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An investigation has been carried out to determine the constants that characterize the flow properties of a non-newtonian liquid which also possesses a wall effect, such that these constants can be used to predict pressure drop or flowrate for engineering purposes. It has been found that if the wall effect is neglected, as is usually the case, then the errors involved in a design calculation can be considerable. A relationship has been derived between pressure drop and flowrate for this type of liquid, on the assumption that it behaves as a generalized-power-law liquid, using four constants which can be determined either on a rotational viscometer or in a tube viscometer. The results as computed from the relationship have been compared with experimental results and have been found to agree to well within acceptable tolerances for engineering applications.

Fully developed steady flows of purely viscous non-Newtonian fluids with viscous heat generations are usually governed by non-linear Sturm-Liouville differential system. It is shown that under certain conditions that the non-linear eigenvalues of the system take a critical value above which no local solutions are obtainable. The theory is applied to specific flow situations and an upper bound for this critical parameter is evaluated in each case. It is shown that the non-linear heat generation term is responsible for double valued pressure gradient flow rate characteristics in the models of screw extruder systems. Computed values of the upper bound of the critical parameter are given for a specific model of a screw extruder.

A unique shear stress-shear rate relationship exists for laminar flow of any time independent substance in a tube, whereas this is not the case for turbulent flow. In order to obtain a unique relationship for turbulent flow, a new approach based on the elementary theoretical interpretation of experimental data is adopted in the present paper. In particular, wall shear stress is found to be a unique function of a new turbulent pseudo shear rate term. In this relationship there are two parameters which characterize a given substance — the limiting viscosity at high shear rateµ m and a factorα m which takes into account modification of turbulent structure by the non-Newtonian properties. Both of these parameters must be determined experimentally. Methods of predicting pressure gradients and of scaling up are outlined. In applying the approach to suspensions in which the solid phase has a density greater than that of the liquid medium, it may be important to determine the increment in shear stress equivalent to the energy required to maintain the solid particles in suspension. The validity of this approach is confirmed by data for the flow of a variety of substances including kaolin suspensions and Carbopol solutions in tubes ranging in diameter from 1.5 to 20 mm.

Transitional region of pipe flows of dilute polymer solutions is investigated. Experiment is made with HEC water solutions and the following results are obtained. Anomaly of the resistance coefficient is very small in the flow region before transition, though it is comparatively large for PEO solutions as reported in the previous paper. In this flow region a kind of fluctuations of static pressure is recognized and once the fluctuations occur, the appearance of issuing jets shows some anomalous winding effects. An approximate analysis is made on the anomaly of the resistance coefficient in the flow region before transition. The results show that the resistance coefficient becomes larger than that of Poiseuille flow and that if Reynolds numbers and the fluctuations are constant, the rate of increase of the resistance coefficient enlarges as Weissenberg number increases. Comparison between analytical results and experimental data shows an approximate agreement.

Equations are derived to predict the one-dimensional behavior of a hyperbolic tangent fluid model flowing between calendering rollers. The equations were solved numerically on a computer, and results were obtained for maximum roller pressure and the exit thickness of the fluid sheet as a function of roller speed and reservoir size for three polymers. The numerical integration employed Gauss-Legendre quadrature and a cascading iteration scheme to solve equations in which unknowns appeared in the limits of the integrals. Unlike existing power law solutions, these results show a definite relation between roller speed and exit thickness.

Turbulent jets of water, issuing from a nozzle at Re above 7000, remain coherent for some distance. The surface of the jet is covered with protrusions which are caused by the eddies in the turbulent liquid. In pure water, if Re is not too high, the surface tension, acting over the curvature of each protrusion, balances the “dynamic thrust” of the eddies. At very high Re, however, the eddies have enough thrust to splash through the surface of the jet: droplets are then visible. When “Polyox” is present in the jet liquid, however, elastic forces are added to those of surface tension in restraining the eddies. Mass transfer of CO2 into the jet liquid is reduced at moderate Reynolds numbers, and at higher velocities the formation of droplets can be strongly inhibited. We interpret these results quantitatively in terms of the rates of stretching of the protrusions and the elongational viscosity (μel) of the non-Newtonian solutions.

