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IV
Phenomena of Importance in Reduced Gravity
In Chapter III various HEDS technologies are explored and the phenomena likely to affect their operation in
reduced gravity are identified. In this chapter, those phenomena and their dependence on, or importance in,
reduced gravity are discussed, along with research that would be needed to develop an adequate database as well
as predictive models for better characterizing the phenomena. The material in this chapter and in Chapter V
provides the basis for the research recommendations outlined in Chapter VI.
IV.A GENERAL CONSIDERATIONS
Introduction
In microgravity research, one is concerned with how, in various circumstances, the relationships or balances
among various fundamental phenomena depend on the presence or absence of gravity and, consequently, how
particular processes of technical importance depend on gravity level.
Some phenomena can therefore be isolated for study if gravity is eliminated. Moreover, knowledge of the
underlying physics can often be obtained when it is not masked by phenomena that are induced by terrestrial
gravity. This basic scientific goal is emphasized in the NASA microgravity research program (Woodward, 1998~.
Understanding and mitigating the technological consequences of low gravity must be one of NASA's primary
research goals. If the HEDS program is to be successful, however, its research goals must differ from NASA's
purely scientific goals. In particular, the emphasis must be on applied research, which is the focus of this report.
Indeed, the report identifies the kinds of enabling technologies required to meet the goals of the HEDS program.
The following paragraphs provide an overview of the basic effects of a reduction in gravity; those effects are
then treated in more detail in subsequent sections of the chapter. The emphasis is on how the interplay of gravity
and other, competing basic physical processes finds expression in various dimensionless combinations of physical
parameters. Either explicitly or implicitly, these dimensionless groups will govern the interaction of microgravity
with the specific phenomena discussed in this chapter. The overview concludes with some general thoughts about
the need for microgravity research to characterize the relationships among these groups.
111
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2
MICROGRAVITY RESEARCH
Gravity and Density Difference
First, the general importance of the combination of gravity with density difference should be noted. Gravity
is a body force per unit mass (a virtual acceleration, according to Newtonian mechanics). Therefore, the gravity-
force-density is the product of gravity and density (p), and fluid motion is affected by gravity if the density is
nonuniform. Accordingly, one expects any change of gravity level coupled with a density difference (whether
preexisting or due to the flow process) to affect the phenomena of interest. In other words, a product gyp will be
of basic significance, where /\p is a characteristic parameter describing the density difference.
Clearly, the larger the density difference the larger will be the effect of a change in gravity level. The largest
density differences normally encountered are those due to the presence of different phases, especially gas and
liquid, or gas and solids. Thus, one must be especially concerned with multiphase flows. Smaller, but often
important, density differences also occur as the result of thermal expansion of a single phase, as will be discussed
in a later section of this chapter.
Frequently, when gravity is changed, physical phenomena that depend on gravity will change in ways that can
be characterized as a change in stability. Stability is an especially useful concept for engineering purposes, since
one normally intends a system to stay unchanged over time or to maintain a steady motion. There are many
classical books treating stability; for fluids, the work of Chandrasekhar (1961) and of Yih (1965) abounds in
stability problems involving gravity.
A common issue for the fluid systems discussed in Chapter III is a change of local stability when gravity is
reduced or vanishes. In Earth gravity, a liquid/vapor system will tend to stratify stably, with vapor above and
liquid below; then, if the interface is displaced it will tend to return. If gravity is absent, the restoring force is
absent and the position of the interface is indeterminate. This is the situation in a cryogenic storage tank. If the
situation in Earth gravity is initially the reverse, with vapor below the liquid, then the situation is unstable, and the
vapor will tend to rise as a consequence of buoyancy. This instability enables a simple boiler to function. As
gravity is reduced, this helpful instability is reduced, and boiler performance is impaired. Clearly, either stability
or instability associated with gravity on Earth may be desirable, depending on the function considered. In some
cases, such as the dispersal of particulates or droplets, neutral stability would be desired, and in such cases, reduced
gravity could be helpful.
Gravity-Density Coupling in Various Basic Processes
Various basic flow processes may now be related to buoyancy as expressed by the gravity-density difference
product, and typical dimensionless groups may thus be identified. It will be seen that if such groups are numeri-
cally very large or very small, dominance of one process over another is implied. More importantly, such groups
are parameters of any theoretical or computational model for system behavior, with effects depending on numeri-
cal coefficients, or correlations, derived from the appropriate theories or experiments.
· Flow forced by pressure difference. If the velocity of buoyant rise (or fall) through a distance L owing to
a given density difference (/\p) is compared with the velocity produced by a given pressure difference, /\p, acting
as the only other force on the fluid, the comparison can be expressed in terms of the dimensionless group glYpL/lYp.
If this parameter] is much less than unity, then buoyancy is insignificant. This is certainly true in rocket exhausts,
for example, where the pressure drop is very large.
· Capillarity, wetting, and Marangoni;flows. The role of buoyancy in comparison with that of capillarity, or
surface tension (c,j, can be expressed by the dimensionless group g/\pL2/c, when surface tension acts on a curved
interface of scale L. This group is the "static" Bond number (Ostrach, 1982) based on density difference. One sees
that for a given surface tension coefficient Achy, capillarity becomes more important relative to buoyancy if either
1With pressure drop expressed in terms of the flow velocity it produces, this parameter is the reciprocal, squared, of the Froude number as
given by Yih (1965) or the "densimetric" Froude number in the hydraulics literature.
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PHENOMENA OF IMPORTANCE IN RED UCED GRA VITY
113
g or L becomes small and less important if L is large, unless g is especially small or zero. In Marangoni flows, it
is the gradient of c, that is important (Ostrach, 1982), so in the above group, c, would be replaced by a surface-
tension difference /\c,, which in turn varies with temperature or composition. This topic is discussed in more detail
in Section IV.B.
· Diffusion. Transport by buoyancy may compete with molecular diffusion in affecting composition, tem-
perature, or some other local property. If the diffusion coefficient, D, is known for the property difference of
interest, the competition is governed by the dimensionless group g/\pL3/pD2. If the diffusing quantity is thermal
energy, the foregoing group becomes g/\pL3pcp2/k2, where k is thermal conductivity and cp is specific heat. This
subject is discussed more fully in Section IV.B.
· Viscosity. Similarly, buoyant transport of momentum may compete with viscosity, according to the group
glipL3 /pv2, in which v is the kinematic viscosity coefficient of the fluid.2
· Chemical transformation and combustion. Obviously, chemical change results in density change, and
therefore an interaction with gravity can be expected, as discussed above. Consequences of chemical change in
low gravity are also discussed in Section IV.F.
· Electromagnetic forces. Electric or magnetic fields may exert body forces whose effects can be compared
with the gravitational body force, gyp.
· Vibration. Time dependence of motion, or unsteady motion in general, introduces yet another degree of
freedom of fluid or solid motion, and the accelerations involved clearly compete with the virtual acceleration
represented by gravity. Thus, the importance of gravity during vertical vibration of a container with a liquid/vapor
interface (Yih, 1965) will be governed by the group g/\p/p£Co2, where £ is the amplitude of the imposed vibration.
The higher the imposed frequency co, the less important buoyancy becomes relative to forces due to acceleration.
The dynamic behavior of machines or structures also depends on gravitational loading, as a simple pendulum
illustrates. Unsteadiness in competition with gravity is an important topic called "jitter, where gravity itself
appears to fluctuate owing to imposed accelerations (Woodward, 1998) arising from crew motions, rocket firings,
and so on. Acoustics in the presence of gravity furnishes still another important example of time dependence. In
general, if a process is time-dependent, new parameters, and hence additional dimensionless groups, will need
attention.
~ , i, ~
· Phase change. In a subsequent section, the physical process of solidification and melting is discussed, and
it is made clear that the role of buoyancy, and hence of gravity level, is an interactive one: the phase change itself
depends on buoyant transport, which in turn depends on the degree of phase change already achieved.
· Granular behavior. In Section IV.G, problems of particulate or granular flows are described. Density and
gravity are of course coupled in such flows, and particle interactions with each other and with liquid or gaseous
media will furnish examples of the static and dynamic phenomena with which buoyancy must be compared.
Gravity Regime Boundaries
It is obvious from the foregoing discussion that particular combinations of processes must be identified for
specific problems. For example, inertia, viscosity, heat conduction, and capillarity may be simultaneously impor-
tant, along with buoyancy, in some technically important phenomena. Therefore, many specific dimensionless
groups will be needed to describe complex flow regimes of interest (Ostrach, 1982~.
Nevertheless, the crude outline given above shows that, generally, gravity level, g, finds itself in a group that
includes density difference and some positive power (n) of length scale, that is, g/\pLn. Thus, one may infer that
when gravity is reduced, any effect of that reduction will be amplified by large density difference (especially that
due to phase difference) and also by large scale.
2This group may be recognized as the classical Grashof number (Eckert and Drake, 1972), to which the previous two groups are related
through the Schmidt number (v/D) and the Prandtl number (pcpvlk).
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4
MICROGRAVITY RESEARCH
Gravity reduction will be significant only in relation to phenomena competing with buoyancy or settling
processes. Therefore, in any given reduced-gravity situation, one would like to know whether the dimensionless-
group coefficients or correlations used in Earth gravity still apply, or whether a different set of models or
correlations needs to be developed. That is, one asks what numerical levels of the various dimensionless groups
applicable to a specific technical situation represent critical boundary zones between regimes of essentially
terrestrial and essentially microgravity behavior. This is the problem posed by fractional gravity environments,
which is where a large proportion of HEDS operations are expected to be carried out.
