstructure both from microgravity and by use of a magnetic field to quench buoyancy-driven convection. Smith, on the other hand, observed no change in microgravity. Prior theoretical work at Clarkson University showed that buoyancy-driven convection in the vertical Bridgman configuration is not vigorous enough to alter the concentration field in front of a growing eutectic sufficiently to cause a measurable change in microstructure. We assumed that the bulk melt was at the eutectic composition and that freezing occurred at the extremum, i.e. with minimum total undercooling at the freezing interface. There have been four hypotheses attempting to explain the observed changes in microstructure of fibrous eutectics caused by convection: I .A fluctuating freezing rate, combined with unequal kinetics for fiber termination and branching. 2. Off-eutectic composition, either in the bulk melt due to an off-eutectic feed or at the freezing interface because of departure from the extremum condition. 3. Presence of a strong habit modifying impurity whose concentration at the freezing interface would be altered by convection. At the beginning of the present grant, we favored the first of these hypotheses and set out to test it both experimentally and theoretically. We planned the following approaches: I .Pass electric current pulses through the MnBi/Bi eutectic during directional solidification in order to produce an oscillatory freezing rate. 2. Directionally solidify the MnBi/Bi eutectic on Mir using the QUELD II gradient freeze furnace developed by Professor Smith at Queen's University. 3. Select another fibrous eutectic system for investigation using the Accelerated Crucible Rotation Technique to introduce convection. 4. Develop theoretical models for eutectic solidification with an oscillatory freezing rate. Because of the problems with Mir, we substituted ground-based experiments at Queen's University with QUELD II vertical and horizontal, with and without vibration of the furnace. The Al-Si system was chosen for the ACRT experiments. Three related approaches were used to model eutectic solidification with an oscillatory freezing rate. A sharp interface model was used to calculate composition oscillations at the freezing interface in response to imposed freezing rate oscillations.

Wilcox. William R.↗

NASA NTRS1999

Experiments on Nucleation in Different Flow Regimes

The vast majority of metallic engineering materials are solidified from the liquid phase. Understanding the solidification process is essential to control microstructure, which in turn, determines the properties of materials. The genesis of solidification is nucleation, where the first stable solid forms from the liquid phase. Nucleation kinetics determine the degree of undercooling and phase selection. As such, it is important to understand nucleation phenomena in order to control solidification or glass formation in metals and alloys. Early experiments in nucleation kinetics were accomplished by droplet dispersion methods. Dilatometry was used by Turnbull and others, and more recently differential thermal analysis and differential scanning calorimetry have been used for kinetic studies. These techniques have enjoyed success; however, there are difficulties with these experiments. Since materials are dispersed in a medium, the character of the emulsion/metal interface affects the nucleation behavior. Statistics are derived from the large number of particles observed in a single experiment, but dispersions have a finite size distribution which adds to the uncertainty of the kinetic determinations. Even though temperature can be controlled quite well before the onset of nucleation, the release of the latent heat of fusion during nucleation of particles complicates the assumption of isothermality during these experiments. Containerless processing has enabled another approach to the study of nucleation kinetics. With levitation techniques it is possible to undercool one sample to nucleation repeatedly in a controlled manner, such that the statistics of the nucleation process can be derived from multiple experiments on a single sample. The authors have fully developed the analysis of nucleation experiments on single samples following the suggestions of Skripov. The advantage of these experiments is that the samples are directly observable. The nucleation temperature can be measured by noncontact optical pyrometry, the mass of the sample is known, and post processing analysis can be conducted on the sample. The disadvantages are that temperature measurement must have exceptionally high precision, and it is not possible to isolate specific heterogeneous sites as in droplet dispersions.