The steady heat transfer in the incompressible laminar boundary layer flow of power law fluids past a wedge having a step discontinuity in surface temperature is studied analytically. The solution method recently introduced by Chao and Cheema is extended to the present problem. The solution is obtained in terms of an infinite set of similar solutions which are expressible as universal functions. Tabulations of such functions are given. For most of the non-Newtonian fluids commonly encountered having large Prandtl numbers, only a very few terms of the functions are needed to yield accurate results.

An incompressible isotropic visco-elastic liquid is bounded by an infinite rigid horizontal disk atz=0. Both the fluid and the disk are in a state of solid body rotation with a uniform angular velocity Ω about thez-axis. At timet=0+, small amplitude non-torsional oscillations are superimposed on the disk, or the disk is impulsively moved with a constant acceleration so that an unsteady motion is set up in the liquid. An analysis is made of the unsteady flow generated in the visco-elastic liquid. The velocity field is calculated by using the Laplace transform treatment and the structure of the associated boundary layers is determined. This analysis provides the existence of Stokes-Ekman-Elastic boundary layers of thicknesses of the order\(\left( {\frac{v}{\Omega }} \right)^{\tfrac{1}{2}} \frac{1}{{\alpha _r }}, r = 1,2\). Special emphasis is given to the limiting behaviour of the solution ast→∞, and the significant interaction of the elastic parameter and rotation is examined. The surface traction at the disk is found and the effects of elasticity on this quantity are discussed. It is shown that in the absence of the elasticity, the results of this paper reduce to the corresponding results of the Newtonian rotating fluid.

Flow Properties of Polymer Solutions. III. Non-Newtonian Viscosity and Relaxation Mechanisms with Long Relaxation Times

or *-dall/dt (27) Equation (27) correlates the derivative of the normal stress with respect to the plastic strain, along with the time rates of stress and strain. The time rate of strain can be considered to be a parameter specifying the environment. Let e11 = A = const A(dall/dell) = 243wr11' l (dcrll/dt) Let

Diverging radial flow between parallel plates was studied under isothermal steady‐state conditions. Flow rates Q and radial pressure distributions p(r) were measured for one Newtonian liquid and three non‐Newtonian polymer solutions. The powerlaw viscosity model, with parameters evaluated from cone‐and‐plate viscometry, was remarkably good in fitting Q(Δp) data and, in most cases, also for describing p(r). Thus, for these polymer fluids with only moderate viscoelasticity (Weissenberg number

Fur den Fall sechseckigen Umrisses einer Konvektionszelle endlicher Lange werden die Grundgleichungen hergeleitet, die die Warmekonvektion in einer nichtnewtonschen Flussigkeit beschreiben. Ihre numerische Losung zeigt im Falle zweier freier Rander, das die viskoelastischen Effekte die Nusselt-Zahl nur wenig andern. Immerhin ist die Nusselt-Zahl bei gegebener Rayleigh-Zahl durchweg — wenn auch nur wenig — groser als bei einer newtonschen Flussigkeit vergleichbarer Viskositat. Diese theoretischen Voraussagen stehen im Einklang mit neueren experimentellen Ergebnissen im Falle zweier starrer Rander. The fundamental system of equations describing finite amplitude steady convection with hexagonal planform in a non-Newtonian fluid have been derived. The numerical solution of this system, in the case of two free boundaries, reveals that there is little change in the Nusselt number due to visco-elastic effects. However, the Nusselt number at any given Rayleigh number is consistently higher, although only slightly, than that of a Newtonian fluid of comparable viscosity. These theoretical predictions are in line with the recent experimental findings in the case of two rigid boundaries.

