Search NASA⌕ Search

SEARCH · Search NASA

Results for “Microgravity Liquid Helium”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 37 records · Page 2

Testing of a Spray-bar Thermodynamic Vent System in Liquid Nitrogen

To support development of a microgravity pressure control capability for liquid oxygen, thermodynamic vent system (TVS) testing was conducted at Marshall Space Flight Center (MSFC) using liquid nitrogen (LN2) as a LOX simulant. The spray bar TVS hardware used was originally designed by the Boeing Company for testing in liquid hydrogen (LH2). With this concept, a small portion of the tank fluid is passed through a Joule-Thomson (J-T) device, and then through a longitudinal spray bar mixed-heat exchanger in order to cool the bulk fluid. To accommodate the larger mass flow rates associated with LN2, the TVS hardware was modified by replacing the recirculation pump with an LN2 compatible pump and replacing the J-T valve. The primary advantage of the spray-bar configuration is that tank pressure control can be achieved independent of liquid and vapor location, enhancing the applicability of ground test data to microgravity conditions. Performance testing revealed that the spray-bar TVS was effective in controlling tank pressure within a 6.89 kPa band for fill levels of 90%, 50%, and 25%. Tests were also conducted with gaseous helium (GHe) in the ullage. The TVS operated nominally with GHe in the ullage, with performance similar to the tests with gaseous nitrogen (GN2). Testing demonstrated that the spray-bar TVS design was flexible enough for use in two different propellants with minimal hardware modifications.

Flachbart, R. H.↗

Design considerations for a micro-g superfluid helium fluid acquisition system

The general description, the operation, and the design of a superfluid helium (SFHe) fluid acquisition system (FAS) for use under microgravity conditions is presented. For the type of FAS considered here, where fine-mesh woven screens are used to retain flowing SFHe within a gallery arm (flow) channel, those forces which determine the flow dynamics are the micro-g accelerations, liquid surface tension, and tensile strength and cumulative pressure drops along a flow path that begins at the bulk liquid and ends at the entrance to a pump. For this case, the dimensionless number, N(T) is written as the ratio between the pressure drop across the screen and the surface tension forces at the screen for low fluid velocities. Static Bond number measurements have bene taken for SFHe using 325 x 2300 twilled Dutch screen and have indicated a screen pore hydraulic radius of 0.00031 cm.

Lee, J. M.↗

Testing the BUNDLE Experiment for Microgravity Disturbance Characterization

A primary objective of the International Space Station is to provide a premier facility for microgravity research. There are many disturbances on board the Station that can disturb the quality of the microgravity environment, including disturbances caused by both system and payload hardware. Disturbances within experiments themselves can also affect the quality of their own environment. The primary functions of the Bridgman Unidirectional Dendrites in Liquids Experiment (BUNDLE) are to melt, directionally solidify, and perform in-situ quench of metallic alloys, providing data to aid in understanding the effects of the microgravity environment on the characteristics of these processed metals. Quenching of the samples in the BUNDLE furnace is accomplished by rapidly introducing high pressure helium into the furnace chamber. This type of quench will induce sample vibrations which may be large enough to adversely affect sample quality. The primary objective of the test described in this presentation was to determine the acceleration imparted on the BUNDLE sample during helium quench. From this information, the science community can better assess whether this method of quench will allow them to obtain the data they need. Utilizing breadboard hardware, the sample quench sequence was conducted. Data was collected from accelerometers located on the breadboard hardware and inside the sample cartridge. The test data indicated that the maximum acceleration achieved by the sample was 0.325 g, which is much higher than the desired microgravity (10(exp -6) g) environment. However, this acceleration level occurred over a time period of less than 0.1 second. Because this time period is so short, there should be no detrimental effects to dendrite growth. This result was consistent with the observations of the experiment scientists, who are satisfied with the results of previous dendrite growth in this breadboard unit.