Bayuzick, R. J.↗

NASA NTRS1999

Role of Dynamic Nucleation at Moving Boundaries in Phase and Microstructure Selection

Solidification microstructures that form under steady-state growth conditions (cells, dendrites, regular eutectics, etc.) are reasonably well understood in comparison to other, more complex microstructures, which form under intrinsically non-steady-state growth conditions due to the competition between the nucleation and growth of several phases. Some important practical examples in this latter class include microstructures forming in peritectic systems in highly undercooled droplets, and in strip cast stainless steels. Prediction of phase and microstructure selection in these systems has been traditionally based on (1) heterogeneous nucleation on a static interface, and (2) comparing the relative growth rate of different phase/microstructures under steady-state growth conditions. The formation of new phases, however, occurs via nucleation on, or ahead of, a moving boundary. In addition, the actual selection process is controlled by a complex interaction between the nucleation process and the growth competition between the nuclei and the pre-existing phase under non-steady-state conditions. As a result, it is often difficult to predict which microstructure will form and which phases will be selected under prescribed processing conditions. This research addresses this critical role of nucleation at moving boundaries in the selection of phases and solidification microstructures through quantitative experiments and numerical modeling in peritectic systems. In order to create a well characterized system in which to study this problem, we focus on the directional solidification of hypo- and hyper-peritectic alloys in the two-phase region, imposing a large enough ratio of temperature gradient/growth rate (G/V(sub p)) to suppress the morphological instability of both the parent (alpha) and peritectic (Beta) phases, i.e. each phase alone would grow as a planar front. Our combined experimental and theoretical results show that, already in this simplified case, the growth competition of these two phases leads to a rich variety of microstructures that depend sensitively upon the relative importance of nucleation, diffusion, and convection.

Karma, Alain↗

NASA NTRS1999

Experiments on Nucleation in Different Flow Regimes

The vast majority of metallic engineering materials are solidified from the liquid phase. Understanding the solidification process is essential to control microstructure, which in turn, determines the properties of materials. The genesis of solidification is nucleation, where the first stable solid forms from the liquid phase. Nucleation kinetics determine the degree of undercooling and phase selection. As such, it is important to understand nucleation phenomena in order to control solidification or glass formation in metals and alloys. Early experiments in nucleation kinetics were accomplished by droplet dispersion methods [1-6]. Dilitometry was used by Turnbull and others, and more recently differential thermal analysis and differential scanning calorimetry have been used for kinetic studies. These techniques have enjoyed success; however, there are difficulties with these experiments. Since materials are dispersed in a medium, the character of the emulsion/metal interface affects the nucleation behavior. Statistics are derived from the large number of particles observed in a single experiment, but dispersions have a finite size distribution which adds to the uncertainty of the kinetic determinations. Even though temperature can be controlled quite well before the onset of nucleation, the release of the latent heat of fusion during nucleation of particles complicates the assumption of isothermality during these experiments. Containerless processing has enabled another approach to the study of nucleation kinetics [7]. With levitation techniques it is possible to undercool one sample to nucleation repeatedly in a controlled manner, such that the statistics of the nucleation process can be derived from multiple experiments on a single sample. The authors have fully developed the analysis of nucleation experiments on single samples following the suggestions of Skripov [8]. The advantage of these experiments is that the samples are directly observable. The nucleation temperature can be measured by noncontact optical pyrometry, the mass of the sample is known, and post processing analysis can be conducted on the sample. The disadvantages are that temperature measurement must have exceptionally high precision, and it is not possible to isolate specific heterogeneous sites as in droplet dispersions.

Bayuzick, Robert J.↗

NASA NTRS1999

Interaction of Porosity with an Advancing Solid/Liquid Interface: a Real-Time Investigation

Problems associated with formation of porosity during solidification continue to have a daily impact on the metal forming industry. Several past investigations have dealt with the nucleation and growth aspects of porosity. However, investigations related to the interaction of porosity with that of a solidification front has been limited mostly to organic analogues. In this paper we report on real time experimental observations of such interactions in metal alloys. Using a state of the art X-Ray Transmission Microscope (XTM) we have been able to observe and record the dynamics of the interaction. This includes distortion of the solid/liquid interface near a poro.sity, solute segr,egation patterns surrounding a porosity and the change in shape of the porosity during interaction with an advancing solid/liquid interface. Results will be presented for different Al alloys and growth conditions. The experimental data will be compared to theory using a recently developed 2D numerical model. The model employs a finite difference approach where the solid/liquid interface is defined through the points at which the interface intersects the grid lines. The transport variables are calculated at these points and the motion of the solidification front is determined by the magnitude of the transport variables. The model accounts for the interplay of the thermal and solutal field and the influence of capilarity to predict the shape of the solid/liquid interface with time in the vicinity of porosity. One can further calculate the perturbation of the solutal field by the presence of porosity in the melt.