Toothpastes which are considered to be concentrated suspension, are consisted of fine solid particles (abrasives), water soluble gums (binders), surface active agents, polyol (humectants), water and others. Only a couple of papers on rheological properties of toothpastes have been published so far, furthermore those results have not been used effectively in practical production of toothpastes. The difficulties of rheological studies of toothpastes are that, (1) they belong to quasi-stable system (2) they are highly concentrated suspension and contain a lot of fine solid particles, and (3) they are so-called multi-components suspension system. For present study, rotational coaxial cylinder viscometer (Thixotrometer) was used to characterise the rheological properties of the water soluble gums, such as carboxymethyl cellulose and carrageenan.Their flow curves were analysed by means of power law equation, and the discussions on their flow constants such as K and γ on activation evergy were made. Both samples showed pseudplastic flow, however carrageenan had more complicated flow characteristics than carboxymethyl cellulose. Non newtonian index, γ, of carrageenan ranged 0.19-0.38 and that of carboxymethyl cellulose 0.40-0.66 respectively. The analysis of non-Newtonian index, activation evergy, dynamic modulus and dynamic viscosity indicated that carrageenan had more complicated structure than carboxymethyl cellulose.

The peristaltic motion of a non-Newtonian liquid with a power-type rheological law in a slot channel with elastic, deformable walls is considered. Using the asymptotic method of “narrow bands”, the velocity distributions in the channel are derived in terms of certain defining parameters.

An analysis of the laminar-turbulent transition behavior of power law non-Newtonian fluids with yield stresses reveals that considerable interaction occurs between the characterizing parameters m and Hedstrom number for low power law indexes, m. In particular, for m < 0.4 small yield stresses tend to destabilize the flow whereas large ones stabilize it. The results of this analysis suggest that in order to obtain correct predictions and interpretations of transitional and turbulent flows of nonNewtonian fluids, a correct modeling of the viscous rheological behavior of the fluid is essential. Nomenclature A = a dimensionless parameter defined by Eq. (6) D - pipe diameter / = friction factor He - Hedstrom number, defined by Eq. (17) K = laminar-turbulent transition parameter K = maximum value of K m = power law index in rheological model Re - Reynolds number,.defined by Eq. (8) r = radial coordinate u = v/(u}, dimensionless velocity v = velocity (v} ~ mean velocity T = -du/d£ j] = consistency index in rheological model k = r/rw £ = value of £ where K = K £o = TQ/TW p — fluid density a = dimensionless parameter defined by Eq. (7) r = shear stress TO = yield stress in rheological model

The present paper is a further development in our effort to describe non-Newtonian flow behavior. As in the past, the thixotropic shear stress or shear rate is characterized by the kinetic theory and currently a modified form ofOldroyds model provides the elastic contribution. Presented here are further modifications of the model in order to predict the first normal stress difference and logical changes to make the model more internally consistent.

The effects of stepped film-thickness on the various characteristics of squeeze films, conical bearings and hydrostatic step seals, using a power law fluid as lubricant have been investigated. It has been shown that the load capacities of squeeze film bearings decrease and those of conical bearings and step seals increase, with the increase in the step height. With a hydrostatic step seal, the load capacity increases as the flow behaviour index of the power law fluid increases.

Theoretical relations are derived for heat transfer into or from turbulent falling films of liquid, flowing both over smooth and over very rough surfaces. Heat transfer from turbulent falling films over smooth plates and ridged plates has been measured, for pure water. The enhancement of hL by the ridges can exceed two times, with suitable dimensions of the ridges. Our results are compared with those of previous workers. Our theory predicts heat transfers in fair accord with our experiments. The addition of ‘Separan’ to render the water non-Newtonian decreases hL for both the smooth and the rough plates. But the enhancing effect of the roughness remains about the same as for pure water. The addition of a surface-active agent to the water usually decreases hL somewhat, though very low concentrations of agent at low angles of inclination can increase hL.