Gattis, Christy↗

Liquid Hydrogen Propellant Tank Sub-Surface Pressurization with Gaseous Helium

A series of tests were conducted to evaluate the performance of a propellant tank pressurization system with the pressurant diffuser intentionally submerged beneath the surface of the liquid. Propellant tanks and pressurization systems are typically designed with the diffuser positioned to apply pressurant gas directly into the tank ullage space when the liquid propellant is settled. Space vehicles, and potentially propellant depots, may need to conduct tank pressurization operations in micro‐gravity environments where the exact location of the liquid relative to the diffuser is not well understood. If the diffuser is positioned to supply pressurant gas directly to the tank ullage space when the propellant is settled, then it may become partially or completely submerged when the liquid becomes unsettled in a microgravity environment. In such case, the pressurization system performance will be adversely affected requiring additional pressurant mass and longer pressurization times. This series of tests compares and evaluates pressurization system performance using the conventional method of supplying pressurant gas directly to the propellant tank ullage, and then supplying pressurant gas beneath the liquid surface. The pressurization tests were conducted on the Engineering Development Unit (EDU) located at Test Stand 300 at NASA Marshall Space Flight Center (MSFC). EDU is a ground based Cryogenic Fluid Management (CFM) test article supported by Glenn Research Center (GRC) and MSFC. A 150 ft3 propellant tank was filled with liquid hydrogen (LH2). The pressurization system used regulated ambient helium (GHe) as a pressurant, a variable position valve to maintain flow rate, and two identical independent pressurant diffusers. The ullage diffuser was located in the forward end of the tank and was completely exposed to the tank ullage. The submerged diffuser was located in the aft end of the tank and was completely submerged when the tank liquid level was 10% or greater. The ullage diffuser tests were conducted as a baseline to evaluate the performance of the pressurization system, and the submerged diffuser tests showed how the performance of the pressurization system was compromised when the diffuser was submerged in LH2. The test results are evaluated and compared, and included in this report for various propellant tank fill levels.

Stephens, J. R.↗

Superfluid Helium Tanker (SFHT) study

Replenishment of superfluid helium (SFHe) offers the potential of extending the on-orbit life of observatories, satellite instruments, sensors and laboratories which operate in the 2 K temperature regime. A reference set of resupply customers was identified as representing realistic helium servicing requirements and interfaces for the first 10 years of superfluid helium tanker (SFHT) operations. These included the Space Infrared Telescope Facility (SIRTF), the Advanced X-ray Astrophysics Facility (AXAF), the Particle Astrophysics Magnet Facility (Astromag), and the Microgravity and Materials Processing Sciences Facility (MMPS)/Critical Point Phenomena Facility (CPPF). A mixed-fleet approach to SFHT utilization was considered. The tanker permits servicing from the Shuttle cargo bay, in situ when attached to the OMV and carried to the user spacecraft, and as a depot at the Space Station. A SFHT Dewar ground servicing concept was developed which uses a dedicated ground cooling heat exchanger to convert all the liquid, after initial fill as normal fluid, to superfluid for launch. This concept permits the tanker to be filled to a near full condition, and then cooled without any loss of fluid. The final load condition can be saturated superfluid with any desired ullage volume, or the tank can be totally filed and pressurized. The SFHT Dewar and helium plumbing system design has sufficient component redundancy to meet fail-operational, fail-safe requirements, and is designed structurally to meet a 50 mission life usage requirement. Technology development recommendations were made for the selected SFHT concept, and a Program Plan and cost estimate prepared for a phase C/D program spanning 72 months from initiation through first launch in 1997.

Eberhardt, Ralph N.↗

Thermocapillary flow with evaporation and condensation and its effect on liquid retention in low-G fluid acquisition devices

The steady motion, thermal and free surface behavior of a volatile, wetting liquid in microgravity are studied using scaling and numerical techniques. The objective is to determine whether the thermocapillary and two-phase convection arising from thermodynamic nonequilibrium along the porous surfaces of spacecraft liquid acquisition devices could cause the retention failures observed with liquid hydrogen and heated vapor pressurant. Why these devices seem immune to retention loss when pressurized with heated helium or heated directly through the porous structure was also examined. Results show that highly wetting fluids exhibit large negative and positive dynamic pressure gradients towards the meniscus interline when superheated and subcooled, respectively. With superheating, the pressure variation and recoil force arising from liquid/vapor phase change exert the same influence on surface morphology and promote retention. With subcooling, however, the pressure distribution produces a suction that degrades mechanical equilibrium of the surface. This result indicates that thermocapillary-induced deformation arising from subcooling and condensation is the likely cause for retention loss. In addition, increasing the level of nonequilibrium by reducing accommodation coefficient suppresses deformation and explains why this failure mode does not occur in instances of direct screen heating or pressurization with a heated inert gas.

