6 results
Particle orbits in a rotating liquid
- Glyn O. Roberts, Dale M. Kornfeld, William W. Fowlis
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- Journal:
- Journal of Fluid Mechanics / Volume 229 / August 1991
- Published online by Cambridge University Press:
- 26 April 2006, pp. 555-567
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Monodisperse latex microspheres ranging in size from submicrometer to several micrometers in diameter can be prepared in the laboratory. The uniformity of diameter is important for instrument calibration and other applications. However it has proved very difficult to manufacture commercial quantities of mondisperse latex microspheres with diameters larger than about 3 micrometers owing to buoyancy and sedimentation effects. In an attempt to eliminate these effects NASA sponsored a Space Shuttle experiment called the Monodisperse Latex Reactor (MLR) to produce these monodisperse microspheres in larger sizes in microgravity. Results have been highly successful.
Using technology gained from this space experiment, a ground-based rotating latex reactor has been fabricated in an attempt to minimize sedimentation without using microgravity. The entire reactor cylinder is rotated about a horizontal axis to keep the particles in suspension.
In this paper we determine the motion of small spherical particles under gravity, in a viscous fluid rotating uniformly about a horizontal axis. The particle orbits are approximately circles, with centres displaced horizontally from the axis of rotation. Owing to net centrifugal buoyancy, the radius of the circles increases (for heavy particles) or decreases (for light particles) with time, so that the particles gradually spiral inward or outward.
For a large rotation rate, the particles spiral outwards or inwards too fast, while for a small rotation rate, the displacement of the orbit centre from the rotation axis is excessive in relation to the reactor radius. We determine the rotation rate that maximizes the fraction of the reactor cross-section area that contains particles that will not spiral out to the wall in the experimental time (for heavy particles), or that have spiralled in without hitting the wall (for light particles). Typically, the rate is close to 1 r.p.m., and design rotation rate ranges should span this value.
Numerical solutions for the spin-up of a stratified fluid
- Jae Min Hyun, William W. Fowlis, Alex Warn-Varnas
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- Journal:
- Journal of Fluid Mechanics / Volume 117 / April 1982
- Published online by Cambridge University Press:
- 20 April 2006, pp. 71-90
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Numerical solutions for the impulsively started spin-up of a thermally stratified fluid in a cylinder with an insulating side wall are presented. Previous experimental and numerical work on stratified spin-up had not provided a comprehensive and accurate set of flow-field data. Further, comparisons of this work with theory showed, in general, a substantial discrepancy. The theory was scaled using the homogeneous meridional-flow spin-up time scale and thus viscous-diffusion effects were excluded from the interior. It was anticipated that these effects could only be significant on the larger viscous-diffusion time scale. However, the comparisons with theory showed a faster rate of decay for the measurements even over the shorter meridional-flow spin-up time scale. Previous workers had suggested a number of explanations but the cause of the discrepancy was still unresolved. To provide data to extend the previous work, a numerical model was used. The model was first checked against accurate experimental measurements of stratified spin-up made using a laser-Doppler velocimeter. New accurate results which cover ranges of Ekman number (5·92 × 10−4 ≤ E ≤ 7·24 × 10−4), Rossby number (0·019 ≤ ε ≤ 0·220), stratification parameter (0·0 ≤ Sa−1 ≤ 1·03), and Prandtl number (5·68 ≤ σ ≤ 7·10) are presented. These results show the radial and vertical structure of the decaying azimuthal and meridional flows. The inertial–internal gravity oscillations excited by the impulsive spin-up are clearly seen. By making use of conclusions from the previous work and the results presented in this paper, it is established that viscous diffusion in the interior is the cause of the discrepancy with theory. Stratification causes the meridional spin-up flow to be confined closer to the boundary disks. This results in non-uniform spin-up of the interior and hence flow gradients in the interior. These gradients introduce viscous diffusion into the interior sooner than anticipated by the theory. A previous suggestion that the faster decay rate is due to angular momentum being injected into the interior from an oscillation of the meridional corner-jet flow is shown to be untenable.
Numerical solutions for spin-up from rest in a cylinder
- Jae Min Hyun, Fred Leslie, William W. Fowlis, Alex Warn-Varnas
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- Journal:
- Journal of Fluid Mechanics / Volume 127 / February 1983
- Published online by Cambridge University Press:
- 20 April 2006, pp. 263-281
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Numerical solutions for the impulsively started spin-up from rest of a homogeneous fluid in a cylinder for small Ekman numbers are presented. The basic analytical theory for this spin-up flow is due to Wedemeyer (1964). Wedemeyer's solution shows that the interior flow is divided into two regions by a moving front which propagates radially inward across the cylinder. The fluid ahead of the front remains non-rotating, while the fluid behind the front is being spun up. Experimental observations have shown that Wedemeyer's model captures the essential dynamics of the azimuthal flow, but that it is not a quantitative model. Wedemeyer made several assumptions in formulating an Ekman compatibility condition, and inconsistencies exist between these assumptions and his solution. Later workers attempted to improve the analytical theory, but their work still included the same basic assumptions made by Wedemeyer.
