The shape of a fluid drop approaching an interface

The shape of a fluid drop approaching an interface does not change appreciably with time and is very close to the equilibrium dimensions, in spite of the large pressure gradient which is present in the draining film. This is because the net vertical force due to the excess pressure in the draining film above that in the drop is identical with that for an equilibrium film being zero for a plane interface and —2Rσ sin² ? for a deformable interface. Employing this result in a force balance around the drop which is independent of the bulk interface shows that the area A of the draining film between a fluid drop of volume V and a deformable fluid-liquid interface is given by where σ is the interfacial tension and Δρ the density difference between the drop and surrounding fluid, 1/b is the curvature at the top of the drop and h is the distance between this point and the edge of the draining film which is inclined to the horizontal at an angle ?. When the interface is a rigid plane the overall curvature 1/R of the draining film and the volume v enclosed by it, together with the angle ? are all zero. The limiting cases of the expression for very small and very large drops agree with those previously established for both deformable and rigid interfaces. An approximate expression which applies when cV^2/3 (where c = Δρg/σ) is between 0.6 and 13.5 and which gives A to within ± e% is where for a rigid plane interface and for a deformable interface When the densities of the drop and bulk heavy fluids are equal, but their respective interfacial tensions σ₁₂ and σ₂₃ with the light fluids are different, the expression becomes which estimates A/V^2/3 to within about ± 25% for σ₁₂/σ₂₃ in the range 0.11 to 9.0 and cV^2/3 (where c = Δρg/σ₁₂) between 0.6 and 13.5.     

The rheology and spin coating of polyimide solutions

The rheological properties of five types of concentrated polyamic acid and polyimide solutions are characterized by non-Newtonian shear viscosity η(\documentclass{article}\pagestyle{empty}\begin{document}$ {\rm \dot \gamma} $\end{document}) and primary normal stress coefficient Ψ₁(\documentclass{article}\pagestyle{empty}\begin{document}$ {\rm \dot \gamma} $\end{document}) measurements over a wide range of shear rates. Onset of non-Newtonian flow of the polyamic acid solutions was observed in the shear rate range 30 to 400s^?1 and of the fully imidized polyimide solution at below 3 × 10^?2s^?1. Significant viscoelastic properties exemplified by normal stresses were observed in all the solutions. The solution rheology results are discussed in the context of spin coating for the deposition of thin films. The relative magnitude of effects of non-Newtonian flow on the dynamics of spin coating is assessed with a Deborah number characteristic of the flow.     

Power consumption of helical ribbon mixers in viscous newtonian and non-newtonian fluids

Power input measurements are reported for helical ribbon impellers for two scales; a 0.15 m diameter and a 0.4 m diameter tank. Data for viscous Newtonian and non-Newtonian fluids are brought together by use of the average apparent viscosity concept and the following equation:

where k_s is the shear rate constant, c is the clearance between vessel wall and impeller with diameter D.Power measurements from this work combined with relevant information extracted from the published literature indicate that impeller geometry has a profound effect upon power requirement, particularly in the laminar region, where the reported data can be described by:

where K_p is a geometric constant and all the other symbols have their usual significance. Theoretical models which fail to allow for system geometry and fluid properties give values which may be seriously in error.     

Mass transfer in the entrance region for axial and swirling annular flow

Rates of mass transfer to the inner core of an annular flow system have been determined for the mass transfer entry region using the limiting current density method. Both in laminar and turbulent flow, the hydrodynamic and concentration boundary layers were not fully developed. The variation of the mass transfer coefficients with length of core section has been demonstrated, and the data for swirling flow correlated by the equation for 1500<Re < 14000, 1500 < Sc < 6200 and 1.75 < L/D^e < 10.14. The data for axial flow in the entry region have been correlated by the equation. for 1800 < Re < 12500 and Sc = 2604. Mass transfer enhancement in the entry region due to swirl when compared to axial flow-systems with a jetting transverse inlet is only noticeable for Re > 6000, but is very significant compared to systems with fully developed boundary layers in axial or swirling flow.     