Research Issues
It would be useful for NASA to develop a catalog of the regime-change zones for the dimensionless param-
eters of all relevant fundamental phenomena. This would presumably entail reviewing and extending the experi-
ments performed in microgravity. This effort would provide the basis for assessing the computational design
capabilities in the future, capabilities essential for comprehensive and credible designs of efficient, reliable
systems and components for HEDS.
Clearly the number and the ranges of relevant parameters are very great. Therefore, the development of the
suggested catalog would require effort carried out over a long period of time, focused on a great variety of difficult
issues and problems.
References
Chandrasekhar, S. 1961. Hydrodynamic and Hydromagnetic Stability. International Senes Monographs on Physics. Oxford: Clarendon Press.
Ostrach, S. 1982. Low-gravity fluid flows. Annul Rev. Fluid Mech. 14:313-346.
Woodward, D. 1998. NASA's Microgravity Research Program. NASA Report TM 1998-208418. Huntsville, Ala.: NASA Marshall Space
Flight Center.
Yih, C.S. 1965. Dynamics of Nonhomogeneous Fluids. London: Macmillan.
IV.B INTERFACIAL PHENOMENA
Interfacial or capillary phenomena generally refer to the broad field of surface-tension-related phenomena
(Maxwell, 1878; Gibbs, 1878; de Gennes, 1985; Haynes and Langbein, 1987~. The terminology derives from the
surface-tension-induced rise (fall) of liquid in a capillary tube for contact angles less than (greater than) 90°. These
phenomena are not directly influenced by gravity level, but they become increasingly important in determining the
configuration and movement of liquid as gravity level is reduced, and they may become dominant in microgravity
(Ostrach, 1982~. In some cases, the effects may be utilized to compensate for the loss of gravity in the manage-
ment of liquids in microgravity; examples are wicked structures as in heat pipes (Faghri, 1995; also see Figure
III.A.5 of this report), capillary pumped loops, and vane structures as in cryogenic storage tanks (Dodge, 1990~.
Because of their dominance and practical importance in reduced gravity, the following interracial phenomena are
discussed in this section: static and dynamic capillary configurations, wetting, and the Marangoni effect (arising
from gradients of the surface free energy).
Capillary Equilibrium and Dynamic Forms
Most of the problems and work described in this section assume a uniform surface tension and therefore omit
Marangoni effects, but in many practical situations these must be included; they are discussed later on. The
dimensionless Bond number measures the ratio of (gravity forces)/(capillary forces) and is given by B = pgL2/c,,
where p and c, are, respectively, the density and surface tension of the liquid, g is the gravitational strength, and L
is a characteristic length; for example, the ratio of height to radius of a liquid in a capillary tube is inversely
proportional to B. Similarly, the Bond number enters into a description of equilibrium shapes determined by the
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PHENOMENA OF IMPORTANCE IN RED UCED GRA VITY
115
competition of gravitational and capillary forces, as exemplified by sessile and pendant drop shapes (Antar and
Nuotio-Antar, 1993; Haynes and Langbein, 1987; Concus and Flinn, 1990~.
There is a large literature devoted to static equilibrium capillary shapes in the absence of gravity; these are
shapes that minimize the area of the surface of a mass of liquid subject to boundary conditions of a fixed volume
and given contact angles at the perimeter of the surface (Antar and Nuotio-Antar, 1993; Haynes and Langbein,
1987; Concus and Flinn, 1990; Carter, 1998~. In some cases (as for a box with a shallow layer of partially wetting
liquid), there are multiple solutions: the liquid may cover a face of the box or collect along the edges and/or
corners (Martinez et al., 1987~. In addition, there is the question of the stability of equilibrium shapes, as
illustrated by the Rayleigh instability of a cylinder or the stability of liquid bridges (Koster, 1990), that arise in
containerless processing of materials.
In addition to static problems, there are many dynamic problems in capillary theory exemplified by Benard
convection, the oscillations of liquid drops or bubbles, resonances between the applied forces or accelerations
(e.g., "jitter), and capillary modes of motion of a mass of liquid in a container (Zhang and Vinals, 1997), including
so-called sloshing problems (Antar and Nuotio-Antar, 1993~. Benard convection, arising from the heating from
below of a layer of liquid with a free surface and leading to a pattern of hexagonal convection cells, is often
dominated by Marangoni convection (Antar and Nuotio-Antar, 1993) rather than the buoyancy-driven convection
analyzed by Rayleigh. In particular, Benard cells have been studied under microgravity conditions.
The stability and dynamics of capillary-dominated configurations of liquids are expected to play an important
role in the management of liquids and in the boiling/condensation process in heat exchangers under reduced
gravity conditions (Westbye et al., 1995~. As mentioned above, they also underlie the operation of heat pipes,
capillary pumped loops, microgrooved heat pipes, and veined structures.
Surface-tension-driven free surface flows arise in nature, science, and technology. An important technologi-
cal application of free surface flows is the laser printer, and HEDS applications include the liquid-droplet surface
radiator and the electrostatic liquid film radiator. It is recognized that drops generally result from the motion of
free surfaces. The dynamics and breakup of drops and of other free surface flows, including theoretical develop-
ments and experimental work, have recently been reviewed and unresolved problems have been outlined (Eggers,
1997; Stone, 1994~.
Research Issues
A body of classical knowledge and current results is available, as indicated above, but there are many
unsolved problems of capillarity-determined liquid configurations involving practical boundary conditions (e.g.,
vessel and conduit shapes) that require additional modeling and experiments. Dynamic problems involving
resonances between imposed vibrations and accelerations (e.g., "jitter) and capillary modes of liquid masses are
important because of the possibility of uncontrolled excursions of the liquid mass; such problems need to be
extended. In addition, the inclusion of Marangoni effects will require computer modeling of complex fluid flows
as well as a greatly improved knowledge of the Marangoni parameters (e.g., temperature and composition depen-
dence of the surface tension).
Wetting
Wetting is a phenomenon in which one condensed phase spreads over the surface of a second condensed phase
(Adamson, 1982; Findenegg and Telo de Gama, 1987~. If the spreading stops at some equilibrium configuration
with the surfaces of both phases exposed, it is called partial wetting; this is illustrated in Figure IV.B.1 for the case
of a drop of liquid L on the surface of a rigid solid S. both in contact with the vapor V. The contact angle ~ is
determined in the classical description by the balance of surface tensions (indicated in the figure) described by the
Young equation:
~sv = Ads + Rev cosO.
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116
FIGURE IV.B. 1 A liquid drop in equilibrium with a rigid solid sur-
face and the vapor; interface free energies are labeled by the adjacent
phases.
MICROGRAVITY RESEARCH
ASH ILLS
If the spreading does not stop, which occurs when cash 2 Cal S + Cal v, then complete wetting is said to occur; the
liquid phase spreads indefinitely on (i.e., it wets) the solid surface as long as it is thick enough to be described as
a macroscopic liquid. In general, the two phases may be liquid or solid. A related phenomenon is the partial or
complete penetration of a second phase into a grain boundary between two crystals. Although wetting is a
capillary or surface phenomenon not significantly affected by gravity level, it becomes increasingly important
when the gravity level is reduced, which makes it one of several phenomena that dominate events under micro-
gravity conditions, and for this reason it is included in this report.
Wetting, partial or complete, underlies such important technologies as soldering and welding (Nance and
Jones, 1993~; liquid-phase sintering (German et al., 1995~; the operation of wicks in capillary pumped loops
(Anatar and Nuotio-Antar, 1993) and vanes in the cryogenic storage of liquids (Dodge, 1990~; heat pipes (Faghri,
1999; Peterson et al., 1998~; boiling/condensation heat transfer, including the rewetting of a hot surface (Westbye
et al., 1995~; and lubrication. Usually, small contact angles or complete wetting are desired in these techniques.
To understand the meaning of complete wetting (de Gennes, 1985; Dietrich, 1988) in the context of the
previous illustration, let W = cask + Cal v - Cal S be the work per unit area required to separate the liquid from the
solid. Then, the condition for complete wetting becomes W 2 2c,~ v, which means that the adhesion of the liquid
to the solid is greater than the cohesion of the liquid. Complete wetting is favored for liquids of low surface energy
in contact with solids to which they are strongly attracted. A film of oil or grease that interferes with adhesion will
favor partial wetting with a large contact angle.
Extensions and modification of the Young description are necessary to account for several important features
of wetting that include (1) hysteresis of the contact angle, (2) dynamics of wetting, and (3) breakdown of the
macroscopic description on sufficiently small scales. Hysteresis of the contact angle (Decker and Garoff, 1997;
Rame and Garoff, 1996) refers to the greater value of the contact angle obtained from measurement if the contact
line is advancing (extending the liquid) than if it is receding; the difference between the advancing and receding
contact angles makes it possible for a drop on a tilted solid surface in 1 go to be stationary (e.g., raindrops on a
window pane). The extent of hysteresis is affected by the microscopic topography and condition of the surface.
The dynamics of wetting is exemplified by dependence of the geometry of the liquid configuration in the
immediate vicinity of a moving contact line on the velocity of the contact line; in particular, the effective contact
angle depends on that velocity. This is particularly true of the ebullition cycle during boiling, where the dynamic
contact angle is required to adequately model the nucleation of bubbles on the heated wall. There is also evidence
that a spreading liquid drop on a solid surface is preceded by a thin film (Marsh et al., 1993~. The dynamics of
wetting has become an important area of investigation (Decker and Garoff, 1997; Rame and Garoff, 1996~.
The thermodynamic description of wetting breaks down when the thickness of the liquid becomes comparable
to or less than the correlation length in the liquid, i.e., when the liquid in a thin film or filament can no longer be
described in macroscopic thermodynamic terms. This can occur if the liquid is imbibed into a porous structure (as
in a wick) whose pore dimensions are comparable to or smaller than the correlation length in the liquid (Wiltzius
et al., 1989~.