No previous work has provided a comprehensive and accurate set of three-dimensional flow-field data for this spin-up problem. We chose to acquire such data using a numerical model based on the Navier–Stokes equations. This model was first checked against accurate laser-Doppler measurements of the azimuthal flow for spin-up from rest. New flow-field data over a range of Ekman numbers 9·18 × 10−6 [les ] E [les ] 9·18 × 10−4 are presented. Diagnostic studies, which reveal the various contributions to spin-up of the separate inviscid and viscous terms as functions of radius and time, are also presented. The plots of the viscous-diffusion term reveal the moving front, which is identified as a layer of enhanced local viscous activity. Immediately after the impulsive start, viscous diffusion is seen to be the major contributor to spin-up, then the nonlinear radial advection term takes over, and, finally, when spin-up is well progressed, the linear Coriolis force dominates. In the vicinity of the front, the inward radial flow is a maximum, and the vertical velocity is very small. Strong radial gradients of the vertical velocity are observed across the front and behind the front at the edge of the Ekman layer, and the azimuthal flow behind the front shows strong departures from solid-body rotation. These results enable us to fill in details of the flow not accurately given by Wedemeyer's model and its extensions.
Three-dimensional baroclinic instability of a Hadley cell for small Richardson number
- Basil N. Antar, William W. Fowlis
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- Journal:
- Journal of Fluid Mechanics / Volume 137 / December 1983
- Published online by Cambridge University Press:
- 20 April 2006, pp. 423-445
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A three-dimensional linear stability analysis of a baroclinic flow for Richardson number Ri of order unity is presented. The model considered is a thin, horizontal, rotating fluid layer which is subjected to horizontal and vertical temperature gradients. The basic state is a Hadley cell which is a solution of the Navier–Stokes and energy equations and contains both Ekman and thermal boundary layers adjacent to the rigid boundaries; it is given in closed form. The stability analysis is also based on the Navier–Stokes and energy equations; and perturbations possessing zonal, meridional and vertical structures were considered. Numerical methods were developed for the solution of the stability problem, which results in an ordinary differential eigenvalue problem. The objectives of this work were to extend the previous theoretical work on three-dimensional baroclinic instability for small Ri to a more realistic model involving the Prandtl number σ and the Ekman number E, and to finite growth rates and a wider range of the zonal wavenumber. The study covers ranges of 0.135 [les ] Ri [les ] 1.1, 0.2 [les ] σ [les ] 5.0, and 2 × 10−4 [les ] E [les ] 2 σ 10−3. For the cases computed for E = 10−3 and σ ≠ 1, we found that conventional baroclinic instability dominates for Ri > 0.825 and symmetric baroclinic instability dominates for Ri < 0.675. However, for E [ges ] 5 × 10−4 and σ = 1 in the range 0.3 [les ] Ri [les ] 0.8, conventional baroclinic instability always dominates. Further, we found in general that the symmetric modes of maximum growth are not purely symmetric but have weak zonal structure. This means that the wavefronts are inclined at a small angle to the zonal direction. The results also show that as E decreases the zonal structure of the symmetric modes of maximum growth rate also decreases. We found that when zonal structure is permitted the critical Richardson number for marginal stability is increased, but by only a small amount above the value for pure symmetric instability. Because these modes do not substantially alter the results for pure symmetric baroclinic instability and because their zonal structure is weak, it is unlikely that they represent a new type of instability.
Numerical solutions and laser-Doppler measurements of spin-up
- Alex Warn-Varnas, William W. Fowlis, Steve Piacsek, Sang Myung Lee
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- Journal:
- Journal of Fluid Mechanics / Volume 85 / Issue 4 / 27 April 1978
- Published online by Cambridge University Press:
- 12 April 2006, pp. 609-639
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The spin-up flow in a cylinder of homogeneous fluid has been examined both experimentally and numerically. The primary motivation for this work was to check numerical solution schemes by comparing the numerical results with laboratory measurements obtained with a rotating laser-Doppler velocimeter. The laser-Doppler technique is capable of high accuracy with small space and time resolution, and disturbances of the flow are virtually negligible. A series of measurements was made of the zonal flow over a range of Ekman numbers (1·06 × 10−3 ≤ E ≤ 3·30 × 10−3) and Rossby numbers (0·10 [les ]|ε| [les ] 0·33) at various locations in the interior of the flow. These measurements exceed previous ones in accuracy. The weak inertial modes excited by the impulsive start are detectable. The numerical simulations used the primitive equations in axisymmetric form and employed finite-difference techniques on both constant and variable grids. The number of grid points necessary to resolve the Ekman layers was determined. A thorough comparison of the simulations and the experimental measurements is made which includes the details of the amplitude and frequency of the inertial modes. Agreement to within the experimental tolerance is achieved. Analytical results for conditions identical to those in the experiments are not available but some similar linear and nonlinear theories are also compared with the experiments.
Evaporation Kinetics in the Hanging Drop Method of Protein Crystal Growth
- James K. Baird, Richard W. Frieden, E. J. Meehan, Pamela J. Twigg, Sandra B. Howard, William A. Fowlis
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- Journal:
- MRS Online Proceedings Library Archive / Volume 87 / 1986
- Published online by Cambridge University Press:
- 26 February 2011, 231
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- 1986
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We present an engineering analysis of the rate of evaporation of solvent in the hanging drop method of protein crystal growth. Our results are applied to 18 different drop and well arrangements commonly encountered in the laboratory. We take into account the chemical nature of the salt, the drop size and shape, the drop concentration, the well size, the well concentration, and the temperature. We find that the rate of evaporation increases with temperature, drop size, and with the salt concentration difference between the drop and the well. The evaporation possesses no unique half-life. Once the salt in the drop achieves about 80% of its final concentration, further evaporation suffers from the law of diminishing returns.