The steady laminar boundary layer flow along a semiinfinite flat plate parallel to the main stream velocity is considered when the shear stress is assumed to be proportional to then-th power of the velocity gradient wheren

Study of the dependence on concentration of the reduced viscosity of carboxymethylcelluloses (CMC), in the absence of added salts and down to relative viscosity 1·-01, leads to two typical parameters of the polyion. The molecular dimensions of H-CMC and Na-CMC have been determined from these two parameters using a rod-like model. The gradient dependence of viscosity has been studied; the two samples of CMC in water (M = 41,000 and 180,000) show the non-Newtonian behaviour of solutions of rigid particles.

The system of hydrodynamic equations describing the spreading of a thin film of a non-Newtonian liquid between two solid colliding surfaces is solved. The temperature profiles along the film radius and thickness are found under the assumption of adiabatic compression.

Theoretical investigations of viscous heating in the flow of fluids with an exponential dependence of viscosity on temperature have shown that, for a given shear stress, two shear rates are possible. Above a critical value, the stress decreases as the shear rate increases. The present work is an experimental study of this phenomenon in plane and circular Couette flows and in cylindrical Poiseuille flow. ArochlorR 1260, a high viscosity Newtonian fluid with an extremely sensitive viscosity-temperature dependence is used as the test fluid. The results clearly show that two shear rates for Couette flow exist for one measured wall shear stress. Because of the viscosity-pressure dependence of the fluid, the Poiseuille flow results are inconclusive.

An experimental investigation of local free convection heat transfer rates from a horizontal, isothermal cylinder to non-Newtonian, power-law fluids has been conducted. Experimental data were obtained employing a cylindrical test section 2.304 in, dia and 8.018 in. long, constructed of twenty aluminum segments each independently, electrically heated. Non-Newtonian fluids utilized included four aqueous, carboxypolymethylene solutions. Both local and average Nusselt number results were compared with theoretical and empirical models.

The gradient dependence of viscosity was studied for fractionated cellulose in cadoxene and in cadoxene/water (1:1). The present experiments were performed in the shear stress range 1,4 to 1 000 dyn/cm2 (1,4.10−5 to 10−2 newton/cm2). The experimental results, viewed in the light of various theories, show that the excluded volume effects and the internal viscosity play an important role in non-Newtonian behavior of cellulose solutions. Es wurde die Schergradientenabhangigkeit der Viskositat von fraktionierter Cellulose in Cadoxen und in Cadoxen/Wasser (1:1) untersucht. Die vorliegenden Experimente wurden im Scherspannungsbereich von 1,4 bis 1 000 dyn/cm2 (1,4.10−5 bis 10−2 Newton/cm2) durchgefuhrt. Die experimentellen Ergebnisse werden mit verschiedenen Theorien verglichen und zeigen, das Effekte des ausgeschlossenen Volumens und der inneren Viskositat eine wichtige Rolle im nicht-newtonschen Verhalten von Celluloselosungen spielen.

Non-Newtonian fluid flow in an eccentric annulus was studied for high polymer aqueous solutions of CMC, HEC and MC. The macroscopic relations between the pressure drop and the flow rate were found to be in fairly good agreement with the results obtained by application of the conventional variational principle. Velocity profiles were measured by a hydrogen bubble method in which photographs were taken of the hydrogen bubbles. The authors observed very interesting secondary flows in the eccentric non-Newtonian fluid flow which may be due to the viscoelastic effect of the high polymer aqueous solution. The authors could not observe such secondary flows for non-Newtonian fluids in a concentric annulus or Newtonian fluids in an eccentric annulus. The authors present an equation giving the relation between the flow rate and the pressure drop for the flow of a non-Newtonian fluid in an eccentric annulus in terms of the experimental data for flow in a circular tube.