Two additional developments in wetting science in recent years are the following: (1) A phenomenon known
as premelting (Frenken and van der Veen, 1985) can occur in which the surface layer of a solid loses translational
long-range order below the melting point of the solid. As the melting point is approached, the thickness of the
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PHENOMENA OF IMPORTANCE IN RED UCED GRA VITY
T
To
Tw
A
~1
Ct ~ OC
117
BFIGURE IV.B.2 Miscibility gap with critical and wetting tem-
peratures Tc and Tw.
layer diverges to become a bulk liquid layer that completely wets the solid at the melting point. Premelting has
been confirmed in several systems. (2) A theory of Cahn (1977), now confirmed in several systems, shows that in
a binary system with a critical temperature Tc, as shown schematically in Figure IV.B.2, there is a temperature Tw<
Tc in the two-phase region above which one of the two phases (say or, rich in A) will be completely wetted by the
other (or', rich in phase B). Further, as the two-phase field is approached from the single-phase A side (shown by
the arrow), an adsorbed layer of B on the A phase occurs and increases in thickness until at the two-phase
boundary, wetting of or by or' occurs.
Research Issues
Technologies that depend on wetting generally require good or complete wetting of a fluid on solid surfaces
(i.e., low or zero contact angle). Based on this requirement and the preceding discussion, areas that would profit
from research are the following: (1) the hysteresis effect, which can inhibit the spread of the wetting liquid, (2) the
dynamics of wetting, which determines the rapidity with which wetting or rewetting will take place, (3) the
1 ~
~7 ~7 1
description of wetting in porous materials when the scale of the pores is comparable to or smaller than the
correlation length in the liquid, so that a bulk description of the liquid is no longer appropriate, and (4) develop-
ment of the molecular theory of wetting, which would enlarge the knowledge base on material combinations and
conditions for good wetting and on wetting or tensioactive agents (Schrader and Loeb, 1992; Eustathopoulos et al.,
1998~.
Marangoni Effect
The Marangoni effect (Hondros, 1998; Antar and Nuotio-Antar, 1993; Legros et al., 1987, 1990; Ostrach,
1982) commonly refers to liquid convection caused by surface tension gradients at the free surface of a liquid or at
the interface between two liquids. Because this dependence of surface tension on position arises in the presence of
temperature or composition gradients along the surface when the surface tension depends on the temperature and/
or on composition, the effect may also be called the thermosolutal, thermocapillary, or solutocapillary effect.
When the surface tension is a function of position along a surface or interface, there is a resultant force on an
element of the surface or interface. Since the net force on the element must vanish to avoid essentially infinite
acceleration of the atomically thin element, a velocity gradient perpendicular to the surface is generated that
1 41 ~ _1_ 1 ~ ~1 1 ~ 41 1 ~ 1 21 21 ~ 41 ~1 21 ~1
supplies the counterbalancing VISCOUS force. l he result IS a tnermosolutal-lnclucecl convection In the ilUlC`. l he
Marangoni effect occurs in the absence of gravity and is a dominant cause of convection in microgravity. A
special publication of the Philosophical Transactions of the Royal Society, London (Hondros et al., 1998) is
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118
MICROGRAVITY RESEARCH
devoted entirely to the Marangoni effect and provides a comprehensive look at what is known about this phenom-
enon, especially as it affects materials processing. It has been suggested (Ostrach, 1977a,b, 1982) that, for
historical reasons, the term Marangoni instability should be used to describe those effects resulting from gradients
initially normal to an interface. Terms such as thermocapillary and solutocapillary can be applied to effects
resulting from gradients of the corresponding variables initially parallel to the interface. However, the term
Marangoni effect (or flow) is commonly applied to the generality of flows induced by any gradient that produces
a variation of surface tension with position on the interface, whether directly or through the effect of perturbations,
and this is the usage that has been adopted in this report.
As an example of the Marangoni effect, consider a single component liquid with a liquid/vapor surface lying
in the xy plane with a surface tension c,. Suppose there is a temperature gradient ~T/0x at the surface. Then, the
force on a surface element in the x direction per unit length along y is dc,/0x = (0c,/0T)~0T/0x). To balance this,
there must be a velocity gradient in the z direction, normal to the surface, of magnitude given by ,u~v/0z =
(0c,/0T)~0T/0x), where ,u is the coefficient of viscosity. This velocity gradient serves as a boundary condition that
generates a convection pattern in the fluid. It is clear from this simple expression that the induced velocity gradient
is proportional to the temperature gradient (0T/0x) and inversely proportional to the viscosity. The actual flow in
any given system may be very complicated because the convection affects the surface gradients of temperature and
composition which, in turn, generate the convection. This coupling of Marangoni-induced convection with the
surface tension via the surface temperature and composition can lead to oscillatory flows (Verlarde,1998~. In the
case of a liquid/solid interface, there is no Marangoni effect since the solid exerts forces that balance interface
tension gradients (and cause a zero velocity at the interface).
In general, the dependence of c, on position arises from both temperature gradients and composition gradients
(the latter is exemplified by the tearing of wine in a glass), since c, generally depends on both variables; often the
effects are mixed, as in the combustion or gasification of liquid drops (Zhang et al., 1996; Aharon and Shaw,
1996~. In a multicomponent system, temperature gradients may have both a direct effect on c, and an indirect
effect through the temperature dependence of the surface composition. Thus, whereas dc,/0T is negative for a pure
component, it may be positive in a multicomponent system, because as T increases, c, may increase if the
adsorption of a surface-active component decreases sufficiently; a notable example is sulfur in stainless steel
above 40 ppm (Mills et al., 1998~. Thus, in multicomponent systems, the Marangoni effect can have either sign
(defined as the sign of dc,/0T).
The Marangoni effect can lead to the migration of drops and bubbles (Verlarde,1998~. For example, consider
a liquid drop with dc,/0T < 0. If it is placed on a plate with a temperature gradient, it will move toward the cold
end. If it is suspended in a fluid with a temperature gradient, however, it will move toward the hot end (assuming
the same sign for the Marangoni coefficient); in the latter case, the return flow along the drop's center line pushes
the nose of the drop forward toward the hot end. Similar phenomena happen with bubbles and can have a strong
effect on pool and forced convective boiling heat transfer.
Marangoni convection usually dominates gravity-induced convection in weld pools in Earth gravity; it is
undiminished in microgravity. When the sign of the effect is negative, the liquid surface is pulled toward the
cooler outer edges of the pool, and when it is positive, the reverse is true. The shape of the weld pool is affected
(Mills et al., 1998~. The Marangoni effect dominates gravity-induced convection in the Benard instability arising
from heating a liquid layer from below, provided the layer is not too deep. The threshold AT required to initiate the
flow that was calculated by Rayleigh based on gravity-induced convection is 104 to 105 times larger than that
found experimentally by Benard (Legros et al., 1987~; the discrepancy was attributed to the Marangoni effect,
which causes warm liquid rising to the top center of a cell to be pulled outward by surface tension to the cooler cell
edges, where it then sinks. As the gravity level is reduced, Marangoni convection increasingly dominates gravity-
induced convection.
The relative strengths of Marangoni convection and gravity-induced convection are quantified by the dimen-
sionless Marangoni (Ma) and Rayleigh (Ra) numbers:
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PHENOMENA OF IMPORTANCE IN RED UCED GRA VITY
119
Ma= c,TpL2/pvk
Ra= goc/\TL3/vk
(Legros et al., 1990; Verlarde, 1998), where, in the first formula, (;ST= dc,/0T, ~ = VT, p is the density, v is the
kinematic viscosity, k is the thermal diffusivity, and L is a characteristic distance, and in the second formula, g is
the gravitational constant, or is the thermal expansion coefficient, AT is a temperature difference over a character-
istic distance L, and v and k are defined as above. To initiate surface-tension-induced Benard flow, that is, flow
due to Marangoni instability, a critical value of Ma ~ 80 is required, and to initiate buoyancy-induced Benard flow,
a critical vale of Ra ~ 1,100 is required. Other characteristics of the flow (e.g., oscillatory and turbulent) are
determined by these and other dimensionless numbers of fluid dynamics.
As indicated by the above examples, Marangoni effects are ubiquitous wherever liquid/fluid interfaces are
subject to temperature and composition gradients. The effects become dominant in reduced gravity, as in the
stirring of a weld pool, the migration of liquid in spills, fire control, and two-phase fluid transport, and in capillary-
operated devices such as heat pipes or capillary pumped loops (referred to in the subsection on wetting). More-
over, multicomponent mixtures may have significantly higher critical heat fluxes than occur for single-component
boiling, and this observed increase in critical heat flux is apparently due to the Marangoni-induced flows.
Research Issues
Based on the previous discussion, the research areas of particular relevance are (1) the modeling and experi-
mental study of the complex convection flows induced by the Marangoni effect, (2) the experimental determina-
tion of the parameters that enter into the Marangoni and other relevant dimensionless numbers (Egry et al., 1998),
(3) investigation of tensioactive agents (Eustathopoulos et al., 1998; Verlarde, 1998) to control the magnitude and
sign of the effect, and (4) inclusion of thermal and concentration gradients to assess the merits of designs where the
Marangoni-induced flow of fluids can be useful or detrimental.
References
Adamson, A.W. 1982. Physical Chemistry of Surfaces, 4th Ed. New York: Interscience.
Aharon, I., and B.D. Shawl 1996. Phys. Fluids 8:1820.
Antar, B.N., and V.S. Nuotio-Antar. 1993. Liquid gas capillary surfaces. Fundamentals of Low Gravity Fluid Dynamics and Heat Transfer.
Boca Raton, Fla.: CRC Press.
