G. S. Beavers

Boundary conditions at a naturally permeable wall

Experiments giving the mass efflux of a Poiseuille flow over a naturally permeable block are reported. The efflux is greatly enhanced over the value it would have if the block were impermeable, indicating the presence of a boundary layer in the block. The velocity presumably changes across this layer from its (statistically average) Darcy value to some slip value immediately outside the permeable block. A simple theory based on replacing the effect of the boundary layer with a slip velocity proportional to the exterior velocity gradient is proposed and shown to be in reasonable agreement with experimental results.

Breakup of a liquid drop suddenly exposed to a high-speed airstream

The breakup of viscous and viscoelastic drops in the high speed airstream behind a shock wave in a shock tube was photographed with a rotating drum camera giving one photograph every 5 ms. From these photographs we created movies of the fragmentation history of viscous drops of widely varying viscosity, and viscoelastic drops, at very high Weber and Reynolds numbers. Drops of the order of one millimeter are reduced to droplet clouds and possibly to vapor in times less than 500 ms. The movies may be viewed at http://www.aem.umn.edu /research/Aerodynamic_Breakup. They reveal sequences of breakup events which were previously unavailable for study. Bag and bag-and-stamen breakup can be seen at very high Weber numbers, in the regime of breakup previously called ‘catastrophic’. The movies allow us to generate precise displacement‐time graphs from which accurate values of acceleration (of orders 10 4 to 10 5 times the acceleration of gravity) are computed. These large accelerations from gas to liquid put the flattened drops at high risk to Rayleigh‐Taylor instabilities. The most unstable Rayleigh‐ Taylor wave fits nearly perfectly with waves measured on enhanced images of drops from the movies, but the eAects of viscosity cannot be neglected. Other features of drop breakup under extreme conditions, not treated here, are available on our Web site. # 1999 Elsevier Science Ltd. All rights reserved.

Non-Uniqueness and Stability of the Configuration of Flow of Immiscible Fluids with Different Viscosities,

Abstract : The arrangement of components in steady flow of immiscible liquids is typically nonunique. The problem of selection of arrangements is defined and studied by variational methods under the hypothesis that the realized arrangements are either those which maximize the speed on exterior boundaries for prescribed boundary tractions or those which minimize the tractions for prescribed speeds. The arrangements which minimize tractions also minimize dissipation by putting low viscosity liquid (LVL) in regions of high shear. The variational problem is used as a guide to intuition in design and interpretation of experiments when results of analysis of stability are unavailable. We always observe some kind of shielding of high viscosity liquid (HVL). This can occur by sheet coating in which LVL encapsulates HVL, or through the formation of rigidly rotating masses of which we call rollers. In other cases we get emulsions of LVL in a high viscosity foam. The emulsions arise from a fingering instability. The LVL fingers into the HVL and then low viscosity bubbles are pinched off the fingers. The emulsions seem to have a very low effective viscosity and they shield the HVL from shearing. In the problem of Taylor instability with two fluids low viscosity Taylor cells are separated by stable high viscosity rollers.

Vortex growth in jets

Observations are reported on the growth of vortices in the vortex sheets bounding the jet emerging from a sharp-edged two-dimensional slit and from a sharp-edged circular orifice. A regular periodic flow is observed near the orifice for both configurations when the Reynolds number of the jet lies between about 500 and 3000. The two-dimensional jet produces a symmetric pattern of vortex pairs with a Strouhal number of 0·43. Vortex rings are formed in the circular jet with a Strouhal number of 0·63. Computer experiments show that a growing pair of vortices in two parallel vortex sheets produces a symmetric pattern of vortices upstream from the original disturbance.

The rotating rod viscometer

This paper reports the development of practical methods of viscometry to characterize non-Newtonian fluids in slow flow. It is shown that measurements of the free surface near rods rotating in STP and polyacrylamide are accurate, reproducible, and in excellent agreement with a theory of rod climbing. Results are presented that establish the theory and experiment as a viscometer for determining the values of certain (Rivlin-Ericksen) constants that arise in the theory of slow flow. The variation of these constants with temperature in our sample of STP has been explicitly and accurately determined. The experiments in STP show that there is a range of rotational speeds for which STP may be well described by the fluids of grade four. Depth-averaged equations are derived from the equations governing steady axisymmetric flow of any incompressible simple fluid. From the depth-averaged equations, we prove a theorem about the variation of the torque required to turn the rod.