Die vorliegende Arbeit bringt eine theoretische Untersuchung der laminaren Grenzschichtstromung einer dem Potenzgesetz gehorchenden nichtnewtonschen Flussigkeit hinter einer beliebigen zweidimensionalen Flache. Es wird eine Form der Grenzschichtgleichung hergeleitet, die in der Hinsicht universell ist, das weder die Gleichung noch die Randbedingungen von den besonderen Bedingungen des Problems abhangen. Dies wird dadurch erreicht, das in die Zahl der unabhangigen Variablen eine Schar von Parametern einbezogen wird, die den Einflus der fur ein spezielles Problem charakteristischen Bedingungen ausdrucken. Die universelle Gleichung wird durch eine Parameterapproximation numerisch gelost. Mit Hilfe dieser Losung werden mehrere spezielle Probleme im Detail diskutiert. This paper presents a theoretical analysis of the laminar boundary layer flow of a non-Newtonian fluid, obeying the power-law model, past an arbitrary two-dimensional surface. The universal form of the boundary layer equation, such that neither the equation nor the boundary conditions depend on the particular conditions of the problem, is derived. This universality is achieved by introducing a set of parameters expressing the influence of external conditions characteristic of a particular problem into a number of independent variables. The universal equation is solved numerically by a parameter approximation. Using this solution several particular problems are analysed in detail.

The problem of an elastico-viscous fluid resting on a plate which moves with a time dependent velocity in its own plane (along negative x direction) and rotating with a constant velocity Iomega; along with the fluid as a rigid body has been discussed, It is found that the elastic property of the fluid increases the drag and the lateral stress on the plate, The rotation introduces fluid motion in the y direction, For a fixed time and distance from the plate, the velocity fluctuates with decreasing magnitudes with increasing Iomega;, with the result that the boundary layer thickness reduces as Iomega; increases, The stresses increase parabolically with respect to Iomega;, The solution for an arbitrary velocity of the plate is also presented, In case of impulsive flows a secondary boundary layer whose thickness is of order √(vt) (v is the kinematic viscosity) is found to develop for very short times

The relation between shear stress and shear rate has been determined for elastohydrodynamic oil films. At low values the rate of shear is directly proportional to the shear stress, but at higher values the shear rate increases more rapidly than the stress. It is shown that the critical factor is the magnitude of the shear stress, not the shear rate, and that this critical magnitude depends upon the pressure and the molecular size. Above the critical stress, in the non-Newtonian region, the shape of the curve relating the stress to the rate of shear depends upon the distribution of the sizes of the molecules in the oil. It is shown that in elastohydrodynamic conditions the limits of Newtonian behaviour are frequently exceeded and that this is liable to influence the pressure distribution, the magnitude of the traction, the generation of heat, and, at high speeds, the value of the film thickness.

A method for the determination of power consumption of industrial agitators in non-Newtonian fluids on the basis of measurements on a model device is presented. The applicability of this method for purely viscous, time dependent and viscoelastic fluids is discussed. The method was tested experimentally using agitators of two different sizes and results of measurements confirmed the applicability of the present method for engineering calculations.

A model is developed for the phenomenon of non-Newtonian secretion in tubes. The motivation for this study is the problem of glandular secretion, particularly in the pancreas. Power-law fluids are considered in some detail, as being biologically appropriate. It is found that for a power-law fluid whose exponent is less than unity (the biological case) two types of flow occur. For a sufficiently high secretion pressure, all of the tube is used for secretion, and a nonlinear pressure profile results. Numerical solutions are obtained for the pressure and rate of efflux. When the secretion pressure parameter falls below a certain critical value, the upper end of the tube begins to be choked off, only part of the tube being used for secretion. This phenomenon does not occur for exponents greater than or equal to unity. Physiological implications are considered, and a qualitative discussion given for the case of non-power-law fluids.