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de Gennes, P.G. 1985. Wetting: statics and dynamics. Rev. Mod. Phys. 57:827.
Decker, E., and S. Garoff. 1997. The need for new experimental and theoretical models. J. A&es. 63: 159.
Dietrich, S. 1988. P. 1 in Phase Transitions and Critical Phenomena, Vol. XII. C. Domb and J.L. Lebowitz, eds. New York: Academic Press.
Dodge, F.T. 1990. Fluid management in low gravity. P. 369 in Low-Gravity Fluid Dynamics and Transport Phenomena: Progress in Astronau-
tics and Aeronautics, Vol. 130. J.N. Koster and R.L. Sani, eds. New York: American Institute of Aeronautics and Astronautics.
Eggers, J. 1997. Nonlinear dynamics and break-up of free-surface flows. Rev. Mod. Phys. 69:865-929.
Egry, I., M. Langen, and G. Lohofer. 1998. Measurements of thermophysical properties of liquid metals relevant to Marangoni effects. Philos.
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Eustathopoulos, N., J.P. Garandet, and B. Drevet. 1998. Influence of reactive solute transport on spreading kinetics of alloy droplets on
ceramic surfaces. Philos. Trans. R. Soc. London, Ser. A 356(1739):871.
Faghri, A. 1995. Heat Pipe Science and Technology. Washington, D.C.: Taylor and Francis.
Faghri, A. 1999. Recent advances in heat pipe analysis and simulation. Annual Review of Heat Transfer, Vol. 8. C.-L. Tien, ed. New York:
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Findenegg, G.H., and M.M. Telo de Gama. 1987. Wetting and adsorption phenomena. P. 191 in Fluid Sciences and Materials Science in
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Frenken, J.W.M., and J.F. van der Veen. 1985. Phys. Rev. Lett. 54:134.
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German, R.M., R.G. Iacocca, J.L. Johnson, Y. Liu, and A. Upa&yaya. 1995. Liquid-phase sintering under microgravity conditions. J. Met.
47(8):46-48.
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Haynes, J.M., and D. Langbein. 1987. Fluid statics and capillarity. P. 53 in Fluid Sciences and Materials Science in Space. H.U. Walter, ed.
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Hondros, E.D., M. McLean, and K.C. Mills, eds. 1998. Marangoni and interracial phenomena in materials processing. Philos. Trans. R. Soc.
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Legros, J.C., A. Sanfeld, and M. Verlarde. 1987. Fluid dynamics. P. 109 in Fluid Sciences and Materials Science in Space. H.U. Walter, ed.
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Nance, M., and J.E. Jones. 1993. Welding in space and low-gravity environments. P. 1020 in ASM Handbook, Vol. 6: Welding, Brazing and
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IV.C MULTIPHASE FLOW
Both single and multiphase3 fluid flows may be used in microgravity environments. While single-phase
flows can behave somewhat differently in space (owing, for example, to the absence of natural-convection-
induced flows), they can be reliably calculated in most cases.
The microgravity issues associated with single-phase flows are primarily those related to heat-transfer-
induced density changes. These phenomena are discussed in Section IV.D, which is concerned with heat transfer
phenomena. Multiphase flows are inherently more complicated than single-phase flows. Because of differences
in phasic density and inertia, multiphase flows may exhibit pronounced phase separation and distribution. Indeed,
3The simultaneous flow of several phases or components, for example, vapor/liquid flows and solid/fluid flows.
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multiphase flows normally configure themselves into distinctive flow regimes in which the various phases are
nonuniformly distributed across the duct through which they flow. Significantly, the modeling of turbulence,
pressure drop, heat transfer, and stability must take flow regimes into consideration. Moreover, surface-tension-
induced (e.g., Marangoni) forces and surface phenomena are likely to be much more important in space than they
are on Earth. It should be noted that virtually all flow-regime-specific phenomena will be influenced by gravity
level during both normal forced-flow operating conditions and various postulated accidents.
Although NASA has so far been able to design around the need for the extensive use of multiphase systems
and processes in its missions, the agency is well aware of the fact that there are numerous issues and concerns
facing space system designers attempting to utilize multiphase flow and heat transfer processes and systems in
space (McQuillen, 1999~. However, it is inevitable that such systems and processes will be needed for HEDS
proposed interplanetary missions and extraterrestrial colonies. Examples, which are discussed in Chapter III,
include Rankine cycle power plants, boilers, evaporators, condensers, multiphase thermal buses, electrolysis units,
and various other life support and materials processing systems for which performance and weight are important
design considerations. In addition, there is a need to better understand multiphase particulate/fluid systems (e.g.,
dust transport and deposition) in reduced gravity if extraterrestrial colonies are to be established.
Unfortunately the current state of knowledge concerning how multiphase systems behave in reduced and
microgravity environments is inadequate to support NASA's proposed missions and goals. Indeed, if one is to
have confidence in the performance of multiphase systems and processes in space and at extraterrestrial sites, then
a well-coordinated research and development effort will be required to provide the needed mission-enabling
technology.
To understand why this effort is so crucial, it is important to note that multiphase flow and heat transfer
technology is a mature one that has been widely used on Earth during the last century. Nevertheless, it is a field
that has been empirically based. Unfortunately, many of the design rules and correlations that are valid on Earth
are invalid for microgravity applications. One reason for this is that the flow regimes (i.e., how the phases
configure) are quite different on Earth and in space. Moreover, natural circulation and buoyancy are suppressed in
space while they play an important role in the performance of many multiphase systems on Earth.
It will not be economically viable to develop multiphase systems for space in the same way as these systems
have been developed on Earth. That is, it will not be practical to test different configurations in space until an
acceptable design is achieved. Rather, physically based analytical models (which can be used in computer codes
for design purposes) should be developed to take into account all relevant aspects of reduced-gravity phenomena.
These analytical models could then be used to optimize designs and scale up small scale microgravity data to full
scale.
Phase Separation and Distribution
It is well known (Heppner et al., 1975) that buoyancy plays an important role in the phase distributions that
lead to the development of the various flow regimes and that, as shown in Figure IV.C. 1, the regimes can be very
different in microgravity environments.
This subsection focuses on some multiphase flow phenomena that are considered to be of vital importance to
the HEDS technologies. Many important and challenging problems in multiphase flow and heat transfer have to
do with multidimensional (i.e., three-dimensional) phenomena, in particular, phase separation and distribution
phenomena.
When a flowing multiphase mixture (vapor/liquid or solid/fluid) changes direction or is subject to other types
of accelerations, the phases may separate nonuniformly. This is because of the relatively large differences in the
inertia of each phase for sufficiently large dispersed particles or bubbles. A good example of this phenomenon can
be seen in Figure IV.C.2, which illustrates phase separation for a bubbly gas/liquid mixture in a piping Tee
(Hwang et al., 1988~.
Because the gas phase has a lower density, and thus a lower momentum flux, than the liquid phase, it has an
easier time changing direction from the inlet section into the side branch of the Tee. Thus the position of the
dividing streamline for the gas (6G) is farther into the incoming stream than that for the liquid phase (by. Hence
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Bassler, B.T., W.H. Hofmeister, and R.J. Bayuzick. 1995. Examination of solidification velocity determination in bulk undercooled nickel.
Materials Research Society Fall Meeting, Boston, Mass. Warrendale, Penn.: Materials Research Society.
Billia, B., and R. Trivedi. 1993. Pattern formation in crystal growth. Pp. 899-1073 in Handbook of Crystal Growth, Vol. IB. D.T.J. Hurle, ed.
Amsterdam: North-Holland.
Coriell, S.R., and G.B. McFadden. 1990. Instability during directional solidification: Gravitational effects. P. 369 in Low-Gravity Fluid
Dynamics and Transport Phenomena. J.N. Koster and R.L. Sani, eds. New York: American Institute of Aeronautics and Astronautics.
Coriell, S.R., and G.B. McFadden. 1993. Morphological stability. P. 785 in Handbook of Crystal Growth, Vol. 1B. D.T.J. Hurle, ed. New
York: Elsevier.
Curreri, P.A., and D.M. Stefanescu. 1988. Low-Gravity Effects During Solidification. P. 147 in Metals Handbook, 9th Ed., Vol. 15. Metals
Park, Ohio.: ASM International.
German, R.M., R.G. Iacocca, J.L. Johnson, Y. Liu, and A. Upa&yaya. 1995. Liquid-phase sintering under microgravity conditions. J. Met.
47(8):46-48.
Glicksman, M.E., and S.P. Marsh. 1993. The dendrite. Pp. 1075-1122 in Handbook of Crystal Growth, Vol. IB. D.T.J. Hurle, ed. Amsterdam:
North-Holland.
Glicksman, M.E., E. Winsa, R.C. Hahn, T.A. Lograsso, R. Rubinstein, and M.E. Sellick. 1987. Isothermal dendritic growth. P. 37 in Materials
Processing in the Reduced Gravity Environment of Space. R.H. Doremus and P.C. Nordine, eds. Materials Research Society (MRS)
Symposia Proceedings, Vol. 87. Warrendale, Pa.: Materials Research Society.
Glicksman, M.E., M.B. Koss, L.T. Bushnell, and J.C. LaCombe. 1995a. The isothermal dendritic growth experiment: Implications for theory.
P. 663 in Modeling of Casting, Welding, and Advanced Solidification Processes VII. M. Cross and J. Campbell, eds. Warrendale, Pa.:
Minerals, Metals, and Materials Society.
Glicksman, M.E., M.B. Koss, L.T. Bushnell, J.C. LaCombe, and E.A. Winsa. 1995b. Dendritic growth of succinontrile in terrestrial and
microgravity conditions as a test of theory. ISIJ International 35(6):1216.