Operating modes of the clarinet

Operating modes of the clarinet are investigated via a simple model consisting of a uniform flat reed coupled to a constant‐area tube as a resonator. The analysis shows that, for lightly damped reeds, the operating frequency which is closest to the natural frequency of the reed is excited at the lowest blowing pressure. Further, the reed must be very heavily damped in order for the lowest resonance frequency of the tube to be preferentially excited. Experiments on a model clarinet, using metal reeds with different amounts of damping applied to the reed, confirm the pattern predicted by the analysis. There is close agreement between the measured and predicted values of the frequency ratio, but in many of the experiments the observed blowing pressure ratios were somewhat higher than the predicted values, the deviation being as high as 40% for the more heavily damped reeds.

Experiments on the Fluid Mechanics of Whistling

Experiments to investigate the fluid mechanics of whistling are reported. A model, consisting of a cylindrical cavity with rounded holes at each end, is used to simulate human whistling. It is found that the frequency is very near the Helmholtz resonator frequency, and that the resonator can be excited by flow through the smooth‐edged orifices bounding the resonant cavity. Furthermore, it is found that the flow velocity of the jet which excites the resonator must lie between limits that are proportional to frequency and that increase with both diameter and thickness of the orifice. It is concluded that whistling can be included in the same class of sound sources as the Rayleigh bird call and the Pfeifentone, since the essential mechanism for exciting them depends on the instability of a jet to the formation of vortex rings and the interaction of the rings with a rigid boundary in the flow.

Rayleigh{Taylor instability of viscoelastic drops at high Weber numbers

Movies of the breakup of viscous and viscoelastic drops in the high-speed airstream behind a shock wave in a shock tube have been reported by Joseph, Belanger & Beavers. They performed a Rayleigh-Taylor stability analysis for the initial breakup of a drop of Newtonian liquid and found that the most unstable Rayleigh-Taylor wave fits nearly perfectly with waves measured on enhanced images of drops from the movies, but the effects of viscosity cannot be neglected. Here we construct a Rayleigh-Taylor stability analysis for an Oldroyd-B fluid using measured data for acceleration, density, viscosity and relaxation time λ 1 . The most unstable wave is a sensitive function of the retardation time λ 2 which fits experiments when λ 2 /λ 1 = O(10 -3 ). The growth rates for the most unstable wave are much larger than for the comparable viscous drop, which agrees with the surprising fact that the breakup times for viscoelastic drops are shorter. We construct an approximate analysis of Rayleigh-Taylor instability based on viscoelastic potential flow which gives rise to nearly the same dispersion relation as the unapproximated analysis

Fluid mechanics of the edgetone

A model is proposed to explain the means by which an edgetone transforms the energy of a fully developed plane jet into energy which is radiated as sound. The edgetone configuration considered consists of a flat plate located in the medial plane of a fully developed two‐dimensional jet. The flow is modeled as follows. A periodic disturbance at the jet origin leads to the formation of an asymmetric vortex street which propagates downstream with a fixed convection velocity and wavelength. The vortex strength, convection velocity, and wavelength are determined as functions of the Strouhal number by applying conservation laws and kinematic relationships. A potential flow analysis of the interaction of the vortices with the edge is used to estimate the nearfield oscillating flow at the jet exit which, in turn, is used to calculate the phase of the feedback mechanism. The phase then determines the operating frequency as a function of jet velocity and edge stand‐off distance. It is shown that the proposed model is capable of predicting the major observed features of edgetone operation. The frequency predictions of the theory are compared with experiments for a wide range of jet parameters in both air and water. The comparison indicates that the frequency predictions are as good or better than previous empirical or semiempirical formulas.

Infiltration of initially dry, deformable porous media

The present paper studies the infiltration of an incompressible liquid in an initially dry (or partially dry), deformable sponge-like material made of an incompressible constituent in the slug-flow approximation having in mind the application to some industrial processes involving flow through sponge-like materials and, in particular, some composite materials manufacturing processes. The resulting initial-boundary value problem is of Stefan type, with suitable interface conditions and evolution equations describing the position of the interfaces delimiting the saturated region within the porous material. Different models are then suggested in the saturated region, depending on the importance of the inertial terms and on the constitutive equation for the stress. Comparison of the simulation with known experimental results is satisfactory.