Shear rate dependence of the viscosity of cellulose nitrate solutions in the dilute concentration regime revisited

A detailed rheological study of cellulose nitrate in ethylacetate had been carried out in the dilute concentration (c) regime, covering a degree of polymerization (DP) range between 300 < DP_η < 7000 and shear rates ($ \dot \gamma $) between 100 s^?1 < $ \dot \gamma $ < 2000 s^?1. The results show a strong dependence of the transition Newtonian to non-Newtonian behavior on the three variables $ \dot \gamma $, DP, and c, similar to that found recently on solutions of synthetic polymers. Emphasis has been put on the critical concentrations corresponding to the standard shear rate 1000 s^?1 to correspond to the standard conditions ($ \dot \gamma $ ? 1000 s^?1; 0.3 < [η] · c < 0.6; DS = 2.90 ± 0.02) proposed for the determination of the intrinsic viscosity [η] of cellulose nitrates. It is shown that solutions with concentrations adjusted according to the above given conditions still exhibit Newtonian behavior, up to the highest range of DP. It follows, therefore, that applying the standard conditions, an extrapolation to $ \dot \gamma $ = 0 as has been proposed often for the intrinsic viscosity determination of cellulose nitrate is not advisable and results in considerable error. Considering the relationship between [η] and DP, the present results indicate that the decrease of the exponent ( a ) from a = 1.0 to a = 0.76, taking place above a DP ? 1000, is not a consequence of the applied shear rate but rather of the molecular properties of the solutes themselves.     

Thermal reactions between MII sulphates and acid orthophosphates in the solid state

Two general types of high temperature reactions between M^II sulphates and acid orthophosphates have been found and a survey of some representative examples has been made. The reactions are typified by two examples using Ca²⁺ salts: and . Similar reactions have been found for other alkaline earth sulphates and phosphates and for corresponding mixed alkaline earth systems. The products have been found to be largely determined by the phase relationships in the respective M^IIO–P₂O₅ systems.     

Rheological behavior of ultrahigh molecular weight polyethylene semidilute solutions. I. Solvent effect

The rheological behavior of ultrahigh molecular weight polyethylene (UHMWPE) semidilute solutions with different solvents has been studied. In a temperature range of 150–185°C, the viscosity of a UHMWPE semidilute solution with paraffin oil as solvent (soln 1) is more temperature-independent and viscosity-stable than that with decalin as solvent (soln 2). Usually the reduction of the molecular entanglement density in solution causes a rapid reduction in viscosity. Apparently soln 1 has a different entanglement density compared with soln 2. The activation energy of UHMWPE semidilute solutions changes with both shear stress and shear rate. The regressive equations for both solutions in between E_σ and σ or E_? and $ {\dot \gamma } $ have been obtained from a least-squares method. Finally, there is a discontinuity in the non-Newtonian flow region of η vs. $ \dot \gamma ^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} $ curve, no matter which temperature or solvent was used. The discontinuity occurs at a shear rate of about 70 s^?1, where a transformation from a qualitative change in entanglement in the solution takes place.     

Free coating of non-newtonian liquids onto a vertical surface

The coating of non-Newtonian liquids onto a vertical surface continuously withdrawn from the liquid bath is considered. An analytic treatment is presented for purely viscous non-Newtonial liquids using the Ellis and generalised Bingham models both of which may be reduced to a new theory for power-law fluids. The theories give a relationship between the dimensionless film thickness, T₁, and the Capillary number, C₁, as a function of the fluid physical properties and the parameters of the viscous model. The dimensionless groups have been generalised to allow for non-Newtonian behaviour. The power-law and Ellis model predictions are compared with previous theoretical studies and shown to be consistent with known limits. Experimental data are also presented for a wide range of non-Newtonian fluids and compared with the new theories.     

Transferts thermiques entre un serpentin et des fluides newtoniens ou non—newtoniens agités par une turbine à pales inclinées dans une cuve

The present work deals with the heat transfer from a hélicoïdal worm to newtonian or non-newtonian liquids, which and stirred by a 4 pitched blade-turbine in a cylindrical tank. The effective viscosity of the pseudoplastic liquids is determined by the Metzner-Otto model combined with the Ellis rheological equation. The heat transfer in the newtonian fluids in a turbulent system is most precisely represented by the following equations: (a) (b) In the case of non-newtonian fluids and of downward pumping, the following equation enables to obtain a very precise evaluation of the heat-transfer coefficient in a transitional flow.      

Power numbers,Taylor numbers and Taylor vortices in viscous newtonian and non-newtonian fluids

Power numbers (Po) as a function of Taylor numbers (Ta) have been investigated for the flow between concentric cylinders with the inner cylinder rotating and the outer cylinder stationary. Both Newtonian and non-Newtonian fluids were used and torque measurement was done by strain gauges. Flow visualization enabled Taylor vortices to be studied too. Three pairs of cylinders were used: two plain and one where the outer cylinder had vertical strip baffles. The nominal outer dimensions were 0.1 m diameter × 0.2 m high. The maximum values of Ta measured were 90 for width of gap (d)/inner cylinder radius (R_i) ratio = 0.9 and approximately 400 for d/R_i = 0.7.Experimental data for all the systems used agreed well with theory in the laminar region (Ta<Ta_c), giving Po ∝ Ta⁻¹. For the laminar Taylor vortex region (Ta>Ta_c),

. In the latter case, the laminar shear rate was used to calculate the Taylor number for the non-Newtonian fluids. Critical Taylor number Ta_c for non-Newtonian fluids was found to be greater than the Newtonian one.The results are useful for estimating the power consumption of potential industrial devices utilizing the advantages of laminar Taylor vortex flow.     