Glicksman, M.E., M.B. Koss, L.T. Bushnell, J.C. LaCombe, and E.A. Winsa. 1995c. Dendritic growth in terrestrial and microgravity condi-
tions. P. 13 in Fractal Aspects of Materials: Proceedings of Materials Research Society (MRS) Fall 1994 Symposium, Vol. 367. F.
Family, P. Meakin, B. Sapoval, and R. Wool, eds. Warrendale, Pa.: Materials Research Society.
Herlach, D.M., R.F. Cochrane, I. Egry, H.J.Fecht, and A.L. Greer. 1993. Containerless processing in the study of metallic melts and their
solidification. Int. Mat. Rev. 38:273-347.
Hofmeister, W., M.B. Robinson, and R.J. Bayuzick. 1987. Undercooling of bulk high temperature metals in the 100 meter drop tube. P. 149 in
Materials Processing in the Reduced Gravity Environment of Space: Proceedings of the Materials Research Society (MRS) Symposium,
Vol. 87. R.H. Doremus and P.C. Nordine, eds. Warrendale, Pa.: Materials Research Society.
Hurle, D.T.J. 1995. Crystallization processes. European Low-Gravity Physical Sciences in Retrospect and in Prospect. ELGRA Report. Paris:
European Low Gravity Research Association (ELGRA).
Hurle, D.T.J., G. Muller, and R. Nitsche. 1987. Crystal growth from the melt. P. 313 in Fluid Sciences and Materials Science in Space. H.U.
Walter, ed. New York: Springer-Verlag.
Langer, J.D. 1980. Instabilities and pattern formation in crystal growth. Rev. Mod. Phys. 52: 1-28
Larson, D.J., and R.G. Pirich. 1982. Influence of gravity driven convection on the directional solidification of Bi/MnBi eutectic composites. P.
523 in Materials Processing in the Reduced Gravity Environment of Space: Proceedings of the Materials Research Society (MRS)
Symposium. G.E. Rindone, ed. Warrendale, Pa.: Materials Research Society.
Naumann, R.J., and D.D. Elleman. 1986. Containerless processing technology. P. 294 in Material Science in Space. B. Feuerbacher, H.
Hamacher, and R.J. Naumann, eds. New York: Springer-Verlag.
Sekerka, R.F. 1986. Phase interfaces: Morphological stability. P. 3486 in Encyclopedia of Materials Science and Engineering. M.B. Bever, ed.
New York: Pergamon.
Shong, D.S., J.A. Graves, Y. Ujiie, and J.H. Perepezko. 1987. Containerless processing of undercooled melts. P. 17 in Materials Processing in
the Reduced Gravity Environment of Space: Proceedings of the Materials Research Society (MRS) Symposium, Vol. 87. R.H. Doremus
and P.C. Nordine, eds. Warrendale, Pa.: Matenals Research Society.
Tabeling, P. 1995. Solidification and nucleation. European Low-Gravity Physical Sciences in Retrospect and in Prospect. ELGRA Report.
Pans: European Low Gravity Research Association (ELGRA).
Wang, S.-L., and R.F. Sekerka. 1996. Computation of the dendntic operating state at large supercoolings by the phase field model. Phys. Rev.
E 53(4):3760.
IV.F CHEMICAL TRANSFORMATION
Combustion
Behavior of Combustion Phenomena in Microgravity
Gravitational acceleration influences combustion phenomena because of the large density differences that
appear as a consequence of the large temperature differences that result from the exothermic chemical reactions
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PHENOMENA OF IMPORTANCE IN RED UCED GRA VITY
157
that characterize combustion processes. Density changes of nearly a factor of ten in combustion gases are not
uncommon.
That buoyancy forces are important in many combustion phenomena on Earth is evident through examination
of the Grashof number, the ratio of buoyancy to viscous forces. For Earth gravity and a density ratio of about 10,
this force ratio is not small for physical scales of the size of about 0.1 m or more. Because the Grashof number
increases as the cube of the physical scale, the influence of buoyancy increases rapidly compared to viscous effects
as the scale approaches the usual laboratory scales of experimental investigation. Consequently, "quiescent"
combustion experiments in earthbound laboratories are nearly impossible to conduct unless some element of free
fall is present. Drop towers, for example, are earthbound laboratories in which quiescent experiments can be
conducted, but they are limited to experimental times of approximately 10 s and less, generally between 2 and 5 s.
For forced flows to overwhelm buoyancy and eliminate its influence, i.e., for the Reynolds number to be much
larger than the square of the Grashof number, forced-flow velocities on the order of a meter per second or more are
needed to suppress the influence of buoyancy. Slow-flow combustion phenomena are therefore difficult to
investigate on Earth without interference from buoyancy.
While some combustion phenomena are not influenced by buoyancy, several important ones are: mixture
flammability, instability, gas diffusion flames, droplet combustion, particle cloud combustion, smoldering, and
flame spread (Law, 1990; Sacksteder, 1990; NRC, 1995~. The importance of these phenomena to HEDS and the
associated reduced-gravity environments stems either from their role in fire safety for space travel and the
habitation of distant planets or from their use in processes such as materials production or construction during
space missions. Each of these combustion phenomena is addressed separately below. Because those aspects of the
phenomena that are of HEDS interest are interrelated, the research recommendations are grouped together at the
end of the section rather than listed under each phenomenon.
Mixture Flammability
Whether a premixed mixture of fuel and oxidizer is flammable following a sufficient input of ignition energy
is a question of importance to fire safety. Mixtures exhibit both lean and rich flammability limits on Earth, with
the limits for upward-propagating flames being wider than those for downward-propagating flames. For down-
ward-propagating flames the burnt gases are above the unburnt gas, while for upward-propagating flames the
opposite is true, which leads to a curvature in flame shape as the burnt gases tend to rise under buoyancy into the
unburnt gases to enhance flammability. This curvature produces flame stretch, which influences the flame
temperature and hence flame propagation. Depending on the magnitude and sign of the product of the strain rate
with the transit time through the flame and the Lewis number (the ratio of the thermal diffusivity to the mass
diffusivity of the less abundant reactant), flame stretch can widen or narrow the flammability limits (Law, 1990~.
At reduced gravity and reduced flame stretch, it is possible for the limits to be outside the normal-gravity limits
(Law, 1990; Sacksteder, 1990~.
At one time, because flammability limits were thought to exist only as a result of the influence of gravity, it
was thought that there would be no flammability limits at zero gravity. However, flammability limits at reduced
gravity have been found, and they are hypothesized to exist because of the relatively enhanced influence of
radiative losses at reduced gravity and/or the effects of chemical kinetics; as such, they would be fundamental
limits (Law, 1990~.
Near-limit premixed flames at reduced gravity exhibit unusual behavior not observed at normal gravity. For
Lewis numbers less than unity, spherically expanding flames propagate and then extinguish. The extinction occurs
as a result of enhanced radiative loss and the reduced effects of flame stretch as the flame radius increases. Under
certain circumstances, "stationary" spherical flames, or flame balls, have been observed (Ronney et al., 1998~.
Because these flame balls, which require radiative heat losses for their stability, are "convectionless," they cannot
exist at normal gravity, where there is convective flow through buoyancy (Law, 1990; Sacksteder, 1990~.
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Flame Instabilities
MICROGRAVITY RESEARCH
Flame front instability, which results in flames whose surfaces are not smooth but instead contain cellular
structures, is due to effects associated with heat and mass diffusional processes and hydrodynamic effects that give
rise to the curved shape of upward-propagating flames. Buoyancy is stabilizing for downward propagation, and its
removal makes the remaining effects dominant, thus allowing for the near-limit phenomena observed in reduced
gravity such as the flame balls mentioned above (Law, 1990~.
Gas Diffusion Flames
in gas jet diffusion flames, in which the flame is formed from a jet of gaseous fuel issuing from a burner tube
into an oxidizer, buoyancy is generally important. The flame stabilizes as a premixed flame near the burner rim,
where the flow speed and flame speed match. Because buoyancy affects the flow speed near the burner rim,
removal of buoyancy affects the stabilization mechanism. Additionally, laminar flames are longer and wider in
reduced gravity than in normal gravity and generate more soot. Radiation losses increase, and flame temperatures
decrease (Sacksteder, 1990~.
Droplet Combustion
Diffusion flames surrounding liquid fuel droplets become spherical in the absence of gravity, mirroring the
configuration employed in classical droplet-burning theory in which the square of the droplet diameter decreases
nearly linearly with time and the flame diameter decreases as the droplet diameter decreases. At reduced gravity,
however, unsteady effects are observed in which burning rates and flame diameters initially increase slowly with
time. Additionally, soot production is enhanced, and a soot shell may form at the location where the thermophoretic
transport of soot back toward the droplet surface is balanced by the outward drag on the soot particles from the
outward fuel flow (Law, 1990; Nayagam et al., 1998~. At normal gravity, the soot mantle is swept away from the
lower, windward side of a droplet and consumed in the upper reaches of the flame plume above the droplet.
Cloud Combustion
Arrays or clouds of droplets or combustible particles may exhibit different flame propagation characteristics
at normal and reduced gravity. With settling at normal gravity, upward-propagating flames propagate through an
initially richer mixture while downward-propagating flames propagate through an initially leaner mixture. Clouds
that may not be flammable because of such settling (they are either initially too rich or too lean) may sustain
propagation when uniformly dispersed under reduced gravity (Sacksteder, 1990~.
Smoldering
Smoldering combustion, i.e., the slow surface oxidation of a combustible solid, has practical ramifications for
safety. In normal gravity, buoyancy enhances oxygen transport to and product removal from the reacting surface.
In reduced gravity, this transport mechanism is absent. Results to date show that carbon monoxide production in
smoldering combustion is enhanced substantially in reduced gravity (NRC, 1995; Stocker et al., 1996), but the
prevalence of this effect is unknown.
