On the stability of elastico-viscous flow between roating cylinders

Summary Thomas andWalters (9, 2, 3) have recently shown that for a certain elastico-viscous liquid, the presence of elasticity has a destabilizing effect on the flow. The problem is reconsidered in the present paper for another idealized elastico-viscous liquid which is as simple as that considered byThomas andWalters. It is shown that in this case, the presence of elasticity has a stabilizing influence. It is concluded that a stability experiment would be a useful addition to the experiments usually set up to characterise elastico-viscous liquids.     

On the stability of elastico-viscous flows

Summary The stability ofCouette flow and planePoiseuille flow of an elastico-viscous liquid is analyzed. It is found that in both cases the flow is less stable than that of Newtonian liquids. The critical wave number is also changed due to the elasticity of the liquid. These changes depend on the type of flow under consideration.This work forms a part of the D. Sc. thesis of the first author. The work was supported by the United States Office of Naval Research under Contract N 62558-4093 and by the Technion Research Found.     

Heat transfer by laminar flow of an elastico-viscous fluid in posttreatment analysis of wire coating with linearly varying temperature along the coated wire

The present study focuses on the heat transfer by the laminar flow of an elastico-viscous fluid in posttreatment of wire coating analysis with linearly varying temperature on the surface of coated wire. The surface of wire (uncoated) and the surface of coated wire were subjected to two thermal boundary conditions. The constitutive equation of motion and equation of energy have been solved by using perturbation theory for velocity, pressure distribution along the radial direction and temperature distribution. The theoretical analysis of flow rate, average velocity, shear stress, thickness of coated wire, and force on the total wire were also derived. Moreover, the flow phenomenon has been studied under the influence of elastic number R _e velocity ratio U and the dimensionless number S in the ranges 0?≤?R _e ?≤?20, 0.2?≤?U?≤?1.4 and 0?≤?S?≤?20. We noticed that with the increase in elastic number R _e velocity decreases whereas thickness of the coated wire and force on the total wire increases. Also temperature profile decreases with the increase of non-dimensional parameter S.     

Secondary flows in pipes of square cross-section and the measurement of the second normal stress difference

In this paper, we are concerned with the possibility of determining the second normal stress difference v₂ for elastic liquids from a study of the direction and strength of the secondary flows in pipes of non-circular cross-section.Experimental results show that the direction of the secondary flows in pipes of square cross-section is the same for six different elastico-viscous systems, although there is some evidence to suggest that the sign of v₂ may not be the same for all the test liquids. The situation is clarified by a theoretical analysis which involves the numerical solution of coupled partial differential equations. From the analysis, it is concluded unequivocally that the direction of the streamlines is no indication of the sign of v₂.     

Edge fracture in cone-plate and parallel plate flows

Edge fracture is an instability of cone-plate and parallel plate flows of viscoelastic liquids and suspensions, characterised by the formation of a `crack' or indentation at a critical shear rate on the free surface of the liquid. A study is undertaken of the theoretical, experimental and computational aspects of edge fracture. The Tanner-Keentok theory of edge fracture in second-order liquids is re-examined and is approximately extended to cover the Criminale-Ericksen-Filbey (CEF) model. The second-order theory shows that the stress distribution on the semi-circular crack is not constant, requiring an average to be taken of the stress; this affects the proportionality constant, K in the edge fracture equation −N _2c = KΓ/a, where N _2c is the critical second normal stress difference, Γ is the surface tension coefficient and a is the fracture diameter. When the minimum stress is used, K = 2/3 as found by Tanner and Keentok (1983). Consideration is given to the sources of experimental error, including secondary flow and slip (wall effect). The effect of inertia on edge fracture is derived. A video camera was used to record the inception and development of edge fracture in four viscoelastic liquids and two suspensions. The recorded image was then measured to obtain the fracture diameter. The edge fracture phenomenon was examined to find its dependence on the physical dimensions of the flow (i.e. parallel plate gap or cone angle), on the surface tension coefficient, on the critical shear rate and on the critical second normal stress difference. The critical second normal stress difference was found to depend on the surface tension coefficient and the fracture diameter, as shown by the theory of Tanner and Keentok (1983); however, the experimental data were best fitted by the equation −N _2c = 1.095Γ/a. It was found that edge fracture in viscoelastic liquids depends on the Reynolds number, which is in good agreement with the inertial theory of edge fracture. Edge fracture in lubricating grease and toothpaste is broadly consistent with the CEF model of edge fracture. A finite volume method program was used to simulate the flow of a viscoelastic liquid, obeying the modified Phan-Thien-Tanner model, to obtain the velocity and stress distribution in parallel plate flow in three dimensions. Stress concentrations of the second normal stress difference (N ₂) were found in the plane of the crack; the velocity distribution shows a secondary flow tending to aid crack formation if N ₂ is negative, and a secondary flow tending to suppress crack formation if N ₂ is positive. Received: 4 January 1999 Accepted: 19 May 1999     