Schmidt, George R.↗

Critical Phenomena in Microgravity: Past, Present, and Future

This review provides an overview of the progress in using the low-gravity environment of space to explore critical phenomena and test modern theoretical predictions. Gravity-induced variations in the hydrostatic pressure and the resulting density gradients adversely affect ground-based measurements near fluid critical points. Performing measurements in a low-gravity environment can significantly reduce these difficulties. A number of significant experiments have been performed in low-Earth orbit. Experiments near the lambda transition in liquid helium explored the regime of large correlation lengths and tested the theoretical predictions to a level of precision that could not be obtained on Earth. Other studies have validated theoretical predictions for the divergence in the viscosity as well as the unexpected critical speeding up of the thermal equilibrium process in pure fluids near the liquid-gas critical point. We describe the scientific content of previously flown low-gravity investigations of critical phenomena as well as those in the development stage, and associated ground-based work.

renormalization-group theory↗

Some Recent Observations on the Burning of Isolated N-Heptane and Alcohol Droplets

In a joint program involving Prof. F.A. Williams of the University of California, San Diego and Dr. V. Nayagam of the National Center for Microgravity Research on Combustion and Fluid Dynamics, the combustion of liquid fuel droplets of n-heptane, n-decane, methanol, methanol-water, ethanol and ethanol-water having initial diameters between about 1 mm and 6 mm continues to be studied. The objectives of the work are to improve fundamental knowledge of droplet combustion dynamics for pure fuels and fuel-water mixtures through microgravity experiments and theoretical analyses. The Princeton contributions to the collaborative program supports the engineering design, data analysis, and data interpretation requirements for the study of initially single component, spherically symmetric, isolated droplet combustion studies through experiments and numerical modeling. The complementary UCSD contributions apply asymptotic theoretical analyses and are described in the published literature and in a companion communication in this conference. The combined program continues to focus on analyses of results obtained from Fiber Supported Droplet Combustion (FSDC) experiments (FSDC-2, STS- 94) conducted with the above fuels in shuttle cabin air and Droplet Combustion Experiment (DCE) data obtained for unsupported and fiber supported droplets of n-heptane in Helium-Oxygen mixtures and cabin air (STS-83, STS-94). The program is preparing for a second DCE experimental mission using methanol/methanol-water as fuels and helium-oxygen-nitrogen environments. DCE-2 is to be conducted aboard the International Space Station. Emphases of recent Princeton work are on the study of simple alcohols (methanol, ethanol) and alcohol/water mixtures as fuels, with time-dependent measurements of drop size, flame-standoff, liquid-phase composition, and finally, extinction. Ground based experiments have included bench-scale studies at Princeton and collaborative experimental studies in the 2.2 second drop tower at NASA-Glenn Research Center.

Dryer, F. L.↗

Effects of Ambient Inert Gas Properties on Cool-Flame Combustion of Normal-Alkane Droplets in Microgravity

Cool-flame-supported quasi-steady droplet combustion was first observed during droplet combustion experiments conducted onboard the International Space Station (ISS) in the Combustion Integrated Rack (CIR) with n-alkane fuels. These cool flames, not visible to the naked eye, result from the existence of the negative-temperature-coefficient (NTC) region during the combustion n-alkane fuels when the chemical-kinetic pathways experience a transition from the low-temperature chemistry to the normal, high-temperature-controlled chemistry. In this study, we present results for the cool-flame-supported droplet burning rates and extinction diameters for n-alkane fuels (n-heptane, n-octane, n-decane, and n-dodecane) in inert-diluent-substituted ambient environments. In these ISS experiments, some of the nitrogen in the ambient gas mixture is replaced by an inert gas, namely, carbon dioxide, helium, or xenon. Different diluents were observed to exert remarkably different influences on the cool-flame combustion of fuel droplets. The results show that the cool-flame extinction diameter is drastically increased with helium dilution compared to N2, CO2 or Xe, and Xe dilution leads to longer burning with a smaller extinction diameter. Unlike hot flames, the cool-flame burning rates are controlled by the cool-flame temperature established in the NTC region by the chemical kinetics, rather than by equilibrium thermodynamics, and the thermo-physical properties of the ambient gases, which control the heat feed back to the liquid droplet. The cool-flame extinction is triggered by the instability that occurs at the low-temperature end of the NTC region, below which cool diffusion flames become statically unstable. A quasi-steady theoretical model, presented earlier, which involves a six-step reduced-chemistry mechanism, valid around the lower end of the NTC region, is shown to explain the observed experimental trends qualitatively.