Flame Spread
Flame spreading over solid and liquid surfaces has direct implications for fire safety and materials selection.
Generally, it is classified as either opposed flow or concurrent flow (with respect to the oxidizer flow). Upward
spread in normal gravity is concurrent and tends to be acceleratory. Opposed flow spread tends to allow steady
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PHENOMENA OF IMPORTANCE IN RED UCED GRA VITY
159
spread to develop. In the absence of a wind, flame spread at reduced gravity is of the opposed flow type, with the
opposing flow with respect to the flame equal to the flame propagation speed (West et al., 1996~.
In opposed-flow flame spread, the spread rate is determined by the upstream transfer to the unburnt fuel of the
heat needed to vaporize it and how this heat transfer is affected by chemical kinetics. Flame extinction occurs at
high velocities as a result of kinetic effects and flame blowoff. Flame extinction has also been found to occur at
reduced gravity in quiescent environments as a result of radiation loss effects that reduce the flame temperature
and propagation speed (this effect is suppressed in normal gravity because of the necessary presence of the induced
flow). The reduced speed makes it difficult for oxygen to be transported to the flame, and extinction occurs
(Altenkirch et al., 1998~.
For thin fuels, the limiting oxygen concentration below which propagation will not occur is higher in reduced
Gravity than in normal Gravity. However a low-speed opposing flow enhances flammability for thin fuels at
.
. . . . . .. . . . . . . . ~ . . . .
reduced gravity, such that the limiting oxygen concentration Is below that for normal-grav~ty downward spread
(Law, 1990; Olson, 1991~. Consequently, there appears to be a minimum oxygen concentration below which
spread does not occur, approximately 15 percent for cellulosic materials (Olson, l991~. The limiting concentration
is higher in normal gravity, which indicates that flammability at reduced gravity may be greater, although flame
spread rates are lower, under certain circumstances, than at normal gravity.
For spread over thick fuels, there appears to be no steady state at reduced gravity. The increased in-depth
conduction needed to raise the temperature of the heated layer in the solid causes the flame to spread more slowly
than for thin fuels, which enhances heat loss by flame radiation. The enhanced radiation causes a reduction in
flame size such that the flame shrinks into a region of continually decreasing oxygen concentration. Eventually,
the decreased oxygen transport results in flame extinction. Apparently, the higher spread rates for thin fuels
prevent this phenomenon, and so thin fuels exhibit steady spread (Altenkirch et al., 1998~.
For liquid fuels, surface tension gradients cause hot liquid fuel to be drawn out from under the flame and
brought in front of it to establish the spread rate. For shallow pools, in which buoyancy would be absent even at
normal gravity, when the flame spreads at a uniform rate, normal and reduced gravity give the same result.
Numerical modeling predicts that pulsation will occur in microgravity with forced convection (Schiller and
Sirignano, 1996), although experimental results seem not to indicate such pulsation (Ross and Miller, 1996, 1998~.
A tentative explanation for this discrepancy is that modeling is two-dimensional while the scale of the experiments
implies three-dimensionality, and expansion normal to the propagating flame is thought to be responsible for
dampening the pulsation.
Implications of the Behavior of Combustion Phenomena
in Microgravity for Spacecraft Design and Operations
Differences in combustion phenomena at normal and reduced gravity have implications for fire safety,
spacecraft materials selection and utilization, especially interior materials, environment selection, interior environ-
ment exchange and ventilation, fire detection and suppression, propulsion, and (potentially) manufacturing/mate-
rials synthesis that relies on the maintenance of exothermic chemical reactions (NASA 1992a,b). Flammability of
materials is an issue of signal importance to spacecraft fire safety. Because spacecraft inhabitants are virtually
captive within the spacecraft, it is imperative that construction materials be selected so that the threat of fire is
minimized, and care should be taken to ensure that the spacecraft breathable environment also minimizes fire risk.
Methods of installation, for example, that preclude electrical overheating, smoldering, and flaming, are necessary
. . . ^. .
to minimize fire ns a.
While the respiratory system responds to the partial pressure of oxygen, fires respond to the concentration of
oxygen. Consequently, judicious selection of breathable environments, whether they be in the spacecraft per se or
within inhabitant's space suits. should maintain suitable partial pressures of oven while minimizing, insofar as
~ 1 ~ ~7
possible, oxygen concentration.
Detection systems designed with the assumption of buoyancy in mind (e.g., smoke detectors) need rethinking.
Technologies different from those usually used on Earth (e.g., radiation sensors) or different applications of
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MICROGRAVITY RESEARCH
existing technologies (e.g., the use of forced ventilation for environment throughput, as opposed to reliance on
buoyancy-driven flow for smoke detectors, as is done on Earth) may be necessary. Fire suppression systems must
not produce end products toxic to humans (e.g., Halon extinguishers), and they must not produce situations that
could pose additional fire safety concerns (e.g., liquid invasion of electrical systems). Fire suppression requires
transport of the suppressant to the fire, and this transport often is affected by reduced gravity. It is, for example,
beneficial to deliver water or carbon dioxide to the base of a fire, a process that can be aided by buoyant inflow at
normal gravity.
While it is unlikely that any combustion process in combustion-based propulsion systems will be affected per
se by reduced gravity, because of the relatively high velocities developed compared to buoyancy-generated
velocities, care should be taken to ensure that the overall system is not adversely affected by reduced gravity
without some compensation for that environment. For example, fluid transport of combustibles is affected by
reduced gravity, and that should be taken into account in propulsion systems design.
There may be a potential for combustion to be used in necessary manufacturing processes. Consequently,
depending on the process, reduced gravity may play a role, e.g., in the transfer of heat from flames for processing
and in direct, high-temperature materials synthesis.
Affected Technologies
As discussed in Chapter III, the technologies affected in the presence of reduced gravity that relate to
combustion include fire detection and suppression technologies, as mentioned above; electrical/electronic packag-
ing to minimize the potential for overheating and smoldering combustion; ventilation control in the presence of a
fire; and fluid distribution systems associated with propulsion. In addition, there is the potential, though as yet
undetermined, influence on technology for materials synthesis during exothermic reaction.
Research Issues
The main issue surrounding combustion as it relates to HEDS activities is safety, which has implications for
materials selection, environment selection, fire detection, fire management, and fire suppression. Further work on
materials and environment selection is needed, but the emphasis should be on robust fire detection and suppression
in reduced gravity.
As discussed previously, a number of areas of reduced-gravity combustion research are particularly relevant
to improving fire detection and suppression in a reduced gravity environment, including flammability and flame
behavior (such as flame instabilities and dynamics under different gravity conditions); diffusion-flame structure
and behavior for gaseous, liquid, and solid fuels, especially the production of soot and toxic products in such
flames and conditions necessary for their extinction; smoldering rates at reduced gravity, with special emphasis on
conditions for initiation and termination of smoldering and on products of smoldering combustion; and, finally,
flame spread phenomena for various types of fuels. Besides safety-related issues such as these, there is a need,
albeit a less pressing one, for reduced-gravity research on materials synthesis and materials processing through
combustion.
Without practical, agreed-upon means for detection and suppression, fire looms as a potential cause of
mission failure. Fluid distribution in propulsion systems and the potential for materials synthesis are combustion
areas in which implications for successful HEDS activity may reside, and which also deserve attention, though
certainly not to the same extent as fire safety.
Pyrolysis
As described in Chapter III, pyrolysis is the mechanism by which chemical transformations are brought about
by application of heat. Only a small number of the many pyrolysis processes that could be of interest for HEDS
are described in Chapter III. The large temperature changes in practical pyrolysis imply appreciable gravity
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PHENOMENA OF IMPORTANCE IN RED UCED GRA VITY
161
effects by virtue of the associated changes in density. It may therefore be expected that reduced-gravity issues
arise in pyrolysis. The effects would involve influences of gravity on the transport of reactants and products to and
from regions of pyrolysis, and so pyrolysis processes that may be of use, such as oxygen-production processes,
will need study to determine these effects.
Many pyrolysis processes that occur during combustion are discussed in the preceding section. For example,
soot production in diffusion flames involves pyrolysis of the gaseous fuels in fuel-rich regions of the flow. These
pyrolysis processes are of importance in fire protection aboard spacecraft, for example.
Research Issues
There are many different kinds of pyrolysis processes, with reactants and products in different phases and with
very different temperature thresholds and time scales. For this reason, microgravity studies of pyrolysis would
need to be pursued on individual bases, separate for each process. One potentially relevant process is the high-
temperature recovery of oxygen from silica in lunar regolith. Another is soot production from hydrocarbon fuels.
A third is gaseous fuel production from cellulosic fuels. Each process proceeds at successively lower tempera-
tures. Many other pyrolysis processes with potential relevance to HEDS can be identified, but each would need to
be evaluated separately with respect to its HEDS relevance and its sensitivity to microgravity. However, it can be
noted that general categories of processes affected by gravity include gas production from solids and gas-phase
chemical transformations in a flow field.
Solution Chemistry
While the interaction of individual molecules in a solution is not expected to be directly affected by gravity
levels (except possibly in the case of very large molecules such as proteins), there are numerous ways in which the
effects of gravity on bulk fluid flow might either inhibit or enhance the efficiency of the chemical reactions carried
out on a HEDS mission. Incomplete mixing of reactants is possibly the chief area of concern for solution reactions
in reduced gravity. On Earth, density-driven convection, particularly in reactions requiring heating, is relied upon
as the default method of mixing reactant solutions. While this driving force would be absent or reduced in low
gravity, complete mixing could still be accomplished in most cases by the use of mechanical stirrers and by paying
careful attention to the design of reaction chambers.