Sen, S.↗

NASA NTRS1999

Growth of Compound Semiconductors in a Low Gravity Environment: Microgravity Growth of PbSnTe

The growth of the alloy compound semiconductor lead tin telluride (PbSnTe) was chosen for a microgravity flight experiment in the Advanced Automated Directional Solidification Furnace (AADSF), on the United States Microgravity Payload-3 (USNP-3) in February, 1996 and on USNW- 4 in November, 1997. The objective of these experiments was to determine the effect of the reduction in convection, during the growth process, brought about by the microgravity environment. The properties of devices made from PbSnTe, an alloy of PbTe and SnTe, are dependent on the ratio of the elemental components in the starting crystal. Compositional uniformity in the crystal is only obtained if there is no significant mixing in the liquid during growth. The technological importance of PbSnTe lies in its band gap versus composition diagram which has a zero energy crossing at approximately 40% SnTe. This facilitates the construction of long wavelength (greater than 6 gm) infrared detectors and lasers. The properties and utilization of PbSnTe are the subject of other papers. 1,2 PbSnTe is also interesting from a purely scientific point of view. It is, potentially, both solutally and thermally unstable due to the temperature and density gradients present during growth. Density gradients, through thermal expansion, are imposed in directional solidification because temperature gradients are required to extract heat. Solutal gradients occur in directional solidification of alloys due to segregation at the interface. Usually the gradients vary with both experiment design and inherent materials properties. In a simplified one dimensional analysis with the growth axis parallel to the gravity vector, only one of the two instabilities work at a time. During growth, the temperature in the liquid increases ahead of the interface. Therefore the density, due to thermal expansion, is decreasing in that direction. However, the phase diagram shows that the lighter SnTe is preferentially rejected at the interface. This causes the liquid density to increase with distance away from the interface.

Fripp, A. L.↗

NASA NTRS1999

In Situ Monitoring of Crystal Growth Using MEPHISTO, Mission STS 87-Program USMP-4: Experimental Analysis

This report summarizes the results of the In situ Monitoring of Crystal Growth Using MEPHISTO (Material por l'Etude des Phenomenes Interessant de la Solidification sur Terre et en Orbite) experiment on USMP-4. The report includes microstructural and compositional data obtained during the first year of the post flight analysis, as well as numerical simulation of the flight experiment. Additional analyses are being continued and will be reported in the near future. The experiments utilized MEPHISTO hardware to study the solidification and melting behavior of bismuth alloyed with 1 at% tin. The experiments involved repeated melting and solidification of three samples, each approximately 90 cm long and 6mm in diameter. Half of each sample also included a 2 mm. diameter growth capillary, to assist in the formation of single grain inside. One sample provided the Seebeck voltage generated during melting and freezing processes. Another one provided temperature data and Peltier pulsed demarcation of the interface shape for post flight analysis. The third sample provided resistance and velocity measurements, as well as additional thermal data. The third sample was also quenched at the end of the mission to preserve the interface composition for post flight determination. A total of more than 45cm of directionally solidified alloy were directionally solidified at the end of the flight for post mission structural and compositional characterization. Metallurgical analysis of the samples has shown that the interfacial kinetics play a key role in controlling the morphological stability of faceted alloys. Substantial differences were observed in the Seebeck signal between the ground-based experiments and the space-based experiments. The temperature gradient in the liquid for the ground-based experiments was also significantly lower than the temperature gradient in the liquid for the space-based experiments. Both of these observations indicate significant influence of liquid convection for the ground-based experiments.