microgravity↗

BUNDLE: A Novel Furnace for Performing Controlled Directional Solidification Experiments in a Microgravity Environment

NASA Marshall Space Flight Center has developed a novel directional solidification furnace prototype for processing metals and alloys experiments in a microgravity environment. The BUNDLE (Bridgman Unidirectional Dendrite in Liquid Experiment) furnace is intended to accommodate the science requirements of Flight Definition Principle Investigators studying cellular/dendritic growth in aluminum and lead alloys at processing temperatures up to 1200 C. The furnace implements a number of innovative features to achieve high thermal gradients and quench rates in a low-power, light-weight design. These include a pyrolytic boron nitride/graphite composite heating element surrounded by layers of self-supporting refractory metal shielding, and a graphite fiber enhanced cold zone allowing high levels of heat extraction from the sample crucible. Novel to the BUNDLE design is an in-situ helium gas quench capability that ensures rapid freezing of the solidifying region (mushy zone) of the metal sample within the furnace without sample disturbance prior to quenching; this is a stringent requirement for subsequent analysis and understanding of microstructural development. The experiment hardware concept features multiple furnaces that may be "bundled" together so many samples, currently up to eight, can be processed at one time. The design of BUNDLE is flexible enough to be implemented on the Shuttle and Space Station in a number of locations (SpaceHab, Express Rack, MPESS, ISPR, etc). BUNDLE prototype furnaces have directionally solidified and quenched 1cm diameter lead - 5.8 weight percent antimony and aluminum - 4 weight percent copper alloys. Quenching of the mushy zone, as recorded by in-situ thermocouples, occurred on the order of 0.5 seconds or less, a rate within the PI's requirements. Subsequent metallographic examination revealed the solidified microstructure to be, as expected, unidirectional. Both the dendrite tips and the eutectic reaction were planar in nature indicating uniform axial heat flow. Delineation between the growing dendrites and eutectic structure with the "quenched-in" liquid was sharp, attesting to the efficacy of the helium quench. BUNDLE's conception, development, capability, and adaptability are presented (in view of Flight PI's needs and science requirements) through viewgraphs depicting actual hardware, generated thermal analysis, and micrographs prepared from BUNDLE processed, flight-like samples.

Carrasquillo, Edgar J.↗

BUNDLE - A Novel Furnace for Performing Controlled Directional Solidification Experiments in a Microgravity Environment

NASA Marshall Space Flight Center has developed a novel directional solidification furnace prototype for processing metals and alloys experiments in a microgravity environment. The BUNDLE (Bridgman Unidirectional Dendrite in Liquid Experiment) furnace is intended to accommodate the science requirements of Flight Definition Principle Investigators studying cellular/dendritic growth in aluminum and lead alloys at processing temperatures up to 1200 C. The furnace implements a number of innovative features to achieve high thermal gradients and quench rates in a low-power, light-weight design. These include a pyrolytic boron nitride/graphite composite heating element surrounded by layers of self-supporting refractory metal shielding, and a graphite fiber enhanced cold zone allowing high levels of heat extraction from the sample crucible. Novel to the BUNDLE design is an in-situ helium gas quench capability that ensures rapid freezing of the solidifying region (mushy zone) of the metal sample within the furnace without sample disturbance prior to quenching; this is a stringent requirement for subsequent analysis and understanding of microstructural development. The experiment hardware concept features multiple furnaces that may be "bundled" together so many samples, currently up to eight, can be processed at one time. The design of BUNDLE is flexible enough to be implemented on the Shuttle and Space Station in a number of locations (SpaceHab, Express Rack, MPESS, ISPR, etc). BUNDLE prototype furnaces have directionally solidified and quenched 1cm diameter lead - 5.8 weight percent antimony and aluminum - 4 weight percent copper alloys. Quenching of the mushy zone, as recorded by in-situ thermocouples, occurred on the order of 0.5 seconds or less, a rate within the PI's requirements. Subsequent metallographic examination revealed the solidified microstructure to be, as expected, unidirectional. Both the dendrite tips and the eutectic reaction were planar in nature indicating uniform axial heat flow. Delineation between the growing dendrites and eutectic structure with the "quenched-in" liquid was sharp, attesting to the efficacy of the helium quench. BUNDLE's conception, development, capability, and adaptability are presented (in view of Flight PI's needs and science requirements) through viewgraphs depicting actual hardware, generated thermal analysis, and micrographs prepared from BUNDLE processed, flight-like samples.

