*
M. Lyell and G. B. McFadden ACMD
R. G. Rehm, CAML
R. F. Berg, Chemical Science and Technology Laboratory*

The study of fluid systems near their thermodynamic critical point is a very active research area in condensed matter physics. The critical point is characterized by a unique temperature, pressure, and density; roughly speaking, at temperatures below the critical temperature, two-phase equilibria between the liquid and vapor phases are possible, and above the critical temperature the distinction between liquid and vapor is lost and only one phase occurs. As the critical point is approached, thermophysical properties of the fluid are observed to develop singularities, and this behavior is also predicted theoretically. The Thermophysics Division has a long-standing research program in this area, and has performed many fundamental experimental studies of fluid behavior near the critical point. Current plans include shuttle-based experiments with xenon to measure the behavior of the fluid's viscosity near the critical point.

A complication with terrestrial measurements of critical fluids is the density stratification that occurs in the presence of a gravitational field. Ideally, one would perform measurements on a homogeneous isothermal system of constant density. However, in this situation the compressibility of the fluid is very large, and is predicted to diverge at the critical point; as a result the fluid density stratifies under its own weight. Stratification places a limit on how close to the critical point measurements can be made before deviations from ideal conditions become significant. For example, density stratification allows the fluid to undulate in modes, called internal wave modes, which cannot be sustained in a homogeneous fluid. Therefore, measurements are being performed in microgravity environments, where the effects of gravity are less important.

In experimental work by Robert Berg (838) to measure fluid viscosity near the critical point, a rectangular paddle whose surface consists of a fine-wire mesh that is permeable to the fluid is driven sinusoidally in the fluid with a given frequency. The fluid viscosity can be inferred by measuring the amplitude of the motion and the force necessary to sustain the motion. It is observed in experiments under terrestrial conditions that at certain driving frequencies the response is more complicated than would be predicted by simple models of the fluid response to the paddle; for the success of space-based experiments it is necessary to understand whether or not these anomalies are related to the strength of the gravitational field. In particular, Dr. Berg suggested that a resonance is taking place between natural frequencies of internal waves in the system and the driving frequency of the paddle.

Margaret Lyell, who is on sabbatical from the Mechanical and Aerospace Engineering Department of West Virginia University, has been collaborating with G. McFadden and R. Rehm in CAML with the intention of identifying the nature of the internal waves and the flow field associated with the waves in both a qualitative and quantitative manner. The most general set of equations which govern the continuum fluid behavior are those of conservation of mass, conservation of momentum (Navier-Stokes), and conservation of energy, along with an equation of state. For the isothermal experiments, it was determined that compressibility and viscous effects on the wave motion could be neglected, and a Boussinesq approximation is valid. Expansion around the base state of zero motion leads to a set of perturbation equations. Due to the density stratification, the system in general could not be solved in closed form. The density stratification itself changes depending upon the temperature of the isothermal system. If the temperature of interest is "far" from the critical temperature (and above it), a closed form approximate solution can be obtained. The general approach required a computational solution. A Galerkin collocation scheme involving a Chebyshev expansion was utilized.

Results thus far include the determination of the eigenfrequencies for various modes over a large range of selected temperature values as illustrated in the accompanying figure. Good agreement with the experimental frequency results has been obtained, thus permitting identification and explanation of the experimental observations. Also, visualization of the calculated flow field has been accomplished by utilizing the closed form solution at a selected temperature far above the critical temperature.

**Figure 14:** Comparisons of experimental data of R. Berg for the
measured resonant frequencies of the fluid viscometer as a
function of temperature **T** relative to the critical temperature .
Data for two of the observed modes are shown (solid circles),
and theoretical predictions based on a model of internal-wave motion
are indicated by the two curves.