Abbaschian, Reza↗

NASA NTRS1998

Experiments on Nucleation in Different Flow Regimes

The vast majority of metallic engineering materials are solidified from the liquid phase. Understanding the solidification process is essential to control microstructure, which in turn, determines the properties of materials. The genesis of solidification is nucleation, where the first stable solid forms from the liquid phase. Nucleation kinetics determine the degree of undercooling and phase selection. As such, it is important to understand nucleation phenomena in order to control solidification or glass formation in metals and alloys. Early experiments in nucleation kinetics were accomplished by droplet dispersion methods. Dilitometry was used by Turnbull and others, and more recently differential thermal analysis and differential scanning calorimetry have been used for kinetic studies. These techniques have enjoyed success; however, there are difficulties with these experiments. Since materials are dispersed in a medium, the character of the emulsion/metal interface affects the nucleation behavior. Statistics are derived from the large number of particles observed in a single experiment, but dispersions have a finite size distribution which adds to the uncertainty of the kinetic determinations. Even though temperature can be controlled quite well before the onset of nucleation, the release of the latent heat of fusion during nucleation of particles complicates the assumption of isothermality during these experiments. Containerless processing has enabled another approach to the study of nucleation kinetics. With levitation techniques it is possible to undercool one sample to nucleation repeatedly in a controlled manner, such that the statistics of the nucleation process can be derived from multiple experiments on a single sample. The authors have fully developed the analysis of nucleation experiments on single samples following the suggestions of Skripov. The advantage of these experiments is that the samples are directly observable. The nucleation temperature can be measured by noncontact optical pyrometry, the mass of the sample is known, and post-processing analysis can be conducted on the sample. The disadvantages are that temperature measurement must have exceptionally high precision, and it is not possible to isolate specific heterogeneous sites as in droplet dispersions. Levitation processing of refractory materials in ultra high vacuum provides an avenue to conduct these kinetic studies on single samples. Two experimental methods have been identified where ultra high vacuum experiments are possible; electrostatic levitation in ground-based experiments and electromagnetic processing in low earth orbit on TEMPUS. Such experiments, reported here, were conducted on zirconium. Liquid zirconium is an excellent solvent and has a high solubility for contaminants contained in the bulk material as well as those contaminants found in the vacuum environment. Oxides, nitrides, and carbides do not exist in the melt, and do not form on the surface of molten zirconium, for the materials and vacuum levels used in this study. Ground-based experiments with electrostatic levitation have shown that the statistical nucleation kinetic experiments are viable and yield results which are consistent with classical nucleation theory. The advantage of low earth orbit experiments is the ability to vary the flow conditions in the liquid prior to nucleation. The put-pose of nucleation experiments in TEMPUS was to examine.

Bayuzick, R. J.↗

NASA NTRS2000

Particle Engulfment and Pushing Micro-Gravity Experiments and Mathematical Modeling

The phenomenon of interaction of particles with solid-liquid interfaces that results in particle engulfment or pushing (PEP) has been Studied since mid 1960's. While the original interest stemmed from geology applications (frost heaving in soil), it was recognized early that understanding particle behavior at solidifying interfaces mi ht yield 9 practical benefits in other fields. In metallurgical applications the issue is the location of particles with respect to grain boundaries at the end of solidification. Considerable amount of experimental and theoretical research was lately focused on applications to metal matrix composites produced by casting; or spray forming techniques. Another application of PEP is in the growing of Y1Ba2CU3O7-delta(123) superconductor crystals from an undercooled liquid. The oxide melt contains Y2Ba1CU1O5 (211) precipitates, which act as flux pinning sites. The paper presents results of PEP micro-gravity research performed by the authors on two shuttle missions using metallic and polymeric materials. In addition. a discussion on the theoretical aspects of the physics of PEP is offered. Analytical and numerical models for planar solidification interfaces developed by the authors are used to explain the experimental results. Shortcomings of steady-state models are emphasized. A numerical model that includes the effect of the solutal field and of natural convection is introduced. A discussion of phenomena associated with dendritic solidification based on experimental observations is also offered. A mechanism of engulfment is proposed.

