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! 4 Systems Far from Equilibrium SO[IDII?ICATION PATTERNS ~ dendritic crystal growth, extraneous gravitational effects obscure the underlying physics of a pattern-forrn~ng process. The term "dendrite" refers here to tree-like patterns, which are most familiar in the form of snowflakes, and which are of special tech- nological importance because of the role they play in determining metallurgical microstructures. Conceptually, dendritic solidification of a pure substance is one of the simplest examples of pattern formation in nature. Sim- plicity, in this case, means that we are fairly sure what the ele- mentary physical mode} of the situation must be. We consider a solid forrn~ng within an undercooled liquid, for example, an ice crystal growing inside a sample of pure water cooled to below its freezing point. Growth under these circumstances ~ controlled al- most entirely by the rate at which the latent heat generated at the solidification front can diffuse away through the liquid. A state- ment of thermodynarn~c boundary conditions at the liquid-solid interface completes the specification of the model. We have good reason to believe that, were we able to solve the mathematical equations implied by the above sentences, we would predict the 77

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78 generation of a rich variety of snowflake-like shapes quite similar to those observed in nature. In fact, this solidification problem turns out to be anything but simple. Along with certain roughly analogous problems in hy- drodynamics (Rayleigh-Benard convection; viscous fingering) and in chemistry (reaction-diffusion problems), it seems to epitomize the fundamental question of how complex structures are generated when initially structureless systems are driven strongly away from equilibrium. Even in the superficially simplest of these models, we are only beginning to be able to predict some special properties of emerging patterns. In general, we are still unable even to deter- mine whether an emerging structure will be regular or intrinsically chaotic. Here, as in any developing scientific field of investigation, what is needed is an interplay between theoretical work and care- fully controlled experimentation. The most useful experiments are always the simplest, that is, the ones in which the phenomena of fundamental interest are least obscured by extraneous effects. The problem of dendritic solidification is an especially clear example of a situation where gravity is the leading culprit in producing extraneous effects. A basic feature of dendritic solidification as it occurs under the conditions described above is that at the speed, v, at which the leading tip of the tree-like structure grows, the Julius of curvature, p, of this tip, and the spacing, A, of the first side branches that appear behind it are all well-defined, reproducible functions of only the undercooling of the fluid. Thus, the classic experunents of Glicksman and his colleagues have been aimed at measuring v, p, and A. It is important for a variety of reasons that these observations be made at small undercoolings, that is, at low ther- modynamic driving force. In this regime, v ~ small and there is little chance that poorly understood departures from thermody- namic equilibrium at the interface can play much of a role. Also, in this regime of slow growth p and ~ turn out to be large, and thus one can achieve considerable accuracy in measuring these geometric quantities by photographic means. At this point, the gravitational effects enter. The latent heat released by the tip of the dendrite warms the fluid and causes it to rise. If the dendrite is growing upward, this gravitationally induced convection opposes diffusion of heat away from the tip, and growth is retarded. Conversely, downward-growing dendrites are accelerated. Unfortunately, the effect is most serious at small

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79 undercoolinge where length scales are large and viscous forces are relatively ineffective. As a result, accurate velocity measurements can be made only when undercooling is high enough. However, sharp pictures can be taken only at low undercooling. A fully satisfactory comparison of experiment with theory requires s~rnul- taneous measurements of velocity and tip radius, but there is only an inconveniently small range of intermediate undercoolings in which this can be done. Obviously, some method for appreciable reduction of this gravitational effect would be most useful. A ~rfl- crogravity experiment for the Space Shuttle L8 now in the planning stage. SURFACE TENSION AND CONVECTION EFFECTS Surface-tension-driven phenomena arise whenever a free sur- face interacts with a field (temperature, magnetic, electric fields) that influences surface tension. A fluid heated from below involves a stability problem, with a threshold temperature difference ~ T required for convection. A fluid heated at the sidewalls, on the other hand, yields convection no matter how small AT. Defining a as the thermal expansion coefficient, ~ the kinematic viscosity, arc the thermal diffusivity, p the density, ~ the surface tension, ~ and u' the depth and width of a fluid layer, and ~ the viscosity, Davis and Homsy have shown that in the former case the problem rs governed by no less than seven dimensionless groups. These are: the Rayleigh number Ra = ga^Td3/~`c, the Marangoni number Ma = (-~/3T)ATd/pu~c, the Capillary number Ca = (,~/c/~), the contact angle 7, the Prandt} number P z'/,c, the Bond number B ~ Apgd2/a, the aspect ratio A = d/w, and thermal boundary conditions at all sur- faces. For other fields such as electromagnetism, suniliar groups arise. Gravity enters the problem through Ra (buoyancy-driven effects within the fluid) and B (density differences, Ap at the inter- face). Ca is a measure of interfacial deflection due to convection. Different aspects of the phenomena associated tenth free surfaces involve gravitational acceleration in varied ways. Minimizing Buoyancy-dri~ren Effects In order to do an experiment that is essentially dominated by surface-tension-driven convection far into the nonlinear regime

