Large Viscosity Variations Research Articles

Thermals in extremely viscous fluids, including the effects of temperature-dependent viscosity

The flow induced by injection of a given amount of buoyancy or hot fluid from a localized source in a viscous fluid is investigated for conditions under which the Reynolds number Re is small compared with one, and the dimensionless buoyancy or Rayleigh number Ra is large compared with one. Laboratory experiments show that the buoyant fluid rises in the form of an extremely viscous ‘thermal’ which enlarges with time as a result of entrainment of surrounding fluid. The formation of a stable ‘chemical ring’ or torus of passive tracer similar in appearance to high Reynolds-number vortex rings is a notable feature of the creeping flow for high Rayleigh numbers. The possibility of large variations of viscosity due to temperature differences is included. A self-similar model is developed based on a boundary-layer analysis of a thin diffusive layer surrounding a spherical thermal for which the flow field is given by the exact solution for non-diffusive Stokes’ flow. Experiments at 2.5 × 102 < Ra < 2.5 × 104 and Re < 10−2 demonstrate the nature of extremely viscous thermals, support the similarity solution and enable evaluation of a proportionality constant. Possible applications of the results to dispersion by viscous drops and particularly to thermal convection in the Earth's solid mantle are mentioned.

Convective instabilities in a variable viscosity fluid cooled from above

Whether in the mantle or in magma chambers, convective flows are characterized by large variations of viscosity. We study the influence of the viscosity structure on the development of convective instabilities in a viscous fluid which is cooled from above. The upper and lower boundaries of the fluid are stress-free. A viscosity dependence with depth of the form ν 0 + ν 1 exp(−γ.z) is assumed. After the temperature of the top boundary is lowered, velocity and temperature perturbations are followed numerically until convective breakdown occurs. Viscosity contrasts of up to 10 7 and Rayleigh numbers of up to 10 8 are studied. For intermediate viscosity contrasts (around 10 3), convective breakdown is characterized by the almost simultaneous appearance of two modes of instability. One involves the whole fluid layer, has a large horizontal wavelength (several times the layer depth) and exhibits plate-like behaviour. The other mode has a much smaller wavelength and develops below a rigid lid. The “whole layer” mode dominates for small viscosity contrasts but is suppressed by viscous dissipation at large viscosity contrasts. For the “rigid lid” mode, we emphasize that it is the form of the viscosity variation which determines the instability. For steep viscosity profiles, convective flow does not penetrate deeply in the viscous region and only weak convection develops. We propose a simple method to define the rigid lid thickness. We are thus able to compute the true depth extent and the effective driving temperature difference of convective flow. Because viscosity contrasts in the convecting region do not exceed 100, simple scaling arguments are sufficient to describe the instability. The critical wavelength is proportional to the thickness of the thermal boundary layer below the rigid lid. Convection occurs when a Rayleigh number defined locally exceeds a critical value of 160–200. Finally, we show that a local Rayleigh number can be computed at any depth in the fluid and that convection develops below depth z r (the rigid lid thickness) such that this number is maximum. The simple similarity laws are applied to the upper mantle beneath oceans and yield estimates of 5 × 10 15−5 × 10 16 m 2 s −1 for viscosity in the thermal boundary layer below the plate.

Heat transfer and horizontally averaged temperature of convection with large viscosity variations

It is pointed out that the understanding of convection in large-Prandtl-number Boussinesq fluids with uniform properties and contained in simple geometries is virtually complete. Present efforts are typically directed towards relaxing some of the original assumptions by going to lower Prandtl number, more complicated geometries, variable material properties, or introducing new dynamical processes such as the Lorentz forces. A description is given of experiments which are concerned with the effect on convection of relaxing the assumption of a uniform viscosity. The reported experiments were designed to measure both the horizontally averaged temperature as a function of depth and the heat transfer of convection over a range of viscosity variations up to 100,000.

The effects of a strongly temperature-dependent viscosity on slow flow past a hot sphere

This work determines analytically the drag on, and heat flux out of, a hot sphere that translates steadily in a fluid of strongly temperature-dependent viscosity. There is no dissipative heating. The essentials are illustrated by an exact solution for the flow induced by slowly squeezing two parallel planes together. The lower plane is hot and stationary; the upper is cold and advances in a direction normal to itself at uniform speed U. The gap is completely filled by a fluid of strongly temperature-dependent viscosity. We find the temperature and velocity profiles, and determine the Nusselt number N and Peklet number P as functions of the normal force D on the lower plane. The large viscosity variation tries to concentrate the flow into a relatively thin softened layer in which the viscosity is of order its value μ0 at the hot plane. In the limit of infinite viscosity ratio (fixed P), it succeeds (lubrication limit): if P [Lt ] 1, the width of the softened layer is determined by conduction and D ∝ μ0U; but D ∝ μ0U4 when forced convection is important. If P → ∞ (fixed viscosity ratio), the softened layer is so thin that it chokes, and all the deformation occurs outside the thermal layer in the fluid of uniform viscosity μ∞ (Stokes limit); then D ∝ μ∞U. These mechanisms appear as three distinct legs in our plot of log P against log D. There are similar transitions in the plot of log N against log D. The solution gives an estimate of the drag on a sphere. We test this estimate against an analytical solution for the sphere in the lubrication limit. Then we extend the solution to cover power-law fluids, and apply it to a model (by Marsh) of magma transport beneath island-arc volcanoes. The results suggest that the magma covers the first 50 km of its ascent by an isoviscous mechanism, with the lubrication mechanism operating in the remaining 50 km. To open a fresh pathway from the source to the surface takes about 106 years and uses about 1027 erg.