Stefanescu, Doru M.↗

NASA NTRS2000

Microstructural Characterization of a Directionally-Solidified Ni-33 (at. %)Al-31Cr-3Mo Eutectic Alloy as a Function of Withdrawal Rate

The Ni-33 (at. %)Al-3lCr-3Mo eutectic alloy was directionally-solidified (DS) at different rates, V(sub I), varying between 2.5 to 508 mm/ h. Detailed qualitative and quantitative metallographic and chemical analyses were conducted on the directionally-solidified rods. The microstructures consisted of eutectic colonies with parallel lamellar NiAl/(Cr,Mo) plates for solidification rates at and below 12.7 mm/ h. Cellular eutectic microstructures were observed at higher solidification rates, where the plates exhibited a radial pattern. The microstructures were demonstrated to be fairly uniform throughout a 100 mm length of the DS zone by quantitative metallography. The average cell size, bar-d, decreased with increasing growth rate to a value of 125 microns at 508 mm/ h according to the relation bar-d (microns) approx. = 465 V(sup -0.22, sub I), where V(sub I) is in mm/ h. Both the average NiAl plate thickness, bar-Delta(sub NiAl), and the interlamellar spacing, bar-lambda, were observed to be constant for V(sub I) less than or = 50.8 mm/ h but decreased with increasing growth rate above this value as 0.93 bar-Delta(sub NiAl)(microns) = 61.2 V(sup -0.93, sub I) and bar-lambda (microns) = 47.7 V(sup -0.64, sub I), respectively. The present results are detailed on a microstructural map. Keywords Optical microscopy, microstructure, compounds intermetallic, directional solidification

Raj, S. V.↗

NASA NTRS2000

An Overview of the MSFC Electrostatic Levitation Facility

Electrostatic levitation (ESL) provides a means to study molten materials in a contamination-free environment, including no contact with a container. Many phenomena important to materials science can be studied in the ESL. Solidification of metals, alloys and undercooled materials represent an important topic for research in the ESL. Recent studies of metals and alloys during solidification in the ESL are reported. Measurements include time, temperature and transformation of metallic glass-forming alloys, solidification velocities, and microstructure. This multimedia report includes a video clip showing processing in the ESL, with descriptions of the different segments in the text.

Rogers, J. R.↗

NASA NTRS2000

Mixing Dynamics Induced by Traveling Magnetic Fields

Microstructural and compositional homogeneity in metals and alloys can only be achieved if the initial melt is homogeneous prior to the onset of solidification processing. Naturally induced convection may initially facilitate this requirement but upon the onset of solidification significant compositional variations generally arise leading to undesired segregation. Application of alternating magnetic fields to promote a uniform bulk liquid concentration during solidification processing has been suggested. To investigate such possibilities an initial study of using traveling magnetic fields (TMF) to promote melt homogenization is reported in this work. Theoretically, the effect of TMF-induced convection on mixing phenomena is studied in the laminar regime of flow. Experimentally, with and without applied fields, both: mixing dynamics by optically monitoring the spreading of an initially localized dye in transparent fluids and, compositional variations in metal alloys have been investigated.

Grugel, Richard N.↗

NASA NTRS2000

Microgravity Processing of Oxide Superconductors

The primary goal is to understand the microstructures which develop under the nonequilibrium solidification conditions achieved by melt processing in copper oxide superconductor systems. More specifically, to define the liquidus at the Y- 1:2:3 composition, the Nd-1:2:3 composition, and several intermediate partial substitution points between pure Y-1:2:3 and Nd-1:2:3. A secondary goal has been to understand resultant solidification morphologies and pathways under a variety of experimental conditions and to use this knowledge to better characterize solidification phenomena in these systems.

Hofmeister, William H.↗

NASA NTRS2001

Mixing Dynamics Induced by Traveling Magnetic Fields

Microstructural and compositional homogeneity in metals and alloys can only be achieved if the initial melt is homogeneous prior to the onset of solidification processing. Naturally induced convection may initially facilitate this requirement but upon the onset of solidification significant compositional variations generally arise leading to undesired segregation. Application of alternating magnetic fields to promote a uniform bulk liquid concentration during solidification processing has been suggested. To investigate such possibilities an initial study of using traveling magnetic fields (TMF) to promote melt homogenization is reported in this work. Theoretically, the effect of TMF-induced convection on mixing phenomena is studied in the laminar regime of flow. Experimentally, with and without applied fields, both 1) mixing dynamics by optically monitoring the spreading of an initially localized dye in transparent fluids and, 2) compositional variations in metal alloys have been investigated.