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80 at high Marangoni numbers, we must have Ma >> Ra, or `2 << (-Ba/3T)/~3p/T)g. Using typical values of -~Ba/3 T) = 7 x 102 dynes/cm K, -~3p/3T) = 1 x lO-3 g/cm3 K, 9 ~ 103 cm/s2, << 3 mm. It ~ well known that for most fluids, we must go to submfllimeter length scales before minunizing (but not completely eliminating) buoyancy effects. Such experunents in thin layers have been conducted by several groups, but with some degree of difficulty. Furthermore, most measurements at such length scales are limited to global properties (average surface temperature, aver- age heat flux, etc.~. Field information (e.g., velocimetry or thermal profiling) cannot be obtained when working at such small length scales. A spacecraft experiment is currently under development in an attempt to measure such field information in domains whose characteristic length is of the order of centimeters rather than mil- limeters. Microgravity studies of surface-tension-driven flows thus present some opportunities, but also present some complications regarding the effect of "jitter on interfacial stability and integrity. Mining Free Surface Deflections Free surface deflections as a result of fluid motion make the convection problem a free-boundary one. Challenging numerical difficulties arise if such fre~boundary problems are treated in generality. However, for most fluids, Ca = (~'c/da) ~ 10-4/d, where ~ ~ in cgs units. Thus Ca is small for any reasonable length, it, and steady flows may be computed for comparison with experiment by domain perturbation. However, instability modes may be intimately linked with free-surface deformability and thre - dimensional flows. As a result, experiments involving finite-amplitude traveling waves are the only way of probing non- I~near instabilities; computations of such three-dimensional, time- dependent, free-boundary problems at high Marangoni numbers are simply too difficult. MINIMIZING THE EFFECT OF GRAVITY ON FREE SURFACE SHAPES In order to minimize the effect of gravity, the Bond number, B. must be much less than unity. For most fluids in 1 g, B ~ (1~103) (d2~/50 ~ 2od2

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81 Again B << 1 implies d2 << 0.05 cm2, or d << 2 mm, i.e., length scales of the order of millimeters. Here microgravity experiments offer one means of suppressing static deformation of free surface shapes. E[ECTllOlLINETICS The subject known as electrokinetics lies at an intersection between fluid mechanics and colloid chemistry since charged in- terfaces with diffuse layers of space charge produce phenomena different from those encountered in traditional fluid mechanics. Motion can arise in response to an unposed electric field, as in elec- trophoresm and electroosmosm, or from purely mechanical forces, as in the electroviscous effects. External fields are small a few volts per centimeter when the solution ~ a good conductor. How- ever, when poor conductors are involved, the external field can be several thousand volts per centimeter. Internal fields due to charge located on interfaces can be very large. ions are transported by diffusion and electromigration. Electric fields influence fluid motion in several ways. Multi- phase systems can be manipulated by electric forces acting directly on fluid interfaces. Separation processes such as electrophoresis, isotachophoresis, and isoelectric focusing depend on the action of an externally applied field. Mechanical effects due to intrinsic sur- face charge are always present in colloidal systems. Studies have been carried out for more than a century, and certain aspects of the subject are well understood. However, even though the subject is mature, some fundamental issues remain outstanding. The reasons microgravity experiments may be useful stem from two factors. First, Joule heating due to the passage of cur- rent produces density gradients. These lead to buoyancy-driven convection. Simple scaling arguments show how such motions can be overwheirn~ng. The temperature rise itself scales as the second power of the characteristic length, while buoyancy-driven motion scales as the first power of the temperature rise and the sec- ond power of the length. Accordingly, the strength of convective motions is directly proportional to the gravitational acceleration and to the fourth power of the characteristic length. Using high- viscosity fluids to reduce motion usually compromises attempts to study interesting situations. Thus, until recently, convection could be suppressed only by decreasing the length scale, and most