On the problem of oscillatory laminar flow of elastico-viscous liquids in channels

Summary A possibility is shown for an exact solution by means of tabulated functions of the problem about the velocity profile of oscillatory flow of elastico-viscous liquids in channels with annular cross-section.

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Zusammenfassung Es wird gezeigt, wie man mit Hilfe tabulierter Funktionen das Geschwindigkeitsprofil in einem Rohr mit Kreisring-Querschnitt für die oszillierende Strömung einer viskoelastischen Flüssigkeit bestimmen kann.

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With 1 figure     

The oscillation of a spheroid along its axis in a non-Newtonian fluid

Summary The present paper investigates the nature of the fluid flow when a spheroid is suspended in an infinitely extending elastico-viscous fluid defined by the constitutive equations given byOldroyd orRivlin andEricksen, and is made to perform small amplitude oscillations along its axis. The solution of the vector wave equation is expressed in terms of the solution of the corresponding scalar wave equation, without the use ofHeine's function or spheroidal wave functions. Two special cases (i) a sphere and (ii) a spheroid of small ellipticity, are studied in detail.Née MissRenuka Rajagopalan.     

On unsteady flow of an elastico-viscous fluid past an infinite plate with variable suction

Approximate solutions of the Navier-Stokes equations are derived through the Laplace transform for two dimensional, incompressible, elastico-viscous flow past a flat porous plate. The flow is assumed to be independent of the distance parallel to the plate. General formulae for the velocity distribution, skin friction and displacement thickness as functions of the given free stream velocity and suction velocity are obtained. The response of skin friction to the impulsive perturbations in the stream and suction velocities is studied. It is found that the order of singularity in the skin friction at t=0 increases due to the elastic property of the fluid in the impulsive case. When the stream is accelerated the skin friction still anticipates the velocity but the time of anticipation is reduced from 1/4 to (1/4) (1—k), where k is the elastic parameter of the fluid. It is found that in general the resistance of the elastico-viscous fluids to an impulsive increase in the stream velocity is greater than the viscous fluids, the elasticoviscous fluids also reach the steady state earlier than the viscous fluids.     

Normal stress difference in liquid lubricants sheared under high pressure

Although normal stress differences in liquids have conventionally been associated with polymers, aspects of rheological behavior in lubricated concentrated contacts suggest that normal stress difference may be significant in even low molecular weight liquids sheared under high pressure and high shear stress. A torsional flow rheogoniometer was constructed for use at high (300 MPa) pressure. Four typical liquid lubricants were investigated, including one polymer/mineral oil solution. Shear stress and N ₂-N ₂ are reported as functions of shear rate. The effect of pressure variation is reported for two liquids. Results are compared with predictive techniques and a molecular dynamics simulation. Simple low molecular weight lubricant base oils can generate measurable and significant normal stress differences when sheared at high shear stress.     