Grugel, Richard N.↗

NASA NTRS2001

The Isothermal Dendritic Growth Experiment (IDGE)

Dendritic solidification is one of the simplest examples of pattern formation where a structureless melt evolves into a ramified crystalline microstructure; it is a common mode of solidification in many materials, but especially so in metals and alloys. There is considerable engineering interest in dendrites because of the role dendrites play in the determination of microstructure, and thereby in influencing the physical properties of cast metals and alloys. Dendritic solidification provides important examples of non-equilibrium physics, pattern formation dynamics, and models for computational condensed matter and material physics. Current theories of dendritic growth generally couple diffusion effects in the melt with the physics introduced by the interface. Unfortunately, in terrestrial based experiments, convective effects in the melt alter the growth process in such a manner as to prevent definitive analysis of convective, diffusive or interfacial effects. Thus, the effective elimination of convection in the melt by operating experiments on orbit were required to produce high-fidelity data needed for achieving further progress. This simple fact comprised the scientific justification for the IDGE.

Glicksman, M. E.↗

NASA NTRS2001

Microgravity Science Glovebox Investigations SUBSA

Solidification Using a Baffle in Sealed Ampoules (SUBSA) is a Microgravity Science Glovebox Investigation manifested for the UF2 flight, on the U.S. Orbiter 111, to the International Space Station (ISS). SUBSA complements the "parent" flight investigation CG13 (Space-and Groundbased Crystal Growth Using a Baffle). During directional solidification, the disk-shaped baffle acts as a partition, creating a small melt zone at the solid-liquid interface. As a result, the level of buoyancy-driven convection at the interface is significantly reduced. In space, the baffle will reduce convection driven by residual micro acceleration. The baffle reduces the Rayleigh number (Ra) of the melt by a factor of 103. The combined effect of the baffle and microgravity will yield a reduction in Ra by a factor of 107 to 109 approaching effectively the acceleration conditions in "nanogravity". The results of ground based tests and numerical modeling will be presented. The furnace for directional solidification (flight hardware and the ground unit) was developed by Tec-Masters Inc. The flight ampoules were produced jointly at Rensselaer, Tec-Masters Inc. and Crystallod Inc.

Ostrogorsky, A.↗

NASA NTRS2002

The Fractional Step Method Applied to Simulations of Natural Convective Flows

This paper describes research done to apply the Fractional Step Method to finite-element simulations of natural convective flows in pure liquids, permeable media, and in a directionally solidified metal alloy casting. The Fractional Step Method has been applied commonly to high Reynold's number flow simulations, but is less common for low Reynold's number flows, such as natural convection in liquids and in permeable media. The Fractional Step Method offers increased speed and reduced memory requirements by allowing non-coupled solution of the pressure and the velocity components. The Fractional Step Method has particular benefits for predicting flows in a directionally solidified alloy, since other methods presently employed are not very efficient. Previously, the most suitable method for predicting flows in a directionally solidified binary alloy was the penalty method. The penalty method requires direct matrix solvers, due to the penalty term. The Fractional Step Method allows iterative solution of the finite element stiffness matrices, thereby allowing more efficient solution of the matrices. The Fractional Step Method also lends itself to parallel processing, since the velocity component stiffness matrices can be built and solved independently of each other. The finite-element simulations of a directionally solidified casting are used to predict macrosegregation in directionally solidified castings. In particular, the finite-element simulations predict the existence of 'channels' within the processing mushy zone and subsequently 'freckles' within the fully processed solid, which are known to result from macrosegregation, or what is often referred to as thermo-solutal convection. These freckles cause material property non-uniformities in directionally solidified castings; therefore many of these castings are scrapped. The phenomenon of natural convection in an alloy under-going directional solidification, or thermo-solutal convection, will be explained. The development of the momentum and continuity equations for natural convection in a fluid, a permeable medium, and in a binary alloy undergoing directional solidification will be presented. Finally, results for natural convection in a pure liquid, natural convection in a medium with a constant permeability, and for directional solidification will be presented.

Westra, Douglas G.↗