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82 . . work was done on a microscopic scale. In other situations, den- sity contrasts are intrinsic. With two-phase systems, for example, sedimentation is always present unless isopycnic systems are used. Unfortunately, their use thwarts study of large classes of ~nterest- ing materials. Here again, we are forced to work on small length scales. From these simple arguments we see that buoyancy-driven motion makes it difficult to observe the electrokinetic phenom- ena in fundamental studies, and certainly influences benefits from practical applications. Microgravity experiments enable one to avoid, or at least suppress, effects of nonelectrokinetic motions. Three sorts of problems can be cited as examples of areas where fundamental studies are in order. The first concerns the rheology of two-phase systems where imposition of an external electric field alters the constitutive behavior of the suspension. A second area involves electrically driven flows In nonhomogeneous buffers. In what is known as Isoelectric focusing, for example, the passage of a current itself alters the buffer compositions so as to cause large concentration gradients. Here interest centers on the mechanics of the flow known as electroosmosis. Micrograv- ity experiments are almost essential if modern flow visualization schemes are to be used. The final example concerns the behavior of fluid-fluid interfaces, where experiments with electrically stressed drops and bubbles provide a means to understand the mechanics of the interface. One of the goals of research in electrokinetics ~ to formulate and test theories for the behavior of electrically stressed systems. Consider, for reference, one of the classical models from contin- uum physics used to describe the dynatnics of ordinary fluids such as air or water. The description of homogeneous fluids of this sort involves conservation laws for mass and momentum, consti- tutive relations connecting the internal stress to the strain rate, and specifications as to how boundaries interact with the fluid. For motions involving simple interfaces, nonlinear effects arise in two ways: through spatial accelerationsthe so-called inertial terms in the equations of motion and through the geometry of deformable boundaries. It is generally accepted that this mode} serves to explain a variety of situations. By solving carefully posed boundary value problems, processes ranging from the Brownian motion of small particles to the flow around certain airfoils can be described. Apparently, this mode! is capable of representing

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83 turbulent flows, although such situations are still mathematically intractable. Thus, although the mathematical difficulties may be formidable, the theoretical foundation ~ wed established. This is not the case with flows involving ionic solutions where an adequate mode} for the electromechanical behavior of interfaces is lacking. With fluid interfaces, charge may be transported across the interface by diffusion and migration of ions in an electric field. At the same time, the interface may dilate and contract due to motion of the bulk fluid. Any study of the interface is compli- cated, perforce, by the behavior of the adjacent fluids. Little is known about the nonequilibrium properties of charged interfaces. It ~ clear that the correct constitutive model will be nonlinear, inasmuch as the Maxwell stress tensor for the electrical force nonlinear; but there ~ considerable uncertainty about the details. Experimental results are needed to guide the theoreticians and test alternative theories. On the other hand, much has been learned about the equip rium behavior of interfaces during the past two decades through direct measurement of the attractive (dispersion) forces and the re- puIsive (electrostatic) forces that make up the Derjaguin-Landau- Verwey-Overbeek theory. Theory and experiment are in close agreement. The next stage will include ex~nination of the behav- ior of nonequilibrium systexns, stressed by mechanical and electri- cal forces. Here we are on much less well-prepared ground, since theories of nonequilibrium interfaces are in the embryonic stage. It is possible that the m~crogravity environment can provide the venue for experiments that will enable us to understand the elec- trokinetic properties of interfaces to a degree comparable to that with which we comprehend their electrostatic behavior. COMBUSTIBLE MEDIA The central scientific issues in combustion are: flame ini- tiation, the characteristic temperature-composition structures of flames, flame morphology, propagation speeds, and flame exis- tence limits. The coupling of complex chemical kinetics with fluid mechanics is a general characteristic of flames. At normal grav- ity, buoyancy forces are observed to introduce flame asymmetries and more convoluted structures than is thought to be character- istic of the underlying transport phenomena. Take, for example, the combustion characteristics of uniform, quiescent clouds of fuel