On the other hand, for a multiphase mixture of immiscible phases with different densities, the surface area at
which a reaction could take place might be greatly enhanced in microgravity. After mixing, density differences
would rapidly separate such a solution into layers on Earth, whereas in reduced gravity the dispersed droplets, with
their greater surface area, could remain suspended for a longer period of time, allowing reactants in the two phases
to interact at a higher rate. Surface tension would still be present, however, as a driving force for coalescence of
the phases and thus a reduction in the surface area of reaction.
A better understanding of phase distribution and separation in low gravity is cited as a need in a number of
sections in this report and so is not discussed in detail here, except to note that such an understanding might also
be applied to enhancing the efficiencies of some types of chemical reactions, such as those involving immiscible
phases, in low gravity.
References
Altenkirch, R.A., L. Tang, K. Sacksteder, S. Bhattacharjee, and M.A. Delichatsios. 1998. Inherently unsteady flame spread to extinction over
thick fuels in microgravity. Pp. 2515-2524 in Twenty-Seventh Symposium (International) on Combustion. Pittsburgh: Combustion
Institute.
Law, C.K. 1990. Combustion in microgravity: Opportunities, challenges, and progress. AIAA-90-0120. New York: American Institute of
Aeronautics and Astronautics.
National Aeronautics and Space Administration (NASA). 1992a. 1991 Integrated Technology Plan for the Civil Space Program. NASA TM-
107988. Washington, D.C.: NASA.
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162
MICROGRAVITY RESEARCH
National Aeronautics and Space Administration (NASA). 1992b. Review of NASA's Integrated Technology Plan for the Civil Space Pro-
gram. NASA TM-107966. Washington, D.C.: NASA.
National Research Council (NRC), Space Studies Board. 1995. Microgravity Research Opportunities for the 1990s. Washington, D.C.:
National Academy Press.
Nayagam, V., J.B. Haggard, Jr., R.O. Colantonio, A.J. Marchese, F.L. Dryer, B.L. rehang, and F.A. Williams. 1998. Microgravity n-heptane
droplet combustion in oxygen-helium mixtures at atmospheric pressure. AIAA J. 36(8):1369-1534.
Olson, S.L. 1991. Mechanisms of Microgravity flame spread over a thin solid fuel: Oxygen and opposed flow effects. Combust. Sci. Technol.
76(4-6):233-249.
Ronney, P.D., M.-S. Wu, H.G. Pearl mar, and K.J. Weilar~d. 1998. Expenmental study of flame balls in space: Preliminary results from STS-
83. AIAA J. 36(8):1361-1368.
Ross, H.D., and F.J. Miller. 1996. Detailed experiments of flame spread across deep butanol pools. Pp. 1327-1334 in Proceedings of the
Twenty-Sixth Symposium (International) on Combustion. Pitttsburgh: Combustion Institute.
Ross, H.D., and F.J. Miller. 1998. Flame spread across liquid pools with very low-speed opposed or concurrent airflow. Pp. 2723-2729 in
Proceedings of the Twenty-Seventh Symposium (International) on Combustion. Pitttsburgh: Combustion Institute.
Sacksteder, K.R. 1990. The implications of experimentally controlled gravitational accelerations for combustion science. Pp. 1589-1596 in the
Proceedings of the Twenty-Third Symposium (International) on Combustion. Pittsburgh: Combustion Institute.
Schiller, D.N, and W.A. Singnano. 1996. Opposed-flow flame spread across n-propar~ol pools. Pp. 1319-1325 in Proceedings of the Twenty-
Sixth Symposium (International) on Combustion. Pittsburgh: Combustion Institute.
c7
Stocker, D.P., S.L. Olson, D.L. Urban, J.L. Torero, D.C. Walther, arid A.C. Fernar~dez-Pello. 1996. Small-scale smoldering combustion
experiments in Microgravity Pp. 1361-1368 in Proceedings of the Twenty-Sixth Symposium (International) on Combustion. Pittsburgh:
Combustion Institute.
West, J., L. Tang, R.A. Altenkirch, S. Bhattacharjee, K. Sacksteder, and M.A. Delichatsios. 1996. Quiescent flame spread over thick fuels in
Microgravity Proceedings of the Twenty-Sixth Symposium (International) on Combustion. Pittsburgh: Combustion Institute.
IV.G BEHAVIOR OF GRANULAR MATERIALS
Lunar and Martian Regolith
Characterizing and understanding the behavior of granular materials in different gravitational fields are of
critical importance to the HEDS program. In order to carry out tasks ranging from determination of the energy
requirements for excavation operations and estimation of the load-carrying capacity of extraterrestrial surfaces,
through the handling of granular materials in conveying systems and the prediction of local surface properties of
planetary bodies under various gravitational conditions, the influence of gravity on the behavior of granular
materials must be better understood. Jaeger, Nagel, and Behringer (1996), in their useful overview of many
aspects of granular material behavior, discuss the conditions under which these material systems behave like
solids, liquids, or gases, depending on environmental conditions. Granular materials of interest in this discussion
are soils, regoliths, and other similar resources that can exhibit cohesion and are arrangements of rigid particles in
frictional contact. Because gravity contributes to the normal stress force of interaction between particle surfaces
and frictional forces are typically proportional to the normal (hence gravitational ~ forces, the elastic behavior of
granular soils, which has both axial and radial force components, is strongly dependent on gravity. In fact, when
granular particle assemblies are produced in terrestrial environments with particle density levels that are sufficient
to maintain continuous contact between adjacent particles, gravity creates internal stress distributions that are
highly nonhomogeneous and anisotropic. Many experiments have shown that the internal stress distributions
produced even in highly simplified (identical particle) granular assemblies are distinctly different when test
conditions are repeated using the same apparatus. Great care thus will be required to isolate gravitational effects
from effects produced by random assembly variations. It is also extremely difficult to conduct terrestrial experi-
ments on the behavior of granular systems (other than angle-of-repose experiments) that are not controlled by
container boundaries.
The ability to construct stable structures and roadways that can carry loaded vehicles on planetary surfaces
(geotechnical engineering) is a critical element in designing equipment for testing, processing, and transporting
these soils. The soil properties depend strongly on the shape of the individual particles, which can vary from very
angular to well rounded. Specifically, it is known that owing to the thermal and mechanical bombardment events
undergone by the Moon, its surface materials include nearly spherical glass particles and very irregular fine
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PHENOMENA OF IMPORTANCE IN RED UCED GRA VITY
163
particles. In contrast, because of the apparent hydrologic epoch that characterized some portion of Mars's
geologic history and because Mars continues to possess an atmosphere that sustains aeolian erosion, it is likely that
Mars regolith is composed of less-abrasive granular materials and therefore has a mechanical behavior different
from that of the lunar regolith. It is already known that the lunar surface is characterized by the extreme density
of its soil only a few centimeters beneath its surface (Carrier and Mitchell, 1990), compaction that exceeds that
produced by even heavy compaction equipment on Earth. Hence, even though it will be extremely difficult to
excavate material or even drill holes in the lunar surface without employing such techniques as vibratory excava-
tion (Klosky, 1997), undisturbed lunar soil should be able to carry heavy loads.
The behavior of dust in reduced gravity is also an important consideration. While the composition, particle
shape, and atmospheric loading of Mars dust is neither fully known nor understood at this time, their potential
impact on future HEDS missions must be considered. Lunar dust, whose characteristics were briefly described
above, presents a different set of problems. It is considered to be one of the most difficult design constraints for
lunar base construction since the lunar fines are abrasive minerals that, in the desiccated lunar environment, are
electrostatically sticky and adhere very tenaciously to most surfaces. Furthermore, there are still questions about
what types of attractive forces cause the dust to cling to surfaces and therefore about the cleaning methods that can
be used to manage lunar dust (Perko, 1998~.
The soil's void ratio (the ratio of void to solids), characteristic particle dimensions, and relative density (the
ratio of soil density to that of solid soil without voids) all bear directly on the soil strength. The presence of
smooth inclusions even at a level of a few percent can cause a significant decrease in soil strength (Klosky, 1997~.
Granular materials under self-weight are used terrestrially to construct structures such as dams, road embank-
ments, mine waste dumps, and ore stockpiles. The edges of such piles cannot be steeper than the angle of repose.
If more material is added and the maximum stability angle is exceeded, avalanches occur. Well-established
failure criteria have been discussed in the literature, but the dominant design criteria for frictional, granular
material continue to be Mohr-Coulomb (M-C) failure measurements (Wood, 1990; Craig, 1992~. These measure-
ments of granular shearing usually focus on the mean properties of the system and use techniques such as
conventional biaxial compression (Klosky, 1997~. Shear stress is applied to a sample under normal load, and the
grains respond elastically up to the yield point, where shear displacement occurs. When this failure occurs, the
measured shear force decreases. Simultaneous measurement of shear stress and normal stress define M-C failure
envelopes. A linear connection through the maxima of these plots has a slope that can be interpreted as the
friction angle. The intercept where the normal stress is zero gives c, the apparent cohesion (the shear stress
necessary to overcome the cohesion), which is the force that enables the soil to cling together in opposition to the
forces tending to separate it into parts. Cohesion implies a surface-surface particle interaction such that as
cohesion decreases, particles move more readily and pack more densely under a given set of stress conditions.
The interaction forces between grain particles include hard-body interactions, friction, and inelasticity, as char-
acterized by a coefficient of restitution less than unity. The dissipative nature of these interactions causes even
an energetic (possessing random motion) collection of grains to coalesce into a dense, compact state. Such a
compact state is usually very inhomogeneous with regard to the forces acting on individual grains, and large
fluctuations in the local forces have been observed (Behringer et al., 1999), with recent studies beginning to
provide new insight into their characteristics.