Carrasquillo, Edgar J.↗

Magnetothermal Convection in Nonconducting Diamagnetic and Paramagnetic Fluids

Nonuniform magnetic fields exert a magnetic body force on electrically nonconducting classical fluids. These include paramagnetic fluids such as gaseous and liquid oxygen and diamagnetic fluids such as helium. Recent experiments show that this force can overwhelm the force of gravity even at the surface of the earth; it can levitate liquids and gases, quench candle flames, block gas flows, and suppress heat transport. Thermal gradients render the magnetic force nonuniform through the temperature-dependent magnetic susceptibility. These thermal gradients can therefore drive magnetic convection analogous to buoyancy-driven convection. This magnetothermal convection can overwhelm convection driven by gravitational buoyancy in terrestrial experiments. The objectives of the proposed ground-based theoretical study are (a) to supply the magnetothermohydrodynamic theory necessary to understand these recent experiments and (b) to explore the consequences of nonuniform magnetic fields in microgravity. Even the linear theory for the onset of magnetothermal convection is lacking in the literature. We intend to supply the linear and nonlinear theory based on the thermohydrodynamic equations supplemented by the magnetic body force. We intend to investigate the effect of magnetic fields on gas blockage and heat transport in microgravity. Since magnetic fields provide a means of creating arbitrary, controllable body force distributions, we intend to investigate the possibility of using magnetic fields to position and control fluids in microgravity. We also intend to investigate the possibility of creating stationary terrestrial microgravity environments by using the magnetic force to effectively cancel gravity. These investigations may aid in the design of space-based heat-transfer, combustion, and human-life-support equipment.

Edwards, Boyd F.↗

An Investigation of Flame Spread over Shallow Liquid Pools in Microgravity and Nonair Environments

Experiments of interest to combustion fundamentals and spacecraft fire safety investigated flame spread of alcohol fuels over shallow, 15 cm diameter pools in a 5.2 sec free-fall, microgravity facility. Results showed that, independent O2 concentration, alcohol fuel, and diluent types, microgravity flame spread rates were nearly identical to those corresponding normal-gravity flames for conditions where the normal gravity flames spread uniformly. This similarity indicated buoyancy-related convection in either phase does not affect flame spread, at least for the physical scale of the experiments. However, microgravity extinction coincided with the onset conditions for pulsating spread in normal gravity, implicating gas phase, buoyant flow as a requirement for pulsating spread. When the atmospheric nitrogen was replaced with argon, the conditions for the onset of normal-gravity pulsating flame spread and microgravity flame extinction were changed, in agreement with the expected lowering of the flash point through the thermal properties of the diluent. Helium-diluted flames, however, showed unexpected results with a shift to apparently higher flash-point temperatures and high normal gravity pulsation amplitudes.

Ross, Howard D.↗

An investigation of flame spread over shallow liquid pools in microgravity and nonair environments

Experiments of interest to combustion fundamentals and spacecraft fire safety investigated flame spread of alcohol fuels over shallow, 15 cm diameter pools in a 5.2 sec free-fall, microgravity facility. Results showed that, independent O2 concentrations, alcohol fuel, and diluent types, microgravity flame spread rates were nearly identical to those corresponding normal-gravity flames for conditions where the normal gravity flames spread uniformly. This similarity indicated buoyancy-related convection in either phase does not affect flame spread, at least for the physical scale of the experiments. However, microgravity extinction coincided with the onset conditions for pulsating spread in normal gravity, implicating gas phase, buoyant flow as a requirement for pulsating spread. When the atmospheric nitrogen was replaced with argon, the conditions for the onset of normal-gravity pulsating flame spread and microgravity flame extinction were changed, in agreement with the expected lowering of the flash point through the thermal properties of the diluent. Helium-diluted flames, however, showed unexpected results with a shift to apparently higher flash-point temperatures and high normal gravity pulsation amplitudes.