Experiments on the stability of convection rolls in fluids whose viscosity depends on temperature

The stability of two-dimensional convection rolls has been studied as a function of the Rayleigh number, wavenumber and variation in viscosity. The experiments used controlled initial conditions for the wavenumber, Rayleigh numbers up to 25 000 and variations in viscosity up to a factor of about 20. The parameter range of stable rolls is bounded by a hexagonal-cell regime at small Rayleigh numbers and large variations in viscosity. Otherwise, the rolls are subject to the same transitions as have already been studied in fluids of uniform viscosity. The bimodal instability leading to a stable three-dimensional pattern occurs at smaller values of the average Rayleigh number as the variations in viscosity increase. This appears to be a consequence of the low viscosity of the warm thermal boundary layer associated with the original rolls.

Numerical simulation of finite-amplitude thermal convection with large viscosity variation in axisymmetric spherical geometry: Effect of mechanical boundary conditions

Finite-difference numerical simulations of finite amplitude thermal convection with large viscosity variations in an axisymmetric sphere and spherical shells have been carried out. The mathematical formulation and numerical schemes are described in detail. Effects of the mechanical boundary conditions on the steady state solutions have been investigated. Results show that flows are strongly dependent on the mechanical boundary condition at the outer surface and relatively insensitive to the core boundary condition in the case of spherical shells. A shear stress free condition results in the upward concentration of convection flow, while a no-slip condition tends to push the flow towards the deeper interior. Numerical results obtained in this study imply that mantle convection within the Moon, Mercury and Mars is probably confined at depth because of the lack of horizontal surface tectonic activity on these three planets. However, mantle convection within the Earth probably concentrates near the surface because plate tectonics are active.

Localized radiation grafting of flame retardants to polyethylene terephthalate. II. Vinyl phosphonates

Six vinyl phosphonates have been synthesized and evaluated as grafts on poly(ethylene terephthalate) for their flame retardance properties. Diethylvinyl phosphonate was used as a model for phosphorus-containing flame retardants in developing the methodology for localizing flame retardants either on the surface of the filament or uniformly throughout it. SEM–X-ray microprobe techniques were used in the verification of the location of the flame retardant in the filament. The flame retardance efficiency of poly(diethyl vinyl phosphonate) was then correlated with its location in the filament. Other grafted phosphorus-containing flame retardants showed a range of efficiencies that depended not only upon the location of the graft within the filament but also upon the %P in the compound. The wide variations in flame retardance efficiencies of copolymers and terpolymers were attributed to large variations in the melt viscosity of the different grafted materials. The grafts showed only small changes in tenacity and large increases in elongation.

Localized radiation grafting of flame retardants to poly(ethylene terephthalate). I. Bromine‐containing monomers

AbstractVinyl bromide was used as a model for bromine‐containing flame retardants in developing methodology for localizing flame retardants either on the surface of the filament, uniformly throughout it, or predominantly at the core. SEM–x‐ray microprobe techniques were used in the verification of the location of the flame retardant in the filament. The flame retardance efficiency of PVBr was then correlated with its location in the filament. Grafting other bromine‐containing flame retardants showed a wide range of efficiencies which depended not only upon the location of the grafting within the filament but also upon the structure of the compound. For the various bromine homopolymer grafts, the apparent thermal stability of the graft and its flame retardance efficiency may be simply related to the alpha aliphatic hydrogen‐to‐bromine ratio. The lower this ratio is, the higher the efficiency. VBr copolymers and terpolymers showed wider variation in flame retardance efficiencies. This is attributed to large variations in melt viscosity of the different grafted materials. The grafts showed only small changes in their melting points and minimal changes in tenacity and stiffness. However, grafting induced large increases in the elongation.

Numerical Simulation of Sea-floor Spreading

A two-dimensional time-dependent method that allows large viscosity variations is used to numerically simulate upper mantle convection, including lithospheric plate motion. The numerical operator developed for viscous flow is in self-adjoint form, so that the conjugate gradient iteration method can be used. Convergence is faster than with relaxation methods. The method is applied to sea-floor spreading with the objective of examining the driving mechanism of mid-ocean ridges. In accordance with this objective, deep convection is suppressed in the model. Counterflow below the plates is confined to depths less than 340 km. The model is fit to observed topography at four different ridge locations. It is found that, for a spreading velocity of 1.2 cm/yr, the ridge can produce compressive stress in the lithosphere out to a distance of 1600 km. For a spreading velocity of 6 cm/yr, this model is clearly excluded, for it requires excessively large stress in the lithosphere. Therefore, upwelling material must cross the seismic discontinuity at 400-km depth.