Research Issues
The continued measurement and characterization of lunar and Martian regolith would have a high priority in
any attempt to establish the groundwork needed for the exploration activities that would support the development
of extraterrestrial base stations. While important characteristics of the mean properties of the regolith are mea-
sured by such techniques as Mohr-Coulomb failure criteria, other aspects are much less well understood: the
kinetics of Coulomb friction, the internal variables and energy fluctuations, and the effects of agitation on particle
size separation.
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MICROGRAVITY RESEARCH
Kinetics of Granular Flow
The spontaneous flow of granular materials out of the base of a silo or hopper requires the presence of a
driving force such as gravity. At reduced gravity, such as lunar or Martian gravity, the flow will likely change, but
in ways that are not now obvious; for instance, unlike a true fluid, the hydrostatic pressure is apparently insensitive
to the depth of the granular material in containers. The contact forces between the grains, and the static friction
with the sides of the container, allow the sand in the hourglass to flow through the orifice at a nearly constant rate.
Thus, when granular material is held in a silo, no height-dependent static pressure head occurs, as it would with a
liquid. The pressure reaches a maximum value independent of height, and these flows will require further
investigation of gravitational effects (Jaeger et al., 1996~. Because many factors contribute to the soil properties,
the analysis of these flows is not trivial. Gravity could be replaced by using another driving force such as electrical
or magnetic. The former would require imparting a charge to the particles and then subjecting them to an electric
field. If the particles were ferromagnetic, such as the agglutinates in an Apollo 11 sample, then a magnet could
drive the flow (Agosto, 1985)
While the Mohr-Coulomb failure criteria for granular material has permitted the measurement of important
mean properties of granular material subjected to shear, the measurements are in a three-dimensional system, so
that the normal stress is not separated from the shear. It would be useful to complement this information with
direct measurements of the forces and displacements of individual grains in a shearing experiment. Studies of
three-dimensional systems show that in systems with 10 to 100 grains, fluctuations in stress are enormous, often
more than an order of magnitude greater than the mean stress; this factor is usually ignored in modeling, impairing
the ability to predict the state of such granular materials (Miller et al., 1996~. Stresses developed in static piles of
cohesionless granular material have received considerable attention (Savage, 1997~. Dense, slow flows and rapid,
gaslike flows (Schafer et al., 1996) are useful idealizations for the development of models (Jaeger et al., 1996), and
real systems often display both flows simultaneously in different spatial domains. As discussed above, particles
coalesce into a dense, packed state, so that to maintain granular material flow at low density, energy must be
continuously supplied, for instance, by shaking. To achieve a flow of dense material, enough shear stress must be
applied to exceed some yield point, where grains begin to slide past each other. Applied stress (including gravity)
is therefore an important parameter, but other important effects include convection, size separation, and mixing.
Using a two-dimensional system, where the effects of gravity are effectively removed, some recent experi-
ments have shown that the packing density of grains subjected to shear undergo a novel kind of phase transition at
a critical packing density (Behringer et al., 1999; Veje et al., l999~. In these experiments the grains are simulated
by the use of photoelastic disks that are birefringent under stress/strain so that cross polarizers show light/dark
regions. The axially emitted light shows bright and dark bands in the polariscope, and from this observation, it is
possible to determine the applied shear forces. The disks, which are on a smooth slippery sheet, are confined at
their inner and outer radii by roughened surfaces that apply shear stresses when rotated. The width of the
characteristic shear band that forms near the inner wheel depends slightly on the packing fraction and is about six
disks in radial direction. Virtually all azimuthal motion of the disks occurs in this band, and the remaining outer
disks remain nearly frozen. The disks in the shear band dilate, compacting the disks in the outer region of the
experiment. Rearrangements of this type can also occur at dense packings and influence the statistical and mean
properties of the flow over relatively long times. In studies of shear from the inner radius, it is observed that there
is a critical packing fraction of the grains, Ec, of 0.77, at which point the system undergoes a change from complete
slipping, where the disks can remain indefinitely at rest without shear, to a state of nonslipping dynamics, with
closely packed grains subject to some shear stress at all times. This transition has some resemblance to a phase
transition. Just above Fc, fluctuations are temporally intermittent, and the resulting stress chains tend to be long.
As F is increased further, the system becomes more homogeneous, because strong contacts now deform enough to
allow other contacts to take up the load. The strong dependence of the dynamic behavior on grain packing density
gives new insight into the properties of granular materials.
When granular materials are agitated in Earth gravity, size segregation of grains can occur by several mecha-
nisms, causing preferential filling of the space beneath large particles by smaller particles (Jenkins and Louge,
1997~. Gravity imparts buoyant forces that separate grains of different sizes. These forces compete with gradients
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PHENOMENA OF IMPORTANCE IN RED UCED GRA VITY
165
in concentration and energy and can result in convection cells in which particles with different properties separate
(Knight et al., 1993~. In the absence of gravity, only the simpler balance between gradients in concentration and
in fluctuation energy remains; important studies in this area are in progress (Jenkins and Louge, 1997~. It would
appear that vibrating containers can be designed for operation in microgravity that will be capable of separating
particles by their size (Rosato et al., 1987; Williams, 1976~. An area receiving considerable attention in the
physics community is the stirring of granular materials by means of paddles to provide mixing without segrega-
tion; the design principles have been discussed by Khakhar et al. (1997) and Ottino (1990~. While these systems
would likely be used in pressurized environments other than the vacuum of space, it is important to note that the
process for separating particles by size has not been validated in environments that are at pressures below 10 torr
(Pak et al., 1995~.
Research Issues
Modeling and predicting the behaviour of granular materials is an important activity that has been the subject
of renewed interest. The classical descriptions of dense granular material use static arrays of grains with typically
undetermined Coulomb friction forces. Recent studies, described above, show that local forces have large fluctua-
tions and are very sensitive to small perturbations in packing density, and such effects must be included in
statistical descriptions and models of granular materials. The modeling of granular materials under applied stress
is important to a number of HEDS activities, such as construction, surface transport, and materials processing at
low pressure. Studies that separate or counter the effects of gravity while examining the effects of shearing on
granular behavior in three dimensions would help in understanding the phase transition observed at a critical
packing density of granular material.
References
Agosto, W.N. 1985. Electrostatic concentration of lunar soil minerals. Pp. 453-464 in Lunar Bases and Space Activities of the 21st Century.
W.W. Mendell, ed. Houston: Lunar and Planetary Institute.
Behringer, R.P., D. Howell, L. Kondic, S. Tennakoon, and C. Veje. 1999. Predictability and granular materials. Physica D: Nonlinear
Phenomena 130(1-2): 1-17.
Carrier, W.D., III, and J.K. Mitchell. 1990. Geotechnical engineering on the moon. Pp. 51-58 in de Mello Volume: A Tribute to Prof. Dr.
Victor F.B. de Mello. E. Blucher, ed. Sao Paulo, Brazil: Editora Edgard Blucher.
Craig, R.F. 1992. Soil Mechanics. London: Chapman and Hall.
Jaeger, H.M., S.R. Nagel, and R.P. Behringer. 1996. RMP colloquium: Granular solids, liquids, and gases. Rev. Mod. Phys. 68:1259.
Jenkins, J.T., and M.Y. Louge. 1997. Pp. 539-542 in Powder and Grains 97: Proceedings of the 3rd International Conference. R.P. Behringer
and J.T. Jenkins, eds. Brookfield, Vt.: A.A. Balkema.
Khakhar, D.V., J.J. McCarthy, and J.M. Ottino. 1997. Radial segregation of granular mixtures in rotating cylinders. Phys. Fluids 9:3600-3614.
Klosky, J.L. 1997. Behaviour of composite granular materials and vibratory helical anchors. Ph.D. dissertation. University of Colorado.
Knight, J.B., H.M. Jaeger, and S.R. Nagel. 1993. Phys. Rev. Lett. 70:3728.
Miller, B., C. O'Hern, and R.P. Behringer. 1996. Stress fluctuations and continuously sheared granular materials. Phys. Rev. Lett. 77:3110.
Ottino, J.M. 1990. The Kinematics of Mixing: Stretching, Chaos and Transport. Cambridge: Cambridge University Press.
Pak. H.K.. E. van Doom. and R.P. Behrinaer. 1995. PhYs. Rev. Lett. 74:4643.
. . . ~ ~
Perko, H.A. 1998. Surface cleanliness-based dust a&esion model. Pp. 495-505 in Space 98: Proceedings of the Sixth International Confer-
ence on Engineering, Construction, and Operations in Space. R.G. Galloway and S. Lokaj, eds. Reston, Va.: American Society of Civil
Engineers.
Rosato, A., K.J. Shandburg, F. Prinz, and R.H. Swendson. 1987. Why brazil nuts are on top: Size segregation of particulate matter by shaking.
Phys. Rev. Lett. 58:1038-1040.
Savage, S.B. 1997. Pp. 185-194 in Powder and Grains 97: Proceedings of the 3rd International Conference. R.P. Behringer and J.T. Jenkins,
eds. Brookfield, Vt.: A.A. Balkema.
Schafer, J.J., J.S. Dippel, and D.E. Wolf. 1996. Force schemes in simulations of granular materials. J. Physique 1(6):1751-1776.
Veje, C.T., W. Daniel, W. Howell, and R.P. Behringer. 1999. Kinematics of a two-dimensional granular Couette experiment at the transition
to shearing. Phys. Rev. E. 59(1):739-745.
Williams, J.C. 1976. The segregation of particulate materials: A review. Powder Tech. 15:245-254.
Wood, D.M. 1990. Soil Behavior and Critical State Soil Mechanics. Cambridge: Cambridge University Press.
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Representative terms from entire chapter:
reduced gravity