Linear and rotatory oscillations of a sphere in infinite expanse of elastico-viscous fluids

Summary A number of rotatory oscillating systems have been used in literature in the study of elastico-viscous fluids.Frater suggested another type of oscillating system for the same. In this system a sphere is suspended in an infinite expanse of fluid and is made to perform small linear oscillations about a mean position. In the present paper, we have considered the rotatory and linear oscillations of a sphere in an infinite mass of elasticoviscous fluids. The general solutions obtained cover almost all types of elastico-viscous models, includingWalters liquidB, Oldroyd type fluids and the Second order fluids. Considering two different representations of the same dilute polymer solution, we arrive at such results which make us to suspect whether the experimental values of the material constants obtained for a certain polymer solution through rotatory oscillating experiments can be used to assess or predict correct results for linear oscillatory flow situations.     

Die simultane Translations- und Rotationsbewegung einer Kugel in einer elastoviskosen Flüssigkeit

Zusammenfassung Es wird gezeigt, daß die Messung von Kraft und Moment bei der simultanen Translations- und Rotationsbewegung einer Kugel in einer allgemeinen inkompressiblen elastoviskosen Flüssigkeit die vollständige Bestimmung der Stoffkonstanten der rheologischen Zustandsgleichung bis zur dritten Ordnung ermöglicht. Diese Untersuchung, welche eine Verallgemeinerung des bekanntenStokesschen. Problems auf eine elastoviskose Flüssigkeit bedeutet, wird mit Hilfe eines Störungsverfahrens durchgeführt und bedient sich der Methode der vektoriellen Kugelfunktionen. Die Ergebnisse werden an Hand zweier Modellflüssigkeiten diskutiert. Die Beziehung zu Untersuchungen anderer Autoren wird aufgezeigt.

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Summary It is shown that measuring of the frictional force and couple in the simultaneous translational and rotational motion of a sphere in a general incompressible elastico-viscous fluid allows complete determination of the material constants of the rheological equation of state up to the third order. This investigation, which means a generalisation of the well knownStokes problem with respect to an elastico-viscous fluid, is executed by means of a perturbation method and uses vector spherical harmonics. The results are discussed in terms of two model fluids. Relations to papers by other authors are established.

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Vortrag, gehalten am 14. April 1963 auf der Rheologentagung in Bad Oeynhausen.

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Zum 100jährigen Jubiläum der Farbenfabriken Bayer AG.     

Thermal radiation effects on MHD flow past a semi-infinite inclined plate in the presence of mass diffusion

MHD mixed free-forced heat and mass convective steady incompressible laminar boundary layer flow of a gray optically thick electrically conducting viscous fluid past a semi-infinite inclined plate for high temperature and concentration differences is studied. A uniform magnetic field is applied perpendicular to the plate. The density of the fluid is assumed to reduce exponentially with temperature and concentration. The usual Boussinesq approximation is neglected due to the high temperature and concentration differences between the plate and the ambient fluid. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The boundary layer equations governing the flow are reduced to ordinary differential equations, which are numerically solved by applying an efficient technique. The effects of the density/temperature parameter n, the density/concentration parameter m, the local magnetic parameter M_x and the radiation parameter R are examined on the velocity, temperature and concentration distributions as well as the coefficients of skin-friction, heat flux and mass flux.     

On the use of a rheogoniometer

Summary Further consideration is given to the usefulness of theWeissenberg Rheogoniometer in the characterization of elastico-viscous liquids. The theory for free and forced oscillation experiments in the parallel-plate system is developed. The advantages of the parallel-plate geometry in the case of forced oscillation experiments is stressed, and the limited value offree oscillation experiments is indicated.

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Zusammenfassung Die Brauchbarkeit des Weissenberg'schen Rheogoniometers bei der Charakterisierung viskoelastischer Flüssigkeiten wird einer weiteren Erörterung unterzogen. Es wird die Theorie für Versuche mit freien und erzwungenen Schwingungen im Parallelplattensystem entwickelt. Die Vorteile der Parallelplattengeometrie im Falle von Versuchen mit erzwungenen Schwingungen werden betont, auf den beschränkten Wert von Versuchen mit freien Schwingungen wird hingewiesen.