Ross, Howard D.↗

Theoretical Studies of Liquid He-4 Near the Superfluid Transition

We performed theoretical studies of liquid helium by applying state of the art simulation and finite-size scaling techniques. We calculated universal scaling functions for the specific heat and superfluid density for various confining geometries relevant for experiments such as the confined helium experiment and other ground based studies. We also studied microscopically how the substrate imposes a boundary condition on the superfluid order parameter as the superfluid film grows layer by layer. Using path-integral Monte Carlo, a quantum Monte Carlo simulation method, we investigated the rich phase diagram of helium monolayer, bilayer and multilayer on a substrate such as graphite. We find excellent agreement with the experimental results using no free parameters. Finally, we carried out preliminary calculations of transport coefficients such as the thermal conductivity for bulk or confined helium systems and of their scaling properties. All our studies provide theoretical support for various experimental studies in microgravity.

Manousakis, Efstratios↗

Using Electrostriction to Manipulate Ullage in Microgravity

A report proposes to use electrostriction to manipulate the ullage in a tank containing a dielectric liquid in a microgravitational environment. In the original intended application, the liquid would be a spacecraft propellant and the goal would be to force the ullage (comprising bubbles of noncondensible gas) to coalesce at one end of the tank, to enable use of one of the established means of (1) measuring the position of the gas/liquid interface and (2) inferring the quantity of liquid from the measurement. Electrically insulated wires would be installed in the tank, shaped and positioned so that application of a suitably high potential (e.g., 1 kV) between adjacent wires in successive pairs would give rise to a sufficient electric field gradient along the tank. The resulting electrostriction in the liquid would give rise to a pressure gradient that would force the ullage toward the low-electric-field-magnitude end of the tank. The feasibility of this proposal was demonstrated in an experiment in a tank containing liquid helium aboard an airplane flying a low-gravity arc. The ullage-segregating electrostrictive effect is expected to be considerably greater in other liquids.

Chui, Talso↗

Sloshing dynamics modulated angular momentum fluctuations actuated by gravity gradient acceleration associated with spacecraft slew motion

The mathematical formulation of orbital spacecraft sloshing dynamics for a partially filled cryogenic superfluid liquid helium II in a dewar container actuated by the gravity gradient acceleration associated with slew motion is studied. The Advanced X-Ray Astrophysics Facility-Spectroscopy (AXAF-S) spacecraft is chosen as a practical example in this study. Explicit mathematical expressions which manage orbital galaxy gradient acceleration with slew motion which is acting on the spacecraft fluid systems are derived. The numerical computation of sloshing dynamics is based on the non-inertia frame spacecraft bound coordinates and the solution of time-dependent, three-dimensional formulations of partial differential equations subject to initial and boundary conditions. This study discloses the capillary effect of sloshing dynamics governed liquid-vapor interface fluctuations, angular momentum and moment fluctuations of fluid system, and also buble mass center fluctuations driven by the gravity gradient acceleration associated with slew motion which affects the stability of the orbital spacecraft fluid system in a microgravity environment.

Hung, R. J.↗

Mathematical Model of Bubble Sloshing Dynamics for Cryogenic Liquid Helium in Orbital Spacecraft Dewar Container

A generalized mathematical model is investigated of sloshing dynamics for dewar containers, partially filled with a liquid of cryogenic superfluid helium 2, driven by both gravity gradient and jitter accelerations applicable to two types of scientific spacecrafts, which are eligible to carry out spinning motion and/or slew motion to perform scientific observations during normal spacecraft operation. Two examples are given for the Gravity Probe-B (GP-B) with spinning motion, and the Advanced X-Ray Astrophysics Facility-Spectroscopy (AXAF-S) with slew motion, which are responsible for the sloshing dynamics. Explicit mathematical expressions for the modelling of sloshing dynamics to cover these forces acting on the spacecraft fluid systems are derived. The numerical computation of sloshing dynamics will be based on the noninertial frame spacecraft bound coordinate, and we will solve the time-dependent three-dimensional formulations of partial differential equations subject to initial and boundary conditions. Explicit mathematical expressions of boundary conditions lo cover capillary force effects on the liquid-vapor interface in microgravity environments are also derived. Results of the simulations of the mathematical model are illustrated.

Hung, R. J.↗