Viscosities calculated from immiscible displacement of Newtonian liquids were found to be 10% lower than the reported values for these materials. This is in agreement with Taylor's [ J. Fluid Mech. 10 , 161 (1961)] results in which it was found that the fraction of fluid left in the tube reached a limiting value of 0.55. If the velocity of the meniscus was twice the average fluid velocity this value would be 0.50 and there would be perfect agreement between the immiscible displacement technique and conventional viscometers. At low stresses immiscible displacement data may exhibit a small degree of curvature that is not evident in the cone and plate results. While the viscosity was found to be independent of tube diameter, this slight discrepancy between the two methods is attributed to surface tension, and when measurements were limited to stresses for which T > 3 γ / R immiscible displacement of Newtonian liquids produced linear consistency relations. Comparative rheological measurements of several non-Newtonian liquids indicate that apparent viscosities calculated by immiscible displacement are equivalent to those obtained with a cone and place viscometer if the data are interpreted in terms of the maximum shear stress and shear rate. While it is recommended that measurements be limited to stresses where surface tension effects can be neglected (i.e., T > 3 γ / R ), the two methods generally differed by less than 8% even at low stresses (i.e., 0.3γ/R) as long as the immiscible displacement data did not exhibit a measurable effect of tube size on apparent viscosity. These results, together with the proven ability to detect flow anomalies such as wall slip, adds considerably to our confidence in immiscible displacement as a rheological technique.

Polysaccharide was synthesized by Aureobasidium pullulans (or Pullularia pullulans) 2552 in a sucrose medium. The field apparent viscosity of the culture medium from shake flask experiments rose to 24,500 cP and then dropped toward its initial value as the fermentation progressed. The magnitude of the maximum apparent viscosity depended on the initial pH of the fermentation broth. The inoculum age influenced the cultivation period before which the maximum viscosity was reached. Rheograms of the fermentation broths showed a change in viscosity behavior from Newtonian to pseudoplastic, and then toward Newtonian characteristics during the fermentation. The calculated non-Newtonian index was found to be a sensitive factor for the indication of the non-Newtonian behavior. Such behavior could not be detected from rheograms. Viscosity profiles of polysaccharide isolated from various stages of the fermentation showed a change from Newtonian to pseudoplastic behavior depending on the concentration (0–2%) of polysaccharide.

Viscosity measurements in a wide shear rate range, on suspensions of six Newtonian and non-Newtonian liquids, are reported. The shear rate dependency of suspensions in non-Newtonian liquids is compared with that of suspensions in Newtonian ones and an extension to the former of the Krieger—Dougherty theory, which holds for the latter, is attempted. Although some success is obtained, an alternative explanation of the observed behavior of the suspensions of non-Newtonian liquids is discussed.

It has been experimentally confirmed that the method of calculating the shear rate dependence of the normal stresses from the flow curve, proposed in [1], gives good results consistent with the experimental data. A series of simple relations proposed for estimating the initial normal stress coefficient are compared with the experimental results and it is shown that there is good agreement between calculation and experiment. In the linear region the Lodge relation, which equates the high-elastic strains to half the ratio of the normal to the shear stresses is satisfied.

A theory of the viscosity of raw rubbers is developed on the postulate that raw rubbers, when subjected to continuous shear in the non‐Newtonian region of flow, consist of microscopic rheological units which are semipermanent aggregates of many rubber molecules [M. Mooney and W. E. Wolstenholme, J. Appl. Phys. 25, 1098 (1954)]. The theory treats these units as tacky, elastic solids, whose relative movement and slippage constitute the macroscopic flow of the rubber. The resulting theoretical flow curve differs little from that given by H. M. Smallwood [J. Appl. Phys. 8, 505 (1937)], but the interpretation of the parameters is radically different. The new theory leads to expressions for the number of temporary point attachments per unit area between two touching rheological units and for the mean life of these attachments.Agreement with experimental data is slightly better than in Smallwood. Two types of observed deviation from the theory can be interpreted as due to two plausible phenomena not included in ...

The flow of non-Newtonian liquid near a rotating disk has been discussed by using second order stress strain velocity relations of classical hydrodynamics. It is found that the effect of cross-viscosity depends on a non-dimensional number Rc. The boundary layer thickness decreases and the dimensionless moment coefficient increases with the increase of Rc.