The glassy state,ideal glass transition,and second‐order phase transition

According to Ehrenfest classification, the glass transition is a second‐order phase transition. Controversy, however, remains due to the discrepancy between experiment and the Ehrenfest relations and thereby their prediction of unity of the Prigogine‐Defay ratio in particular. In this article, we consider the case of ideal (equilibrium) glass and show that the glass transition may be described thermodynamically. At the transition, we obtain the following relations: and with Λ = (α_gβ_l − α_lβ_g)²/β_lβ_gΔα²; and The Prigogine‐Defay ratio is with Γ = TV(α_lβ_g − α_gβ_l)²/β_lβ_gΔβ, instead of unity as predicted by the Ehrenfest relations. Dependent on the relative value of ΔC_V and Γ, the ratio may take a number equal to, larger or smaller than unity. The incorrect assumption of perfect differentiability of entropy at the transition, leading to the second Ehrenfest relation, is rectified to resolve the long‐standing dilemma perplexing the nature of the glass transition. The relationships obtained in this work are in agreement with experimental findings. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 71: 143–150, 1999     

Flow transition for natural convective heat transfer in a porous medium saturated with water near 4°C

In this paper, the problem of buoyancy-induced convection flow in water-saturated porous media near 4°C is examined using a numerical model. Darcy's law is used to model flow behavior and a single equation convective heat transfer model is used for the energy equation. As the Boussinesq approximation is not valid for this case, a parabolic dependence of density on the temperature is used. Natural convection is generated and sustained by a uniform heat source. Flow behavior is governed by three natural parameters appearing in the model. They are: (i) dynamical parameter, (ii) geometric parameter, γ = b/a; and (iii) wall temperature, in relation to the reference temperature at the density extremum. For certain ranges of θ_w (<0) and Gr, interesting density inversion effects are possible. Transient solutions are obtained for various aspect ratios and modified Grashof number values. For a wide range of Grashof number, steady state solutions could not be obtained. Flow mutations into periodic and chaotic solutions are investigated for a range of Grashof number (100 to 40,000) and aspect ratio values (1 to 10).     

Viscosity of bead suspensions in polymeric solutions

Viscosity measurements made by a cone-plate viscometer on polyisobutylene in decalin solutions at different concentrations and their corresponding glass bead suspensions with filler loadings up to 40% by volume are reported. The range of shear rate $ \dot \gamma $ investigated is between 0.1 and 1000 sec^?1. The solutions show shear-thinning behavior, and the relative viscosity η_r of the slurries generally decreases with increasing shear rate. The results indicate two different types of mechanism, respectively at high and low shear rates. At low $ \dot \gamma $, the relative viscosity can be correlated extending relations already well known for suspensions in Newtonian liquids which are based on the mechanism of aggregate disruption. The behavior at high $ \dot \gamma $ values is believed to be due to the influence of the filler on the flow properties of macromolecules, in particular on relaxation time. Through a shifting procedure, an increase in relaxation time which depends on filler content and not on polymer concentration is shown.     

Jet penetration measurements in a venturi scrubber

The maximum centreline penetrations, l^**, of cross-current liquid jets in a Venturi scrubber were measured for orifice diameters, d, of 1.397, 2.108, 2.565 and 3.860 mm. The data are correlated by for the range of conditions, 36 ≤ gas throat velocity V_g ≤ 125 m/s; 1.2 ≤ liquid injection velocity V_j ≤ 18 m/s; 0.06 ≤ liquid to gas ratio      

Experimental Correlation between Mechanical Non‐Linearity in LAOS Flow and Capillary Flow Instabilities for Linear and Branched Commercial Polyethylenes

An experimental correlation between the non‐linear behaviour of commercial polyethylene melts in LAOS flow, and the pressure fluctuations associated with melt flow instabilities developed in capillary rheometry are presented. Polyethylene melts with enhanced non‐linear behaviour under LAOS conditions present larger pressure fluctuations during capillary extrusion, and consequently, larger surface distortions on the extrudate. The combination of both methods can be a tool to predict the development of melt flow instabilities in the extrusion process of polyethylene melts, and can elucidate their correlation with material structural properties ( , MWD and topology).