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84 drops or fuel particulates. Gravitationally induced sedimenta- tion processes make it difficult or impossible to experimentally establish such two-phase systems at i g. For both single- and two- phase flame structures, characteristic buoyancy-~nduced flow rates of flame gases are comparable to exper~rnentally observed flame propagation rates in the neighborhood of flame extinction Innits. This results in strikingly different limiting fuel concentrations for upward and downward flame propagation in normal gravity. The essential reasons for flame extinction limits are not adequately understood, either for premixed, gaseous flames or for premixed two-phase flame structures. Another example is the experimental determination of flame propagation rates and flame extinction conditions for a premixed, quiescent cloud of fuel particulates in a gaseous oxidizer. In a terrestrial experiment, uniform clouds of particulates are created through use of suitable mixing processes. Quiescence is achieved only after the decay of mix~ng-induced turbulence and secondary flow. Downward flame propagation cannot occur where particle settling speeds exceed flame propagation rates. At ~ g, settling speeds of large particles are comparable with flame propagation rates, especially near flammability limits (the lower limit of fuel concentration for quasi-steady flame propagation). For example, unit density 5~pm particles settle at about 7 cm/s. Theories that suggest lower flammability limit flame speeds cannot be tested for the larger particle sizes. A rrucrogravity environment would permit the establishment of a quiescent uniform cloud of premixed particulates for study. Another msue in combustion concerns the spherically sym- metric combustion of a fuel droplet ~ an oxidizing atmosphere. The theory aims at predicting ignition, ga~phase unsteadiness, liquid-phase unsteadiness, quasi-steady burning rates, disruptive burning, and extinction processes for the spherically symmetric case. Drop towers have been employed in preliminary studies anned at establishing and burning nearly spherically symmetric drops. However, experimental times needed to observe the entire quasi-steady burning history of a drop vary as the square of the initial droplet diameter. Space-based microgravity environments are needed to permit the long experimental times required for the study of large drop sizes. Other combustion phenomena that are substantially distorted by I-g conditions include autoignition

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85 of large premixed gaseous systems, two-phase combustion phe- nomena involving large liquid-gas or solid-gas interfaces, radiative ignition of solids and liquids, smoldering and its transitions to flaIn~ng or extinction, the structures of lam~nar gas jet flames, and the properties of counterflow-difFusion flames. It ~ noteworthy that computational methods and combustion theory permit the analysis of combustion processes in micrograv- ity much more readily than is possible when buoyancy is present. However, the experunental observations needed for comparison with buoyancy-free combustion theory are generally not available. These are of considerable theoretical importance since theoretical analyses for interpretation of I-g observation generally truncate both the chemical kinetics and the dunensionality of the combus- tion phenomena studied. Thus, greater theoretical and computa- tiona] efforts are needed to support and exploit the results of these experunental studies. Soot formation ~ another area where m~crogravity experi- ments could be useful. In the combustion of hydrocarbons and mete] oxides, particulate formation during the combustion of metallic particulates involves poorly understood high-temperature condensation processes. The experimental characterization of con- deneation kinetics generally requires the analysis of the condensa- tion-susta~n~g flame structures. Flame theories have been con- structed for gravity-free environments, but the ingredients require careful testing even without the complication of gravity. Inasmuch as buoyancy effects at ~ g have been shows to substantiaIly distort counterflow diffusion flame structures, experiments in micrograv- ity are needed to support these studies.