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Paper read at the Annual Meeting of the German Rheologists, Bad Ems, May 18–19, 1967.     

Elastico-viscous squeeze films: Part 2. superimposed rotation

The first part of this paper contains a reconsideration of the conventional squeeze-film situation. It is shown that the Part 1 concentration on the half-time t_{$\text{1}\text{2}$} sometimes hides interesting elastico-viscous effects, since these are most pronounced at short times after the load is applied.The bulk of the paper is concerned with the more general situation in which a rotational flow is superimposed on the basic squeezing flow. This is brought about by rotating the bottom plate. An approximate theoretical analysis is shown to lead to a satisfactory prediction of observed behaviour under conditions of light loading.The experiments show substantial transient stress-overshoot effects under some conditions and there is also a possible indication of long-range memory effects in some of the experiments.     

An unsymmetric constitutive equation for anisotropic viscoelastic fluid

A continuum constitutive theory of corotational derivative type is developed for the anisotropic viscoelastic fluid–liquid crystalline (LC) polymers. A concept of anisotropic viscoelastic simple fluid is introduced. The stress tensor instead of the velocity gradient tensor D in the classic Leslie–Ericksen theory is described by the first Rivlin–Ericksen tensor A and a spin tensor W measured with respect to a co-rotational coordinate system. A model LCP-H on this theory is proposed and the characteristic unsymmetric behaviour of the shear stress is predicted for LC polymer liquids. Two shear stresses thereby in shear flow of LC polymer liquids lead to internal vortex flow and rotational flow. The conclusion could be of theoretical meaning for the modern liquid crystalline display technology. By using the equation, extrusion–extensional flows of the fluid are studied for fiber spinning of LC polymer melts, the elongational viscosity vs. extension rate with variation of shear rate is given in figures. A considerable increase of elongational viscosity and bifurcation behaviour are observed when the orientational motion of the director vector is considered. The contraction of extrudate of LC polymer melts is caused by the high elongational viscosity. For anisotropic viscoelastic fluids, an important advance has been made in the investigation on the constitutive equation on the basis of which a series of new anisotropic non-Newtonian fluid problems can be addressed. The project supported by the National Natural Science Foundation of China (10372100, 19832050) (Key project). The English text was polished by Yunming Chen.     

Experimental investigations of the stability limit of the helical flow of pseudoplastic liquids

The stability of the laminar helical flow of pseudoplastic liquids has been investigated with an indirect method consisting in the measurement of the rate of mass transfer at the surface of the inner rotating cylinder. The experiments have been carried out for different values of the geometric parameter = R ₁/R ₂ (the radius ratio) in the range of small values of the Reynolds number,Re < 200. Water solutions of CMC and MC have been used as pseudoplastic liquids obeying the power law model. The results have been correlated with the Taylor and Reynolds numbers defined with the aid of the mean viscosity value. The stability limit of the Couette flow is described by a functional dependence of the modified critical Taylor number (including geometric factor) on the flow indexn. This dependence, general for pseudoplastic liquids obeying the power law model, is close to the previous theoretical predictions and displays destabilizing influence of pseudoplasticity on the rotational motion. Beyond the initial range of the Reynolds numbers values (Re>20), the stability of the helical flow is not affected considerably by the pseudoplastic properties of liquids. In the range of the monotonic stabilization of the helical flow the stability limit is described by a general dependence of the modified Taylor number on the Reynolds number. The dependence is general for pseudoplastic as well as Newtonian liquids.Nomenclature C _i concentration of reaction ions, kmol/m³ - d = R ₂ –R ₁ gap width, m - F _M () Meksyn's geometric factor (Eq. (1)) - F ₀ Faraday constant, C/kmol - i _l density of limit current, A/m³ - k _c mass transfer coefficient, m/s - n flow index - R ₁,R ₂ inner, outer radius of the gap, m - Re = V _m ·2d·/µ _m Reynolds number - Ta _c = _c ·d^3/2·R ₁ ^1/2 ·/µ _m Taylor number - Z _i number of electrons involved in electrochemical reaction - = R ₁/R ₂ radius ratio - µ apparent viscosity (local), Ns/m² - µ _m mean apparent viscosity value (Eq. (3)), Ns/m² - µ _i apparent viscosity value at a surface of the inner cylinder, Ns/m² - density, kg/m³ - _c angular velocity of the inner cylinder (critical value), 1/s     