     

The relation between melt flow properties and molecular weight of polyethylene

The non-Newtonian behavior of commercial linear polyethylene samples and their fractions were studied at 190°C. The viscosity η versus shear rate \documentclass{article}\pagestyle{empty}\begin{document}$ \dot \gamma $\end{document} curves of whole polymers could be superimposed onto a single master curve despite the variations of their molecular weights and molecular weight distributions. For fractions, however, the same master curve was inapplicable, and the sensitivity of the viscosity to shear rate was found to be greater than those of the whole polymers. The zero-shear viscosities η₀ of fractions were related to the 3.42 power of the weight-average molecular weight M_u as follows: For whole polymers, the zero-shear viscosities were found to be considerably higher at the same M_w and markedly lower at the same z-average molecular weight M_z than those of the fractions. Thus, it was concluded that η₀ corresponds to an average of molecular weight between M_w and M_z. It was found that the molecular relaxation time τ is proportional to M_z^5.3 for whole polymers and to η₀M_w for fractions. Using these relations it was possible to relate the flow ratio, the ratio of flow rates at two different shear stresses, with the molecular weight distribution.     

Drainage in thin planar non-newtonian fluid films

Equations for the radial and linear drainage of non-Newtonian fluids in horizontal and inclined films are presented. For a power law fluid with index m, the variation in dimensionless film thickness Δ with dimensionless time T is given by: where Δ and T are appropriately defined for drainage in radial horizontal and linear inclined films. The corresponding approximate expression for a Bingham plastic fluid is: in which A is the minimum film thickness defined appropriately at the asymptotic limits when Δ » A and Δ ? A.     

Studies on melt spinning. III. Velocity field within the thread

The velocity field within a molten spinning thread was analyzed quantitatively by solving the equations of continuity and momentum for Newtonian liquids. In solving the equations, the viscosity was assumed known and was given by the expression where x and r are distances in cylindrical coordinates. A series solution in velocity v having the expression was obtained when several simplifying assumptions were made on the equations. The series solution was found to converge when cr² < 1 is satisfied. μ₀e^βx and ν₀e^αx above are tangents on semilog paper at x = x to the macroscopic viscosity and velocity profiles μ(x) and ν(x) which were computed separately by means of a technique developed previously by the author.^1,2 The value c was derived from the temperature profile across the thread at x = x computed separately using another technique developed by the author.³ The above series solution showed numerically that under most conceivable spinning conditions the velocity field within the thread is for practical purposes flat across the thread and, in addition, purely extensional.     

Kinetics of ion exchange in monodisperse resin

Ion-exchange kinetics within a conventional strong base resin, Dowexl-8X®, and a resin with uniform particle size, Dowex® Monosphere® Tough Gel® TG550A®, were investigated using neutron activation analysis and radio-tracer techniques. The kinetics of ion exchange were measured in a batch and in a “shallow-bed” flow system. The experimental data were compared with the results of model computations. The diffusivities of several anions within TG550A and Dowexl-8X were deduced. It was found that at 25°C Br^-, Cl^-, OH^-, and Na⁺ diffuse within TG550A with the diffusion coefficients = 6.0 × 10^-7 cm²/s, = 1.2 × 10^-6 cm²/s, = 7.0 × 10^-8 cm²/s, and = 5 × 10^-7 cm²/s. Diffusion of anions within a conventional resin, Dowexl®, was slower: = 3.5 × 10^-7 cm²/s, = 6 × 10^-7 cm²/s, = 2.7 × 10^-8 cm²/s, and = 5 × 10^-7 cm²/s. A higher rate of ion diffusion and the bead-size uniformity may make monodisperse Dowex Monosphere Tough Gel TG550A resin attractive for analytical applications. The difference in properties between conventional and monodisperse resins is not sufficient to affect the large volume applications of resins. © 1997 John Wiley & Sons, Inc. J Appl Polm Sci 65:1271–1283, 1997     

High speed wire coating by withdrawal from a bath of viscoelastic liquid

Data are presented for the thickness H_∞ of liquid coating entrained by continuous withdrawal at speed U of a wire of radius R from the free surface of a large bath. For Newtonian fluids of viscosity μ, density ρ and surface tension σ, the data are carried out to coating speeds beyond the applicability of current theories, to Capillary numbers of nearly one hundred. In the high speed range the data, which cover several orders of magnitude in viscosity, can be well represented by the equation for U_μ/σ = N_ca > 3. All data presented are at an essentially constant Goucher number of 0.08, where N_Go ≡ R(ρg/2σ)^1/2. Data for viscoelastic fluids show phenomena quite distinct, qualitatively and quantitatively, from Newtonian observations. In particular, strongly elastic fluids show a markedly reduced ability to be entrained onto the wire. Further, the coating thickness appears to become independent of Capillary number at high speed.