Numerical simulation of sea-floor spreading

A two-dimensional time-dependent method that allows large viscosity variations is used to numerically simulate upper mantle convection, including lithospheric plate motion. The numerical operator developed for viscous flow is in self-adjoint form, so that the conjugate gradient iteration method can be used. Convergence is faster than with relaxation methods. The method is applied to sea-floor spreading with the objective of examining the driving mechanism of mid-ocean ridges. In accordance with this objective, deep convection is suppressed in the model. Counterflow below the plates is confined to depths less than 340 km. The model is fit to observed topography at four different ridge locations. It is found that, for a spreading velocity of 1.2 cm/yr, the ridge can produce compressive stress in the lithosphere out to a distance of 1600 km. For a spreading velocity of 6 cm/yr, this model is clearly excluded, for it requires excessively large stress in the lithosphere. Therefore, upwelling material must cross the seismic discontinuity at 400-km depth.

Cytochrome c interaction with the mitochondrial membrane: A spin label study

Horse-heart ferrocytochrome c has been labeled with N-(2,2,5,5-tetramethyl-3-pyrrolidinyl-1-oxyl) iodoacetamide at methionine-65. The paramagnetic resonance spectrum of labeled ferricytochrome c indicates a weak immobilization of the radical ( τ c = 9.3·10 −10 sec) which becomes stronger upon binding of labeled cytochrome c to cytochrome c-depleted mitochondrial membranes ( τ c = 3.3·10 −9 sec). The hyperfine coupling constant remains, however, unchanged (16.7 ± 0.1 gauss) indicating that the cytochrome c binding site is highly polar. The region where cytochrome c is bound to the membrane is insensitive to large variations of medium viscosity.

Fluid Rheological Effects in Sliding Elastohydrodynamic Point Contacts With Transient Loading: 2—Traction

This paper describes the results of the traction measurements obtained in the experiment discussed in the companion paper entitled “Fluid Rheological Effects in Sliding Elastohydrodynamic Point Contacts With Transient Loading: I.—Film Thickness.” Under the conditions investigated, the traction values appear to be primarily a function of the sliding velocity. Large variations in fluid composition and inlet viscosity had little influence on the tractive force. It was also found that rapid application of the normal load had a negligible effect on the traction.

Thermal convection with large viscosity variations

The influence of large variations of viscosity on convection in a layer of fluid heated from below has been investigated. Solutions for the flow and temperature fields were obtained numerically assuming infinite Prandtl number, free-surface boundary conditions, and two-dimensional motion of fixed horizontal wavelength. The effects of a temperature-dependent and a depth-dependent viscosity were each studied; calculations were also carried out using a temperature- and depth-dependent viscosity model appropriate to the earth's mantle.

Convection in a saturated porous medium at large Rayleigh number or Peclet number

When the dimensions of a convective system in a saturated porous medium are sufficiently great, diffusion effects can be neglected except in regions where the gradients of fluid properties are very large. A boundary-layer theory is developed for vertical plane flows in such regions. In special cases, the theory is equivalent to that for laminar incompressible flow in a two-dimensional half-jet, or in a plane jet or round jet, for which similarity solutions are well known.A number of experiments have been performed using a Hele-Shaw cell immersed in water, with a source of potassium permanganate solution located between the plates. At very small values of the source strength, a flow analogous to that of a plane jet from a slit is obtained. The distance advanced by the jet front, or cap, is measured as a function of time, and the velocity is found to be nearly proportional to the velocity of the fluid on the axis of the steady jet behind the cap, as given by the similarity law of Schlichting and Bickley. At large values of the source strength, a two-dimensional ‘broad jet’ of homogeneous solution descending under gravity is produced; the shape of the flow region can be calculated with little error from potential theory, neglecting the effect of the mixing layers.A possible example of a mixing layer observed in a geothermal region is examined. The theoretical form of the temperature distribution is calculated numerically, taking into account the large viscosity variation with temperature and also the possibility of a large permeability variation. These effects are found to have less influence upon the solution than might have been expected. Quantitative values obtained for the physical parameters are consistent with other geophysical observations.

Study of gas-lubricated, hydrodynamic, full journal bearings

The two most important characteristics of gas-lubricated, full journal bearings are load-carrying capacity and half-speed whirl. In this paper these two characteristics have been investigated. The gases used as lubricants were mainly hydrogen, nitrogen and neon, giving a large viscosity variation; argon and carbon dioxide were also used to investigate the effect of the ratio of specific heats (γ). The bearings used were plain cylinders with a longitudinal groove cut into the upper surface of each bearing, extending over half the bearing length. It had been shown previously that this groove improved the half-speed whirl characteristics, and tests on bearings without the groove, included in the paper, demonstrate that the groove does not affect the load capacity. The parameters investigated are bearing length, diameter, clearance, rotational velocity, ambient pressure, and the gas viscosity and ratio of specific heats. It is shown that, in general, the existing theories of load performance for compressible lubricants, operating in isothermal conditions, give a satisfactory correlation with the experimental results. The half-speed whirl results are not related to a complete analytical theory, but useful stability criteria are established.