Pure liquids described as concentrated cluster dispersions

The Fulcher-Tammann-Hesse-Vogel equation, ln = A + B/(T – T ₀ ), is shown to be equivalent to the general viscosity-composition relationship, ln _r =k f /(1 – f ), for binary mixtures. The Cailletet-Mathias law of the Rectilinear Diameter is rearranged to represent a density mixture formula for two components. Temperature-independent viscosities and densities can then be calculated for dense, solid cluster fractions, dispersed in a low-density, low-viscosity non-clustered continuous phase. The cluster fraction decreases with temperature. The value ofT ₀ is shown to be related to the liquid- or solid-like behavior of the clusters. For liquids with a vapor pressure < 1 mm Hg at the melting point, the calculated cluster volume fraction suggests close packing of clusters, ranging in shape from monodisperse spheres to polydisperse non-spherical particles. Examples are given for molecular liquids, molten metals, and molten salts. The size of the clusters is estimated from the heat of evaporation.     

On the viscosity-temperature behavior of polymer melts

Based on the free volume concept and the equation by Doolittle, an empirical equation is offered for the flow activation energy, E ^*, for polymer melts for the range of over 150°C above glass transition temperature, T _g. This E ^* represents the temperature coefficient of viscosity for the Newtonian region which is also equal to the value measured at constant shear stress for non-Newtonian flow. Data show that the E ^* of linear polymers approaches a constant value for a temperature range above T _g+150°C. Data on 17 polymers are correlated. The proposed equation for this region predicts the E ^* of polymer melts from the volume expansion coefficient, _l, above T _g and also from the T _g.Correlations have also been developed between E ^* and _l and between E ^* and T _g by simplifying the equation by use of the Simha-Boyer expression. A polymer having a lower _l or higher T _g generally has a higher E ^*. However, more satisfactory results are obtained by calculating E ^* from both _l and T _g. The E ^* calculated is found to agree with measurements within the experimental precision of about ±1 Kcal/mole.The effects of polymer composition, molecular weight, branching and microstructure on E ^* are also discussed. These factors influence E ^* in the way in which they effect _l and T _g.     

Enhanced load bearing by polymer solutions lubricating external near-contact of sliding cylinders

A small fixed cylinder attached to a load cell almost touches a larger, rotating cylinder (axes parallel). Newtonian liquids in the gap give reasonable loads, but viscoelastic liquids give markedly enhanced load bearing in relation to their apparent viscosities. F normal load on small cylinder - h minimum film thickness between cylinders - L length of smaller cylinder - R reduced radius, defined by equation 1/R = 1/R ₁ + 1/R ₂ - R ₁ radius of small cylinder - R ₂ radius of large cylinder - U surface velocity of large cylinder - load factor defined by equation = Fh/URLµ - µ viscosity of liquid     

Differential constitutive equation for entangled polymers with partial strand extension

Beginning with a formal statement of the conservation of probability, we derive a new differential constitutive equation for entangled polymers under flow. The constitutive equation is termed the Partial Strand Extension (PSE) equation because it accounts for partial extension of polymer strands in flow. Partial extensibility is included in the equation by considering the effect of a step strain with amplitude E on the primitive chain contour length. Specifically, by a simple scaling argument we show that the mean primitive chain contour length after retraction is L=L ₀ E ^1/2, not the equilibrium length L ₀ as previously thought. The equilibrium contour length is infact recovered only after a characteristic stretch relaxation time λ _s that is bounded by the reptation time and longest Rouse relaxation time for the primitive chain. The PSE model predictions of polymer rheology in various shear and extensional flows are found to be in good to excellent agreement with experimental results from several groups. Received: 16 July 1997 Accepted: 22 January 1998